Below a material-specific critical temperature \(T_{c}\) the DC
resistivity of a superconductor falls to zero — not to a small residual value,
but to zero within the resolution of any measurement yet performed. Persistent
currents in superconducting rings have been observed for years with no
detectable decay, setting an upper bound on the resistivity of order
\(10^{-25}\ \Omega\,\)m, some \(10^{17}\) times smaller than copper's.
The transition is sharp. In a pure sample the resistance falls from its
normal-state value to zero over a range of millikelvin, in contrast to the
gradual flattening towards a residual resistivity that an ordinary metal shows
(the section in Chapter 6).
Ohm's law \(\vb{E} = \rho\,\vb{J}\) then forces an immediate conclusion. If
\(\rho = 0\) while \(\vb{J}\) remains finite,
\begin{equation}
\vb{E} = 0 \qquad\text{everywhere inside a superconductor carrying a steady current.}
\end{equation}
A steady current flows with no electric field to drive it. This is already
enough to distinguish superconductors from every other conductor, and it has one
consequence we shall need at once: by Faraday's law
\(\nabla\times\vb{E} = -\partial\vb{B}/\partial t\), so
inside the material. The flux through it cannot change.
The Meissner effect
The relation above says only that flux is frozen. A material that lost
its resistance while sitting in a field would therefore keep that flux forever,
and a material cooled in zero field and then placed in one would exclude it. The
final state would depend on the order of operations — which is the signature of
a non-equilibrium, history-dependent system, not of a thermodynamic phase.
Meissner and Ochsenfeld showed in 1933 that superconductors do not behave this
way. Cool a superconductor in a field through \(T_{c}\) and the flux is
actively expelled:
\begin{equation}
\boxed{\;\vb{B} = 0 \quad\text{inside a superconductor}\;}
\end{equation}
regardless of history. The superconducting state is a true equilibrium state,
reached by the same route whatever order the field and the cooling are applied
in — and that is what licenses the thermodynamic treatment of
the section on thermodynamics of the Superconducting Transition.
A superconductor is not merely a perfect conductor
The distinction is worth stating carefully, because it is the single most
commonly examined conceptual point in the subject.
Cool in a field, then…
Perfect conductor
Superconductor
…{}cool through \(T_{c}\)
flux stays trapped
flux is expelled
…{}then remove the field
currents keep flux trapped
\(\vb{B}=0\), unchanged
Final state
depends on history
unique
Governing condition
\(\partial\vb{B}/\partial t = 0\)
\(\vb{B} = 0\)
\(\vb{B}=0\) is a stronger statement than \(\partial\vb{B}/\partial t = 0\): it is
the difference between “the flux cannot change” and “there is no flux”.
Superconductivity requires both zero resistance and perfect
diamagnetism; either alone is insufficient.
Perfect diamagnetism
Setting \(\vb{B}=0\) in the constitutive relation
\(\vb{B} = \mu_{0}(\vb{H}+\vb{M})\) of the relation in Chapter 8 gives \(\vb{M} = -\vb{H}\),
hence
This is the result already quoted in the section in Chapter 8: a
superconductor is the perfect diamagnet, and no material can do better, since
\(\chi < -1\) would mean \(\vb{B}\) antiparallel to \(\vb{H}\).
The flux is excluded by screening currents flowing in a thin surface
layer. They are real currents, they generate a field that exactly cancels the
applied one throughout the interior, and they flow without dissipation. The
layer is not infinitesimally thin — its thickness is the London penetration
depth of the section on london Theory and the Two Length Scales — but for a bulk sample it is a small correction.
Above \(T_{c}\) the field passes through the
sample. Below \(T_{c}\) surface screening currents (red) generate a field that
exactly cancels the applied one inside, and the flux lines are pushed around
the specimen. The expulsion happens on cooling, whether or not the field
was present beforehand — which is what makes it thermodynamic rather than a
consequence of frozen-in flux.
The critical field
A magnetic field destroys superconductivity. Above a temperature-dependent
critical field \(H_{c}(T)\) the material reverts to the normal state, and
the boundary is well described by a parabola:
so that \(H_{c} = H_{c}(0)\) at \(T=0\) and \(H_{c}=0\) at \(T=T_{c}\). The curve
\(H_{c}(T)\) is the phase boundary in the \(H\)–\(T\) plane: superconducting inside,
normal outside.
The \(H\)–\(T\) phase diagram of a
type-I superconductor. The parabola the relation above separates the
superconducting region from the normal one. Crossing it vertically
(raising \(H\) at fixed \(T<T_{c}\)) is a first-order transition with latent heat;
crossing it at the single point \(H=0\), \(T=T_{c}\) is second order — a
distinction developed in the section on thermodynamics of the Superconducting Transition.
Critical current and Silsbee's rule
A current flowing in a superconductor generates its own magnetic field, and if
that field reaches \(H_{c}\) at the surface, superconductivity is destroyed. This
is Silsbee's rule: the critical current is whatever current produces
the critical field at the surface. For a wire of radius \(r\), Ampère's law
gives \(H = I/2\pi r\) at the surface, so
and the critical current density is
\(J_{c} = I_{c}/\pi r^{2} = 2H_{c}/r\).
Three critical parameters, one surface
\(T_{c}\), \(H_{c}\) and \(I_{c}\) are not independent. They are three coordinates on
a single critical surface in \((T,H,I)\) space, and exceeding any one of them
destroys the superconducting state. Raising the temperature lowers \(H_{c}\) by
the relation above, and lowering \(H_{c}\) lowers \(I_{c}\) by
the relation above. A superconducting magnet is designed against all three at
once.
Worked example 9.1.
Lead has \(T_{c} = \text{7.19 K}\). A field of 0.074 T applied at
2.0 K is just sufficient to destroy superconductivity. Find the
critical field at \(T=0\).
Solution. Rearranging the relation above in terms of \(B\),
A current of a few hundred amps in a millimetre wire — a current density of
about \(10^{8}\) A m\(^{-2}\), two orders of magnitude beyond what copper tolerates
— and with no dissipation at all.
\(H\) or \(B\)? Keep track of \(\mu_{0}\)
Sources quote critical fields sometimes as \(H_{c}\) in A m\(^{-1}\) and sometimes
as \(B_{c} = \mu_{0}H_{c}\) in tesla, and the two differ by a factor of
\(4\pi\times10^{-7}\). Silsbee's rule the relation above is written in terms of
\(H_{c}\); if a problem gives the critical field in tesla, divide by \(\mu_{0}\)
first. A quick sanity check: critical fields of elemental superconductors are of
order \(10^{-2}\) T, i.e. \(10^{4}\) A m\(^{-1}\).
Elemental superconductors
Critical temperatures and zero-temperature critical fields of some
elemental (type-I) superconductors, with two important non-elemental cases for
comparison.
Material
\(T_{c}\) (K)
\(B_{c}(0)\) (mT)
Note
Al
1.18
10.5
the qubit material
Sn
3.72
30.5
Hg
4.15
41
Onnes, 1911
Pb
7.19
80.3
highest \(T_{c}\) elemental type-I
Nb
9.25
206
type-II; RF cavities
Nb\(_{3}\)Sn
18.3
—
type-II, \(B_{c2}\approx\text{30 T}\)
YBa\(_{2}\)Cu\(_{3}\)O\(_{7}\)
92
—
cuprate, above liquid N\(_{2}\)
Type-I superconductors are not useless
It is sometimes said that type-I materials have no practical application because
\(H_{c}\) is small. The first half is true — they cannot carry high fields, and
every superconducting magnet is type-II — but the conclusion is wrong.
Aluminium, a type-I superconductor with \(T_{c}=\text{1.18 K}\), is the
material from which essentially every superconducting qubit is fabricated;
niobium and aluminium tunnel junctions make the SQUIDs of
the section on the Josephson Effects and Applications; and type-I films are used in transition-edge sensors and
single-photon detectors. What type-I materials cannot do is carry large currents
in large fields.
Type-I and Type-II Superconductors
Superconductors fall into two families according to how they respond to an
increasing magnetic field. The division is sharp, it is decided by a single
dimensionless number, and it determines which materials are laboratory curiosities
and which are used to build magnets.
Type-I: a single sharp transition
A type-I superconductor obeys the Meissner effect completely up to
\(H_{c}\) and then abruptly becomes normal. The magnetisation curve is a straight
line of slope \(-1\) terminating at \(H_{c}\):
\begin{equation}
M = -H \quad (H<H_{c}), \qquad M = 0 \quad (H>H_{c}).
\end{equation}
There is no intermediate regime and no partial flux penetration. Critical fields
are small — tens of millitesla, as the table above shows — so these
are called soft superconductors. Nearly all superconducting elements are
type-I.
Type-II: two critical fields and a mixed state
A type-II superconductor has two critical fields but still only
one \(T_{c}\).
Below \(H_{c1}\), the lower critical field, it behaves exactly
like a type-I material: complete flux expulsion, \(M=-H\).
Between \(H_{c1}\) and \(H_{c2}\) it enters the mixed state (or
vortex state): flux penetrates in discrete quantised tubes, but the
bulk of the material remains superconducting and the resistance is still
zero.
Above \(H_{c2}\), the upper critical field, superconductivity is
destroyed.
\(H_{c2}\) can be enormous — tens of tesla in Nb\(_{3}\)Sn, over a hundred in some
cuprates — which is why every superconducting magnet ever built, from MRI
scanners to the LHC, uses a type-II material operating in the mixed state. These
are the hard superconductors.
Magnetisation
against applied field. The type-I material expels flux completely until
\(H_{c}\) and then gives up all at once. The type-II material expels completely
only to \(H_{c1}\); between \(H_{c1}\) and \(H_{c2}\) flux enters gradually as
quantised vortices while the material still carries current without
resistance.
Vortices and flux quantisation
In the mixed state the flux does not enter uniformly. It threads the material in
vortices: cylindrical tubes with a normal core of radius \(\sim\xi\) (the
coherence length) surrounded by a circulating supercurrent that screens the field
over a distance \(\sim\lambda\) (the penetration depth). Each vortex carries
exactly one flux quantum,
and they arrange themselves into a triangular Abrikosov lattice, the
spacing shrinking as the field rises until the cores overlap — which is what
happens at \(H_{c2}\).
The \(2e\) in the flux quantum is the evidence for pairing
The flux quantum is \(h/2e\), not \(h/e\). Measuring it — which was done in 1961,
independently by two groups — showed directly that the charge carriers in a
superconductor have charge \(2e\), confirming the Cooper pairing that BCS theory
had proposed four years earlier. It remains among the cleanest confirmations of
a microscopic theory by a macroscopic measurement.
The two lengths and the parameter that decides everything
The reason is the energy of the boundary between a normal and a superconducting
region. Creating that boundary costs condensation energy over a thickness
\(\sim\xi\) (the pairing cannot switch on instantaneously) but saves magnetic
energy over a thickness \(\sim\lambda\) (the field is admitted). If \(\xi > \lambda\)
the wall energy is positive, the material minimises interface area, and flux
stays out until the whole sample gives up at once: type I. If \(\lambda > \xi\)
the wall energy is negative, the material maximises interface area by
subdividing the flux as finely as quantisation allows, and the vortex state is
the result: type II.
Expressions for the critical fields
Ginzburg–Landau theory relates the two critical fields to the two lengths:
\begin{equation}
B_{c} \simeq \sqrt{B_{c1}B_{c2}} \quad\text{(to within logarithmic factors)}.
\end{equation}
The structure of the relation above is worth reading physically. \(B_{c2}\) is
reached when one flux quantum is squeezed into an area \(\sim\xi^{2}\), i.e. when
the vortex cores touch — so a short coherence length gives a
high upper critical field. That is precisely why the cuprates, with
\(\xi\) of order a nanometre, survive fields above 100 T. Conversely
\(B_{c1}\) is set by \(\lambda\): it is the field at which admitting the first
vortex becomes energetically favourable.
with a \(2\pi\), not a \(\pi\). Writing \(\Phi_{0}/\pi\xi^{2}\) — which appears in
several sets of notes — overestimates \(B_{c2}\) by a factor of two, and
correspondingly underestimates \(\xi\) by \(\sqrt{2}\) when the relation is inverted
to extract a coherence length from a measured field. The \(2\pi\) traces back to
the requirement that the vortex phase wind by \(2\pi\) around the core; it is not
a matter of convention.
The lower critical field is likewise \(\Phi_{0}\ln\kappa/4\pi\lambda^{2}\). The
logarithm is often dropped for order-of-magnitude work — it is typically
between \(2\) and \(5\) — but it should be dropped knowingly rather than silently.
Worked example 9.3.
A type-II superconductor has \(B_{c1} = \text{12 mT}\) and
\(B_{c2} = \text{9.0 T}\). Estimate \(\xi\), \(\lambda\) and \(\kappa\), taking
\(\ln\kappa \approx 4\).
Hence \(\kappa = \lambda/\xi = 234/6.05 = 39\), comfortably above \(1/\sqrt2\) and
consistent with the assumed \(\ln\kappa = 3.7 \approx 4\). A ratio of forty between
the two lengths is typical of a good magnet material.
Worked example 9.4.
Show that the ratio \(B_{c2}/B_{c1}\) is essentially \(2\kappa^{2}/\ln\kappa\).
Solution. Dividing the two expressions in the relation above,
For \(\kappa = 39\) this gives \(2(1521)/3.66 = 831\), so the mixed state spans
nearly three decades in field. For a type-I material with \(\kappa < 1/\sqrt2\)
the ratio falls below one, the two fields cross, and the mixed state ceases to
exist — which is another way of stating the criterion the relation above.
Type I versus type II at a glance
{1.2}
Type I
Type II
Critical fields
one, \(H_{c}\)
two, \(H_{c1}\) and \(H_{c2}\)
\(T_{c}\)
one
one (still)
Meissner effect
complete up to \(H_{c}\)
complete only to \(H_{c1}\)
Transition at \(H_{c}\)
sharp
gradual through mixed state
\(\kappa = \lambda/\xi\)
\(<1/\sqrt2\)
\(>1/\sqrt2\)
Wall energy
positive
negative
Field scale
tens of mT
tens of T
Name
soft
hard
Examples
Al, Sn, Hg, Pb
Nb, Nb\(_{3}\)Sn, NbTi, cuprates
Use
detectors, qubits
all magnets
Note the asymmetry that catches people out: type-II materials have two
critical fields but still only one critical temperature.
Flux pinning
A vortex in the mixed state feels a Lorentz force when a transport current
flows, and a moving vortex dissipates energy — which would give the material a
resistance after all. Practical magnet wire is therefore deliberately filled with
defects, precipitates and grain boundaries that pin the vortices in place.
The critical current of a real type-II conductor is set not by Silsbee's rule but
by how strongly its flux lattice is pinned, which is a metallurgical property
rather than a thermodynamic one. This is why NbTi wire is drawn, twisted and
heat-treated with such care, and why two samples of the same compound can carry
very different currents.
Thermodynamics of the Superconducting Transition
Because the Meissner effect makes the superconducting state a genuine
equilibrium phase, ordinary thermodynamics applies. Remarkably, that alone — with
no microscopic input whatever — yields the condensation energy, the entropy
difference, the order of the transition and the size of the heat-capacity jump.
Free energy and the condensation energy
The relevant potential at fixed temperature and field is the Helmholtz free
energy \(F\). The superconducting state is the stable one below \(T_{c}\), so
