Macroscopic superconductivity

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Part 6
Macroscopic superconductivity
571
573
Lecture 6.0
Overview of superconductivity
The lectures on Ginzburg-Landau theory,used for Module III of the Spring 2011 course
\Basic Training in Condensed Matter Theory",are numbered\6.0",etc.,because they
also constitute Unit 6 from the draft text States in Solids,used for the course\Solid
State II".Units 1{5 covered basics,quantum transport,structural order,and mag-
netism;units 7 and 8 will cover the microscopic (BCS) pairing theory of supercon-
ductivity,and\exotic"superconductivity (meaning high critical temperatures and/or
unusual pairing symmetries).Virtually none of this is needed for macroscopic super-
conductivity;the cross-references you'll nd are intended for enrichment,often making
analogies between dierent kinds of ordering.The main prerequisite you will nd useful
is Landau's mean-eld theory of phase transitions,which you should have encountered
in graduate-level statistical mechanics.
The Ginzburg-Landau theory of superconductivity is peculiar in that,on the one
hand,like any other macroscopic theory,it is purely classical { superposition,inter-
ference,indeterminacy are not part of it.But on the other hand the symmetry being
broken has no meaning outside of quantum mechanics.
The fundamental property of superconductivity { or of super uidity in general {
is not the strange transport behavior,but its strange sort of long-range order:that
symmetry breaking involves the quantum phase of a sort of wavefunction,even though
(according to basic quantum mechanics) the value of such a phase never aects any
physical observables.The systems aected are quantum uids:they escape having
a solid ground state because the particle masses are so small and the interpacticle
interactions are so weak that quantum uctuations would destroy it;this is realized (on
earth) only in helium,or the dense electron gas in metals.If those super uids happen
to be charged,they are superconductors.
Electricity and magnetism are irrelevant to the origin of superconductivity,just as
they were to (most) magnetism:as in that case,super uidity is a consequence of the
the quantum statistics and the interparticle potentials.But once superconductivity is
given,electromagnetismdoes play a more central role than in magnetic phenomenology.
(Half the explanation is that the energy scale of superconductivity is smaller by a factor
of up to 1000;the other half,I think,is the direct relationship between the supercurrent
and the order parameter gradients.) Still,in presenting the theory of superconductivity,
I will begin with the special case of a neutral super uid,and subsequently add on the
magnetic eld terms.
Copyright
c
2011 Christopher L.Henley
574 LECTURE 6.0.OVERVIEWOF SUPERCONDUCTIVITY
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 this
background helps explain why the\BCS"theory of superconductivity was the high
point of 20th-century condensed matter theory.Brie y,the phenomenon was initially
discovered by Kamerlingh Onnes in 1911,who pioneered the helium techniques needed
to reach 1K.Soon after,he found superconductivity was destroyed by a not-so-large
magnetic eld,temporarily dashing hopes of building super electromagnets (they were
realized decades later with help of Type II superconductors,Lec.6.6 ).In 1933,the
Meissner eect { the (reversible) expulsion of ux upon entering the superconducting
state { proved this state was a thermodynamic phase.Meanwhile,in 1937 super uidity
was discovered in
4
He and was understood over the next  15 years;this provided many
hints about superconductivity,which were developed independently during the 1950s in
the 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 fully
characterizes a crystalline solid:the majority of experimental phenomena (especially
those involving spatial dependences) can be understood in terms of GL theory:this un-
derlies every lecture in Part 6.Meanwhile,the Bardeen,Cooper,and Schrieer (BCS)
pairing theory of 1957 explained why a metal goes superconducting,and how the values
of the GL parameters are determined frommicroscopics;this is the basis for the lectures
in Part 7.After intense activity in the 1960s,the subject became almost dormant in the
1970s { except for the discovery of super uid helium 3,which (as anticipated) has BCS
pairing but with dierent symmetries and origins than in the original superconducting
metals.
Note that for some years in the late 1950s,the macroscopic theory was developed
by Russians while the microscopic theory was independently developed by Americans.
The latter approach found ways to relate those phenomena directly to the microscopic
theory,but it is a bit formal.For this book,I found it more transparent to relate
the phenomena to the GL theory.(Then GL theory can be related to the microscopic
theory.)
In the original superconductors,the electron pairs had the simplest possible sym-
metries,and the mechanism was due to phonons,giving a traditional upper bound
T
c
< 30K.Starting in the 1980s,several families of superconductors were discovered with
higher T
c
's,unusual pair symmetries,and/or non-standard mechanisms:they typically
have complex crystal structures and are quite poor metals in their non-superconducting
state.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 pair
condensation) in cold dilute gases opened up yet another avenue to realize super uid
order.
Overview of lectures on superconductivity
Certain students disdain the macroscopic theory for the microscopic theory (includ-
ing its exotic extensions).They should recollect that the macroscopic theory tells us why
superconductors superconduct.Indeed,it is the proper venue to explain the majority of
575
experimental observations,particularly anything related to the geometry of the system;
for many purposes,the role of the microscopic theory is only to explain the quantitative
values of parameters in the macroscopic theory.The macroscopic theory also encom-
passes the outstanding applications of superconductors,such as Josephson junctions
(including SQUIDs and superconducting qubits) and superconducting magnets.Hence,
whereas in magnetism we progressed from the microscopic to the macroscopic,here we
shall do the reverse.Indeed,it would be hard to motivate the quantities computed in
Part 7 without a prior introduction to superconducting behavior.
1
Part 6 will be all about the macroscopic theory,in particular the Ginzburg-Landau
theory.This is valid for all known and all imaginable superconductors,in the same
way that elasticity theory is valid for all solids.Most of practical superconductivity
can be understood at this level:the expulsion of magnetic eld,inhomogeneous states,
ux quantization,Josephson junctions,Type II superconductors (and their phase tran-
sitions),and ux- ow resistivity.Most of these phenomena entail some kind of spatial
variation,and so the key parameters in G-L theory will be two length scales,the\co-
herence length" and the\penetration depth".(These are dened in Lec.6.2.)
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 nature
of the order in a superconductor.[There exist,in principle,forms of superconductivity
which 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 gives
quantitative 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 specic heat.It
furthermore it makes specic predictions about dynamical measurements involving the
quasiparticles.These are valuable diagnostics of the precise nature of the ordered state.
Finally,one can consider the specic phonon-mediated microscopics,in particular
the\strong-coupling"modications of the above picture that come in when T
c
is not
so small compared to the phonon energy.
The last unit,Part 8,concerns\exotic"superconductors with a non-phonon mech-
anism,an unusual pairing state,and/or a large T
c
;the materials are dierent from the
traditional elemental metals and alloys,ranging from helium 3 to conducting polymers
to nuclear matter.Of course,the most studied of these are the cuprate high-temperature
superconductors.
I'll tend to downplay the role of electricity and magnetism (but not ignore it,since
E&Mis central to many of the phenomena!),as well as that of thermodynamics.In part
this is because I'll highlight the analogies to neutral super uids,and in part because I'll
emphasize the long range order.
References
The classic texts are M.Tinkham,Introduction to Superconductivity (original edi-
tion 1975),with sometimes fussy experimental details;J.R.Schrieer Theory of Su-
perconductivity (original edition 1964),written for theorists;and (3) P.G.de Gennes,
Superconductivity of Metals and Alloys,1966.The bible of the eld is R.D.Parks Su-
perconductivity,volumes 1 and 2 (1969),which collect review articles on every aspect
of superconductivity known in the 1960s.
1
Nevertheless,this text are designed so you can go directly from this overview to Part 7,and return
to Part 6 after that if desired.
576 LECTURE 6.0.OVERVIEWOF SUPERCONDUCTIVITY
6.0 A Phenomenology
The following are basic observations on superconductivity,which will be explained in
the next three units.
F(T)
F
S
F
cond
F
N
T
c
T
C (T)
el
C
S
C
N
T
c
T
T
c
(T)
T
(a).(b).(c).
S


N

Figure 1:Temperature-dependent phenomena in a superconductor.(b).Resistivity (showing
the jump to zero).(c).Specic heat (electronic).The solid curves in (b) and (c) are observed
along the T axis of (a);the dashed (normal state) data might be measured with the cooling
history represented by the horizontal dotted line in (a),at a eld large enough to destroy
superconductivity but not so large as to change the Fermi sea appreciably.(d).Free energy
obtained by integrating (c).
1.Resistivity
This was,of course,the rst discovered property of superconductors (Kamerlingh
Onnes,1911).Resistivity plummets,ideally,to zero [Fig.1(a)].In practice (Fig.2) it
usually falls gradually and the experimentalist must choose the feature of the fall at
which to place the T
c
.
In the\normal"or non-superconducting phase,resistivity behaves as (T) = 
0
+
O(T
x
),where the power x depends on the dominant scattering mechanism,e.g.x = 5
fromelectron-phonon scattering (dominant at higher T) or x = 2 fromintrinsic electron-
electron scattering (see Lec.1.7 ).Here 
0
is due to disorder:an ideal metal has zero
resistance at T = 0.
But superconductivity is not the same as perfect conductivity.Consider a material
obeying Ohm's Law in which resistivity  suddenly vanishes.The Maxwell equation
states

1
c

@B
@t

= rE = r(J) = 0 (6.0.1)
If  = 0,then this says B(r;t) = B(r;0):whatever instantaneous magnetic eld pattern
you had at the moment T passed below T
c
would be frozen everywhere,forever after.
But in fact superconductors expel the eld (Meissner eect,see below).
2.Electronic specic heat and gap
Specic heat is the rst probe of the thermodynamic state of a material.Asingularity
in C(T) proves that a phase transition is happening (even when we have no idea of its
nature).In the case of superconductors,it was discovered in the 1930s that the electronic
part of the specic heat,C
el
(T),has a jump at T
c
.Qualitatively,this ts the mean-
6.0 A.PHENOMENOLOGY 577
Figure 2:Resistivity in some real superconductors.(The materials illustrated here and in
Fig.4(a) are all\exotic"in the sense that the parent material is not an elemental metal or
alloy;they are discussed further in Unit 8.) (a).Organic superconductors;here (TMTSF)2PF6
is measured at pressure 8 kbar while the other two salts are at atmospheric pressure ( 1 bar).
[From D.Jerome,Science 252,1509 (1991).] (b).High-T
c
cuprate,La
2x
Ba
x
CuO
4
,recorded
for dierent current densities.[FromJ.G.Bednorz and K.A.Muller,Z.Phys.B64,189 (1986),
the discovery paper.] [Missing:Heavy-fermion-type metals.See e.g.gure in L.P.Gor'kov,
Sov.Science Rev.A Phys.9,1 (1987).]
eld theory of phase transitions (better so than most materials);quantitative prediction
of the jump must wait till BCS theory (see Lec.7.3 ).
In the late 1940s it was found that,at low temperatures C
el
(T)  e
=T
at low
T.This shows the superconducting phase has a nonzero gap (0) in the electronic
elementary excitations.T = 0.
2
The normal state,by contrast,of course has the
familiar C(T)  T./T
c
too,which comes out of pairing theory (Lec.7.4 ).
Integrating C
el
(T) carefully yields the condensation energy F
cond
(T),the free energy
dierence between a superconducting and a normal state.[Fig.1(c)].It turns out
F
cond
(0) is only  10
3
k
B
T
c
per electron.That means that only a small fraction of
electrons are really involved in the change of state.One would guess (correctly) this
smallness is related to that of T
c
=E
F
 10
5
.
Fig.1 also includes curves (dashed) for the normal state that would be seen if we
could just turn o the mechanism responsible for supercondivity.These are not merely
hypothetical:one realizes the normal state just by applying H > H
c
,which tips the
balance of energies so as to favor it.(Usually H
c
is so small that this has negligible
eects on the resistivity or thermodynamics.)
3.Systematics of T
c
in the periodic table?
Comparing Tc's
To phonon-based resistance:
The last shall be first.
In fact the critical temperature varies over an enormous range,between dierent
2
In the absence of the usual linear term,C(T) is dominated by the T
3
phonon term at low T,so a
delicate subtraction is necessary to measure .
578 LECTURE 6.0.OVERVIEWOF SUPERCONDUCTIVITY
materials but sometimes for small changes in a material.Among pure elements,T
c
ranges from 9.5 K for Nb and 7.2 K for Pb,down to 0.015 for W (at zero pressure);
there are some 20 more elements that become superconducting only under pressure.
Most tendencies of the superconducting energy scale only get explained in BCS theory.
However,the existence of this small parameter T
c
=E
F
is a foundation of the macroscopic
theory:it implies the corresponding length scales are much larger than the atomic scale
(Lec.6.2 ).
There are correlations of superconductivity with other properties,such as (for ele-
mental metals) their position in the periodic table.The most important correlation
is that bad metals (Hg,Pb,Bi) are good superconductors,whereas good metals (Au,
Cu,and all alkali metals) are bad superconductors.The explanation must wait for
the microscopic theory,specically the pairing mechanism due to the electron-phonon
mechanism.(Brief answer:when that is large,it increases the electron-scattering rate
and hence the normal-state resistivity.)
Another observation is that magnetic moments usually destroy superconductivity.
Thus elemental Mn,Fe,Co,and Ni show no superconductivity at any T { the super-
conducting transition got preempted by a magnetic ordering.
3
4.Meissner eect and critical eld
The most important discovery was the Meissner eect.This is diamagnetism on
a massive scale, = 1=4,large compared to the typical values of order 10
5
(see
Lec.4.0 ).If we increase magnetic eld,at the critical eld H
c
the superconducting
state goes unstable:there is a phase transition to the normal state.Notice that I write
H for the applied eld.The exact statement is that the superconducting phase is in
equilibrium with an adjoining normal phase with local eld H
c
;since eld lines are
concentrated depending on the geometry,the actual H at which superconductivity is
destroyed (and the it goes away as a function of space) depend on the sample shape and
its orientation with respect to the eld.The critical eld is really jHj = H
c
only for a
needle-shaped sample aligned with H.It turns out that H
c
/T
c
,as will be explained
in Lec.6.2.
5.Type II superconductors
A\type II"superconductor,in contrast to type I,does not abruptly transition to a
ux-expelling phase;rst it enters a regime in which a fraction of the eld is admitted,in
the form of quantized ux lines,while superconductivity persists.This Abrikosov phase
is the subject of Lec.6.6.It is also,misleadingly,called the mixed state.Do not confuse
it with the\intermediate state"(see Lec.6.3 ) which is the coexisting mixture of normal
and superconducting phase occurs in Type I superconductors in certain geometries A
useful (!) mnemonic:\the mixed state is intermediate,and the intermediate state is
mixed."
The eld response and phase diagram of a type II superconductor are shown in
Fig.3(e,f).Like the type I,for low elds this is in the Meissner phase that excludes all
(bulk) ux.The Abrikosov phase lies between the lower critical eld H
c1
and the upper
critical eld H
c2
.In fact H
c2
is greater than the H
c
that the same material would have
if it were type I.Thus,the Abrikosov phase permits superconductivity in large eld,
which is why Type II superconductors are the basis of superconducting magnets.
3
There are exceptions in complicated structures,where the magnetism and superconductivity are
centered on dierent ions in the unit cell,and also in some superconductors with exotic mechanisms.
6.0 B.REAL SUPERCONDUCTORS CAN BE MESSY 579












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























N
H
S
H
S
N

H
B
T
H
N
S
B
H
Abr.
N
Meiss.
T
H
Abr.
Meiss.
N
(c).
(e).
(a).(b).
(d).(f).
H
c
T
c
H
c
H
c1
H
c2
T
c
H
c2
H
c1
Figure 3:Critical eld and Meissner eect.The sample geometry is a thin sliver parallel
to the eld.(a).At temperature just above T
c
(H).(b).At T < T
c
(H),showing ux
expulsion.(c).Internal magnetic eld of a Type I superconductor as a function of applied
H,at T < T
c
(solid curve) or at T > T
c
(dashed curve).The arrows indicates the negative
(diamagnetic) susceptibility  in the superconducting phase.(d).Phase diagram of type
I superconductor,as a function of temperature and applied eld.\N"and\S"denote the
normal and superconducting phases.The double line indicates a rst-order (discontinuous)
transition at H
c
(T);the dot marks the critical point transition at H = 0.(e).Same as (c)
for Type II superconductor.(f).Phase diagram of type II superconductor.[This could be
the same material as (a) with microscopic scattering introduced that decreases the coherence
length { see Lec.6.2.] H
c1
(T) and H
c2
(T) are the lower and upper critical elds (continuous
transitions).The T axis in (c) and (e) is vertical line in (d) and (e).
6.0 B Real superconductors can be messy
The ideal pictures in Fig.1 are not what you see experimentally.As Fig.2 and Fig.4(A)
show,the resistatnce drop usually isn't immediate with temperature,and the expelled
ux isn't usually 100%.A common reason for this is that the sample is a mix of
superconducting and normal material;even if an entire sample is single-phase,spatial
variations in the stoichiometry mean that around the average T
c
,the sample is part
normal and part superconducting.There is more than one imaginable morphology.
First,what if you surround grains of normal metal by a network of superconductor
that percolates (connects across the sample) as in Fig.4(b)?For example,when an Al
alloy forms,some pure Al is likely to phase separate which superconducts;Visualize the
sample morphology as a pile of rocks,each wrapped in aluminum foil.) You will see
R(T) = 0 below the T
c
of Al,even though Al forms a negligible fraction of the sample.
On the other hand,the equilibriumvalue of diamagnetismis proportional to the volume
fraction and is negligible.
Second,what if we invert the previous morphology,so that each superconducting
grain is surrounded by normal material?(Visualize lumps of metal,each wrapped in
insulating paper.) Then the entire sample will have a nonzero resistance,since the best
current path consists of normal and superconductor in series.(But R will be small,
since those paths are short.) In high-T
c
cuprates,there is a nite resistance even when
580 LECTURE 6.0.OVERVIEWOF SUPERCONDUCTIVITY
B
N
N
N
N
S
S
B
S
N
S
S
N
S
(b) (c)
(a)
0.1
0.2
0.3
K C
3.5 60
K C
3 60
K C (final)
3 60
0
Shielding fraction
10 20
T(K)
Figure 4:Inhomogeneity and ux pinning in diagmagnetism of bsuperconductors.
(a).Magnetic susceptibility Susceptibility (T) for three samples of superconducting K
x
C
60
powder exhibiting shielding diamagnetism,expressed as fraction of a niobium standard (1:0
represents essentially 100% exclusion of ux, = 1=4).The upper curves are taken after
the\mixing"stage of materials preparation;bottom curve is the same sample as the top curve,
but at the\nal"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).) (b).Cartoon of how ux lines
(arrowed) can get trapped.Morphology of large normal domains,separated by thin super-
conducting sheets.(c).Reverse morphology that also traps ux,of superconducting domains
separated by normal layers.
superconducting grains are touching,if they are dierently oriented.
In any kind of inhomogeneous geometry,the magnetization shows lots of hysteresis.
If a sample is cooled below T
c
in a eld,some ux gets trapped in the normal regions
(Fig.4(b,c)).To get out it must pass through a superconducting region,but there is a
big barrier to doing this.
The Meissner eect is thus a more reliable indicator of superconductivity than the
resistance.Specic heat is less aected by geometry than either resistivity or Meissner
eect.But a specic heat bump can be caused by many kinds of ordering { it is not
particularly 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 (and
some other ones) will get explained.
Recap of topics to appear in Unit 7:
All of these were elaborated in Sec.6.0 A.
Meissner eect ( ux expulsion) and critical eld { Lec.6.3.
Zero resistivity { Lec.6.4
Specic heat jump:Lec.6.1
Left to microscopic theory (Unit 7)
The BCS/Bogoliubov\pairing"theory explains the value of C
s
(T

C
)=C
n
(T
c
) and
the temperature dependence F
cond
(T),and why T
c
(and the gap ) are exponentially
6.0 C.SCORECARD OF FACTS TO BE EXPLAINED 581
small:Lec.7.3 An oshoot of this is why magnetic moments kill superconductivity {
see [around] Lec.7.4 (\pairbreaking"eects).
Furthermore,the pairing theory describes single-electron (\Bogoliubov quasiparti-
cle") excitations,which got left out of the GL picture.This includes the gap,(0) =
(3:5=2)T
c
and the corresponding low-T specic heat C
s
(T).In fact a quasiparticle is
a superposition of an electron and a hole;the relative amplitudes are\coherence fac-
tors"which explained the contrasting behaviors of NMR damping (peak below T
c
);
ultrasound damping (no peak below T
c
),the evidence that originally clenched the BCS
theory.(See Lec.7.4 or Lec.7.5 ).
From the microscopic viewpoint,the phase-angle stiness we see in the GL theory is
a consequence of\o-diagonal long-range order"(Lec.7.1 ).G-L theory can explicitly
be derived from microscopics,see Lec.7.8 [omitted].
The electron-phonon pairing mechanism explains why bad metals tend to be good
superconductors,and the isotope eect:Lec.7.6 (phonon-mediated mechanism).