Chapter 1
Introduction to
Superconductivity
It was the beginning of the 20th century when in the Leiden laboratory of
the Dutch physicist Heike Kamerlingh Onnes the liquefaction of helium, the
last remaining gas to be liquefied, was achieved. The most common method
to liquefy a gas is to compress it; however, if the temperature of that gas is
higher than the critical point, even under strong pressure it is impossibile to
achieve the liquid state.
Figure (1.1) Phases Diagram.
In order to obtain the transition from the gas state to the liquid state it
is important to know the minimun temperature at which this phenomenon
can be seen, that is the critical point temperature, then to cool down the gas
before compressing it.
1
CHAPTER 1. Introduction to Superconductivity
Element Tc [K] at patm Tc [K] pc [atm]
He 4.22 5.19 2.24
H2 20.39 33.24 12.8
N2 77.33 126.2 33.5
O2 90.2 154.6 49.8
H2O 373.15 647.096 217.7
Table (1.1) Critical Temperature and Pressure for some important gasses.
As shown in Table(1.1) the transition temperature for helium is only 4.22
K. This means that the technologies needed to achieve that value should be
very sophisticated and much more complicated than the clever but modest
means needed for the other gasses.
Finally in 1908, after a sixteen-hour long experiment, the history of
searching for ”new liquids” was pushed to its end. As we can always see
in the developing of science, after solving a problem a new one is ready to be
investigated, thanks to the new discoveries. In fact, the liquefaction of he-
lium was the fundamental step for the discovery of strange properties of some
materials at very low temperature, thus opening a new chapter in solid-state
physics.
In 1911 Onnes discovered that at a temperature close to 4 K the electrical
resistance of mercury suddenly dropped to unmeasurable values.
Figure (1.2) Discovery of superconductivity (H. Kamerling Onnes, 1911):
resistance of mercury versus temperature.
2
A large amount of experimental studies followed that incredible discovery
leading to the finding of other materials with the same property but below
different temperatures. It was suddenly seen that the temperature at which
the resistivity drops to zero was a property of each material, which was
called Critical Temperature Tc. However, the absence of any resistance is not
the only fundamental characteristic of superconductors, as they also possess
anomalous magnetic and thermal properties, so that it is more useful to talk
about a new state of matter, as Onnes himself defined it in one of his papers:
[...] the mercury at 4.2 K has entered a new state, which, owing
to its particular electrical properties, can be called the state of
superconductivity [2].
Since its first years this physical state of matter was considered very
attractive by a large amount of physicists and engineers, as it was regarded,
and with some restrictions it could be regarded also nowadays, as the natural
case nearest to the concept of a “Perpetual Motion”.
3
CHAPTER 1. Introduction to Superconductivity
1.1 Properties of Superconductors
1.1.1 Electrotechnical basis
Before entering the world of superconductivity, it is useful to recall the be-
haviour of normal conductors.
An electrical conductor is a material that allows the current to flow eas-
ily through it. A current is nothing else than a net flux of electrons inside a
material (although conventionally the direction of the current is the opposite
with respect to the electrons motion) due to free electrons in the Valence
Band (first species conductors), or thanks to the presence of ionic species
which carry the current (second species conductors). Some examples of con-
ductors are metals, water and the human body. The birth of a current has
to be forced by an electric field, which drives the electrons to move from the
negative potential to the positive one, in order to neutralize the difference of
charge. According to the “Simplified Ohm’s Law” current and voltage are
linked in a proportional way, thanks to the Resistance R :
R = V
I
(1.1)
that is measured in Ohm [Ω] in the S.I.1.
However, the Resistance depends on the geometry of the samples, so it
is more useful to define the Resistivity, the resistance per unit of the cross
section area S and unit of the length l, that is a unique function of the
temperature, characteristic of every material:
ρ = R · S
l
[Ωm] (1.2)
As we can see from inverting the eq.(1.1) at the same applied voltage a
conductor, characterised by a small resistance, will lead a larger current with
respect to a low conductive material. The explanation of this behaviour is
that in a conductor, for example a metal, the outerboundary-electrons are
not bound with a single atom, but they are shared among them, so they
are allowed to flow across the lattice. Without an external force the chaotic
movement will prevail, and no effect will be seen. Whereas in the presence of
an electric field the motion forced by that field will prevail on the Brownian
motion, leading the current to flow. The resistance is due to the loss of
energy that these electrons undergo when they collide with the ions of the
crystal lattice, growing with the presence of impurities inside the material,
and with its temperature, which increases the thermal motion of the ions of
1
International System of Units
4
1.1. Properties of Superconductors
the lattice around their site. Therefore, if we push a metal near the absolute
zero, the resistance will tend to a very small value but not to zero due to the
inevitable presence of some impurities.
Figure (1.3) Resistivity vs Temperature in an impure metal and in a super-
conductor.
Following Matthiessen’s Rule the resistivity could be expressed as the sum
of two contributions:
ρ = ρt + ρr (1.3)
where ρt is the thermal component due to the vibration of the crystal lat-
tice (which vanishes at absolute zero) while ρr is the residual resistivity due
to the imperfections, and therefore it could be considered as a first rough
measurement of the purity of a metal. Thus, an ideal crystal would have
ρ = 0, while a real crystal has ρ = ρr. But the lack of resistance is not
enough to classify a pure crystal as a superconductor. First of all because
superconductors are characterized by a sudden drop of the resistance to zero
regardless of their degree of purity; moreover they eexhibit different thermal
and magnetic behaviours.
5
CHAPTER 1. Introduction to Superconductivity
1.1.2 Meissner-Ochsenfeld Effect
In 1933 Meissner and Ochsenfeld discovered one of the fundamental proper-
ties of superconductors, by measuring the magnetic field distribution outside
tin and lead samples. If we considere a PERFECT CONDUCTOR, with
ρ = 0, we obtain from local Ohm’s equation:
−→E = ρ−→J =⇒ −→E = 0 (1.4)
and from Faraday-Neumann-Lenz equation:
−→∇ ×−→E = −∂
−→B
∂t
=⇒ −→B = cost (1.5)
This means that in a perfect conductor the induction field is constant
in time thus, the magnetisation state does not depend only on the external
conditions, but also on the history of magnetisation.
Figure (1.4) Magnetic behaviour of a perfect conductor.
(a): Sample cooled down in absence of any magnetic field;
(b): Sample cooled down in the presence of a magnetic field.
As we can see from Fig.(1.4) the state of the magnetisation changes with
the history of the magnetisation itself, keeping its initial value unchanged.
In a SUPERCONDUCTOR the behaviour is different and really interest-
ing: the magnetic field does not penetrate the superconducting sample and
moreover it is ejected from it if the transition to superconductive state is
forced to happen in a magnetic field [see Fig.(1.5)].
All metals other than ferromagnetics have zero magnetic induction in the
absence of a magnetic field. The cause of this property is that the magnetic
field due to the magnetic moment of any atom is oriented chaotically, so that
6
1.1. Properties of Superconductors
Figure (1.5) Magnetic behaviour of a superconductor.
(a): Sample cooled down in absence of any magnetic field;
(b): Sample cooled down in the presence of a magnetic field.
the magnetic moments cancel each other out. When an external field −→H is
applied, there appears to be a finite induction field
−→B = µ−→H , where µ is the
magnetic permeability.
In paramagnetic materials µ > 1 so the applied field is enhanced; in
diamagnetic materials µ < 1 and the applied field is weakened (−→B < −→H ). In
a superconductor the Meissner effect implies −→B = 0 also in the presence of−→H , corresponding to zero magnetic permeability. This effect is called Ideal
Diamagnetism.
As
B = Bi +Be (1.6)
where Bi and Be are respectively the absolute vakues of the internal and
external induction field, a superconductor must have an internal flux density
opposite to the external one everywhere:
Bi = −Be (1.7)
or alternatively as He = Be/µ0
M = −He (1.8)
To oppose the applied field, thus having no magnetic induction inside the
sample, an electrical current flowing in the surface layer of the supeconductor
is needed, called Surface Supercurrent, which acts as a magnetic shield.
As the resistance is zero that supercurrent will flow perpetually without
dumping.
7
CHAPTER 1. Introduction to Superconductivity
Figure (1.6) Surface Supercurrents in the surface layer of the sample.
Besides, if we consider the superconductive sample as homogeneous and
amagnetic (µ = 1), assigning the diamagnetic behaviour to the shielding
currents (µ = 0), from the Ampe`re-Maxwell law we find
−→∇ ×−→B = µ0−→J (1.9)
If
−→B is zero inside the superconductors, −→J must be zero, too. Thus, the
current cannot flow inside the superconductors, but only in their surface
layer.
It is almost clear that the current cannot flow only in the surface of a
material, as this implies an infinite current density, but it can penetrate
a thin thickness (10−6 ÷ 10−5 cm) that differs for every material and with
the current density, called Penetration Depth λ. As the magnetic field
decreases with an exponential law, the penetration depth is defined as the
distance at which the field decreases by a factor e.
Like every superconductive parameter the penetration depth varies with
the temperature, tending to infinity for T = Tc, which is when the materials
turn to normal conductive state.
λ(T ) = λ(0)√
1−
(
T
Tc
)4
(1.10)
The penetration depth also grows with the presence of impurities. This
characteristic is very important from an engineering point of view as if the
penetration depth is too small, the current density able to flow in the cable
will be so small that the superconductivity could not have any practical
application.
8
1.1. Properties of Superconductors
Figure (1.7)
Magnetic field B0 at the
surface of the supeconduc-
tor decays to B0/e at a dis-
tance equal to λ inside a
superconductor (shaded re-
gion).
1.1.3 Thermal Properties
We have already stressed that superconductivity is different from being only
the absence of any electric resistance, but it is a real phase transition. The
specific heat plot as a function of the reduced temperature (T\Tc) is a further
evidence of this:
Figure (1.8) Resistivity and specific heat as a function of the reduced tem-
perature.
9
CHAPTER 1. Introduction to Superconductivity
The heat capacity of any metal is made up of two contributions:
1. the electronic heat capacity due to the electrons;
2. the lattice heat capacity due to the crystal lattice.
As the electronic heat capacity decreases linearly with decreasing tempera-
ture, while the lattice one with the third power of the temperature, the latter
one will be negligible for T→0.
The behaviour of a superconductor is rather different, in fact cel does not
decrease linearly with temperature but exponentially, as shown in Fig.(1.8),
in agreement with:
cel
αTc
= ae−b(Tc/T ) (1.11)
where a and b are two material constants independent of the temperature.
This means that even a very small heat loss can lead to a strong cooling.
At T = Tc we can notice a jump in the heat capacity, due to the transition
to the normal conductive state. Nevertheless, the magnitude of this jump
varies in different superconductors.
Also the thermal conductivity is characterized by some rather distinctive
features.
As we know in the presence of a temperature gradient a metal is not in
thermal equilibrium, a heat flux P = −k dT/dx proportional to the tem-
perature gradient arises. The proportionality constant k is the thermal con-
ductivity, which depends on the material and on the temperature and, in
analogy with the heat capacity, it can be written as the sum of an electronic
and a lattice contribution.
There are several mechanisms of thermal conductivity, related to the dif-
ferent kinds of scattering processes, so those two contributions can be calcu-
lated as:
1
kel
' 1
kel−lat
+ 1
kel−imp
+ 1
kel−el
(1.12)
1
klat
' 1
klat−el
+ 1
klat−imp
+ 1
klat−lat
(1.13)
In normal metals the major role is played by the electronic thermal contri-
bution k−1el = aT 2 + b/T , while in superconductors the temperature depen-
dence is completely different and so the lattice contribution is considerably
enhanced at T < Tc.
10
1.2. Theories and Models
1.2 Theories and Models
The discovery of superconductivity aroused an enormous interest. Many
physicists tried their hand at explaining the disappearance of electrical resis-
tance and the other features of the superconductive state, including Einstein,
Heisenberg, Landau, Frenkel and many others.
A number of important phenomenological and microscopic theories were
created, allowing to describe and predict, step by step, an increasing number
of phenomena.
1.2.1 “Two Fluid” Model
The first important model was proposed in 1934 by C.J. Gorter and H.B.G.
Casimir. It was a macroscopic model then called “Two Fluid” Model, be-
cause it assumes the contemporary presence of two electron carriers inside a
superconductor:
1. the “normal” carriers, identical to those of the electron system in a
normal metal;
2. the “superconductive” carriers, responsible for the anomalous proper-
ties.
The superconductive carriers, called Superelectrons, are able to flow
inside the material without interaction with the lattice, that is with no dis-
sipation of their energy.
We could think that for:
• T > Tc there are only electrons;
• T < Tc we are in the presence of both the contribution;
• T = 0 there are only superelectrons.
Actually we are able to have really zero losses only in Direct Current
(DC), because the sample could be modelled as in Fig.(1.9).
In DC the current is carried only by the superconductive branch that
short-circuits the branch of normal electrons. In fact, the electric field is
not allowed inside the material, since it will accelerate the superelectrons
indefinitely leading to a change in the current flow, by hypothesis constant.
As a consequence, remembering eq.(1.4), the current density must be zero,
too.
11
CHAPTER 1. Introduction to Superconductivity
Figure (1.9) Equivalent Circuit of a DC powered superconductor.
In alternate current (AC) on the other hand, the electric field is necessary
to accelerate the carriers, since the electrons change their direction of motion
continuosly. In that case we can model the cicuit as in Fig.(1.10).
Thanks to their greater inertia superelectrons will follow the changes
of
−→E , and behave as an inductance. So they cause a loss of energy due
to the fact that a fraction of the current has to flow in the resistive branch.
However, the magnitude of that phenomenon is much lower than many other
losses inside a real superconductor, since the resistive fraction of current is
very little (the induction in Henry is about 10−12 times the Ohm resistance).
As we can imagine with increasing frequency the normal electron conduction
will also increase, until reaching a critical value fc [see Paragraph (1.3.3)] at
which the sample will go back to the normal conductive state.
Figure (1.10) Equivalent Circuit of an AC powered superconductor.
12
1.2. Theories and Models
1.2.2 Classical Model: the London’s equations
In 1935 the London brothers proposed a macroscopic (non-quantic) model,
that could explain the perfect conductivity and the Meissner effect of super-
conductive materials with two simple equations [3].
Let us consider a linear, homogeneous, isotropic and time-invariant me-
dium with the electrons weakly bounded to the atoms and in thermodynamic
equilibrium. Neglecting the reciprocal interactions between them, every elec-
tron feels a force made of two contributions:
md
−→v
dt
= −→F em +−→F coll (1.14)
where −→F em ' q−→E is the force induced by the electric field and −→F coll '
−m−→v /τtr is the collision term, with τtr mean time among two collisions.
Hence we can write:
md
−→v
dt
= q−→E − m
−→v
τtr
(1.15)
In the presence of an electric sinusoidal forcing field of pulsation ω we have
v = qτtrm
1
1 + jωτtr
E (1.16)
so we can obtain the current density fasor:
J = σ0
1 + jωτtr
E = σ(ω)E (1.17)
where σ(ω) is the complex electrical conductivity. If we consider an electri-
cal conductor as copper, characterized by τtr ' 2.4 · 10−14, we can simply
approximate the complex conductivity with the real value σ0.
An interesting result of this model is that for f 1/τtr the conductivity
becomes purely complex, which means that the material has no losses. Phys-
ically this means that the electrons are oscillating so rapidly that they cannot
cross a distance long enough to collide with any atom, they are fixed around
a point, behaving as a dipole, so the current will flow without losses, as the
only carriers are the superelectrons. The superelectrons must therefore not
be affected by the collisions with the ions of the crystal lattice. Mathemati-
cally this is equivalent to set ω 1/τtr, but it has to be valid also for ω = 0,
so the unique solution is that τtr →∞, so:
σ(ω) = lim
τtr→∞
(
nsq2sτtr
ms
1
1 + jωτtr
)
= 1
jω
nsq2s
ms =
1
jω
·
1
Λ
(1.18)
13
CHAPTER 1. Introduction to Superconductivity
where Λ = ms/ (nsq2s). Applying this limit to the eq.(1.17) we obtain the
First London Equation:
−→E = ∂
∂t
(Λ
−→
J ) (1.19)
and it is easy to see that the factor Λ is striclty connected with the penetra-
tion depth
λ =
√ ms
µ0nsq2s
(1.20)
If we apply the curl operator to eq.(1.19)
−→∇ ×−→E = −→∇ × ∂
∂t
(Λ
−→
J ) (1.21)
remembering eq.(1.5) we obtain:
∂
∂t
[−→∇ × (Λ−→J ) +−→B
]
= 0 (1.22)
Assuming that there was a time when magnetic fields and currents did not
exist we can write the Second London equation:
−→∇ × (Λ−→J ) = −−→B (1.23)
We shall underline that the system formed by the London’s equation does
not explain the origin of superconductivity; it is only a mathematical model
that reproduces the experimental results and the presence of the penetration
depth and the Meissner Effect with very good accurancy. Those equations
do not substitute the Maxwell’s equation but they are restrictive conditions
of them in the superconductive domain. However, they cannot explain the
penetration of the magnetic field in Type II superconductors [see 1.4].
1.2.3 BCS Theory
The BCS Theory was postulated in 1957 by three american physicists, John
Bardeen, Leon Cooper and John Robert Schrieffer, who won the Nobel prize
in 1972 [4]. This was the first microscopic theory which explained every
phenomenon of superconductivity and also nowadays it is considered the
best one to explain the behaviour of the metallic materials (both Type I and
II), even though it shows remarkable disagreements with the results obtained
with the new ceramic High Temperature Superconductors (HTS) materials
as for example YBCO.
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