SOLID STATE PHYSICS
Superconductivity:
Author
Topics includes:
INTRODUCTION TO SUPERCONDUCTIVITY
COOPER PAIR
BCS THEORY
TYPES OF SUPERCONDUCTORS
INTRODUCTION:
Superconductivity is the phenomenon in which electrical
resistance of the materials suddenly disappears below a certain temperature
the materials that exhibit superconductivity are called superconductors.
Superconductivity was discovered in 1911 by Heike kamerlingh arenas it was
just made possible to liquify helium which produced extremely cold
temperatures .
How do superconductors work at quantum level?
1908 Dutch physicist hiking on this figured out how to turn helium gas into
liquefied helium from first time this was quite an achievement because
helium liquefies it only four degrees above absolute zero that’s -269°C or -
452°F this is the lowest boiling point of any known material he later cooled
down a sample of mercury to this temperature and ran electricity through it
to his shock he found that it had no resistance which meant no energy loss
this was highly unusual because normally at least some energy is lost in the
process of electrical flow through materials recognizing the importance of
this phenomenon he called this new state of matter a superconductor.
Onnes received the 1913 Nobel Prize in physics for this discovery under
ordinary conditions when electricity flows through a material there’s always
a resistance to this flow because electrons bump into atoms causing some
energy loss in the process but somehow as if by magic in this new state of
superconductivity the electrons flowed right through the material as if there
weren’t any atoms in their way in fact this material is such that if you put a
current in a superconducting wire in a loop that current will continue to flow
virtually forever with no added voltage or energy source and
superconductors have another seemingly magical feature and that they
expel magnetic fields so if you put a magnet over a superconductor the
magnet will levitate. How is this possible? How can any material carry
current perfectly with no energy loss? to answer this question we have to
dive deep into the subatomic realm which means we have to invoke
quantum mechanics ?
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While the idea of materials reaching a low Resistance at very cold
temperatures was rather well accepted the question was what happens at or
close to absolute 0 Lord Kelvin thought that electrons would completely stop
and thus the resistance would become infinite so when it was first discovered
that a materials electrical resistance can become zero at very low
temperatures this was unexpected in 1911 onus was the first to discover this
in mercury and found it to be superconducting at 4.2° kelvin later it was
found that other materials and alloys can be superconducting much higher
temperatures typical temperatures however are still pretty cold usually lower
than150° k . The next big discovery was made in 1933 by Walter meissner
and Robert oxen Feld they found that when a metal is cooled while in a small
magnetic field the flux is spontaneously excluded it’s the metal becomes
superconducting this is known as the meissner effect normally matter allows
magnetic fields to pass through it however a property of superconductivity is
that superconducting materials expelled magnetic flux fields in other words
magnetic fields cannot pass through it so the magnetic field of the magnet
lifts the material in order for the flux to flow to the opposite pole since it
can’t go through the material this is what causes levitation even after this
discovery still not known what exactly was the cause of superconductivity it
was not until 46 years after the discovery by onus that we got the first real
microscopic theory to describe what’s actually going on in 1957 John
bardeen Leon Cooper and John Robert Schrieffer proposed what is now called
the BCS theory in their honor it ain’t done the 1972 Nobel Prize in physics.
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So what did they figure out to understand how electrons can flow with no
resistance in superconductors we first need to understand what causes
resistance in the 1st place inside of a metal the outermost electrons in the
valence shell being furthest away from the nucleus are so free to move
around that a sample of metal can be treated as a bunch of atoms
surrounded by a sea of electrons.
The electrons can flow in a liquid like manner if we add electricity to one side
of the metal the metal accepts these new electrons easily and it pushes out
some electrons on the other side to make room we interpret this flow as a
current of electricity but the electrons don’t flow perfectly as electrons travel
through the material the atoms which now have slightly positive charges
because they’ve given up an electron in their outer shell are in the way if the
atoms were perfectly still the electrons would have an easier way to sneak
through the material but this is usually not the case the atoms vibrate or
their imperfections in the lattice the electrons collide with the atoms that
may be vibrating this causes the electrons to scatter and it ends up giving up
some of its energy to the atom causing it to vibrate a little bit more this
added vibration causes the whole lattice to vibrate a bit more this higher
vibration results in heating up the metal so this is how energy is lost due to
resistance under normal conditions even the best conductors have some
resistance to electron flow.
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You might even feel this heat loss sometimes if you put your hand over the
power cord of your devices as the temperature rises the atoms vibrate more
strongly causing more energetic collisions and higher resistance the
vibration which causes the electrons to scatter can be reduced by reducing
the temperature of the metal.
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But the nature of quantum mechanics specifically the uncertainty principle is
such that there will always be a little bit of motion this is called 0 point
energy there’s always some motion because the uncertainty principle states
loosely that no quantum object can have precise values for its position and
momentum at the same time.
So now that you understand what causes resistance in the 1 st place let’s
look at how this resistance goes away completely but to understand this I
have to introduce you to two terminologies in particles called fermions and
bosons you know that matter is composed of particles you’ve heard of
protons neutrons and electrons each of these particles has a certain property
related to momentum called spin note that spin doesn’t mean a physical
spinning but is called spin because electrons act like tiny bar magnets with
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an angular momentum these values are spin are multiples of planck’s
constant one of the fundamental constants of the universe which relates the
energy that a photon can carry to its frequency the values of spin are either
integer multiples of planck’s constant for half integer multiples a particle
with a half integer spin is called a fermion named after Italian scientist Enrico
Fermi a particle that has an integer spent is called a boson named after
Indian scientists satyendra Bose an electron can have a spin of plus ½ or -
1/2so it is a fermion a photon can have a spin of plus one or -1 so it is a
boson fermions and bosons behave differently at the subatomic level it turns
out that while any number of identical bosons can occupy the same energy
level in a quantum system this is not the case for fermions 2 or more
identical fermions cannot occupy the same energy level in a quantum
system this is called the Pauli exclusion principle .
This is why you can only have two electrons in any given orbital of an atom
This is why solid objects can’t pass through each other and you don’t pass
right through the floor or through a wall the electrons of the atoms on your
feet cannot occupy the orbitals of the electrons on the atoms of the floor
however bosons are a different kind of beast they don’t have this restriction
in fact it appears that they like to pack together when there are a bunch of
them around especially at low temperatures as an electron moves through a
conductor it is repelled from other electrons due to their mutual negative
charge but it also attracts the positive ions that make up the rigid lattice with
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the metal this attraction distorts the ion lattice moving the ions slightly
towards the electron increasing the positive charge density of the lattice in
the vicinity this positive charge density attracts other electrons at long
distances this attraction between electrons due to the displaced ions can
overcome the electrons repulsion and cause them to join 2 electrons when
joined together in this way are called a Cooper pair and the positively
charged region of the atomic lattice is part of what is called a phonon A
collective motion of multiple ions the same frequency are called phonons,
phonons are important for conducting heat and sound through solids but for
our purposes they are important in understanding superconductivity .
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If the temperature of the material is low enough to Cooper pair remains
together because it does not have enough energy to break apart this peer
can then be treated as a single particle when two electrons come together in
this way they’re half spins interact in such a way that together they form an
integer spin.
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In other words they start behaving like bosons instead of fermions they are
no longer subject to the polyclonal principle what happens now is that since
an arbitrary many bosons can fit into the same low energy state the
collection of Cooper pairs starts acting like 1 entity or unit this state when a
bunch of bosons cooled to low temperatures occupy the lowest quantum
ground state is called a Bose Einstein condensate they act like 1 bosonic
electron all at the same low energy state and it’s negatively charged
because it is made-up of electrons that are negatively charged so this means
they
can conduct electricity normally when an electron collides with an atom and
scatters it loses some energy by going to a lower energy state than before it
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collided this energy is lost at the lattice of the metal but with Cooper pairs
there’s no lower energy state for it to
go to there is no quantum state that these bosons can occupy that is lower
because they are already at their lowest energy state so no energy is lost
because they can’t lose anymore energy the lack of interaction of the Cooper
pairs with the atoms effectively results in no resistance to the flow of
electrons and the material becomes a superconductor the interaction of the
electrons in a Cooper pairing is very weak so it typically only happens at very
low temperatures as the temperature gets above the critical temperature
Cooper pairs get disrupted because there’s enough energy to break them up
and so superconductivity is lost it's the formation of Cooper pairs to the
interaction with the phonons of the material that are the cause of
superconductivity.
BCS THEORY AND COOPER PAIR:
According to classical physics, part of the resistance of a metal is due to
collisions between free electrons and the crystal lattice’s vibrations, known
as phonons. In addition, part of the resistance is due to scattering of
electrons from impurities or defects in the conductor. As a result, the
question arose as to why this does not happen in superconductors?
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A microscopic theory of superconductivity was developed in
1957 by John Bardeen, Leon Cooper and J. Robert Schrieffer, which is known
as the BCS theory. The central feature of the BCS theory is that two electrons
in the superconductor are able to form a bound pair called a Cooper pair if
they somehow experience an attractive interaction between them. This
notion at first sight seems counterintuitive since electrons normally repel one
another because of their like charges. This may be thought of in the following
way and is illustrated in Figure
Fig : Classical description of the coupling of a cooper pair.
An electron passes through the lattice and the positive ions are attracted to
it, causing a distortion in their nominal positions. The second electron (the
Cooper pair partner) comes along and is attracted by the displaced ions.
Note that this second electron can only be attracted to the lattice distortion if
it comes close enough before the ions have had a chance to return to their
equilibrium positions. The net effect is a weak delayed attractive force
between the two electrons.
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This short lived distortion of the lattice is sometimes called a
virtual phonon because its lifetime is too short to propagate through the
lattice like a wave as a normal phonon would. From the BCS theory, the total
linear momentum of a Cooper pair must be zero. This means that they travel
in opposite directions as shown in Figure 8. In addition, the nominal
separation between the Cooper pair (called the coherence length) ranges
from hundreds to thousands of ions separating them! This is quite a large
distance and has been represented incorrectly in many textbooks on this
subject. If electrons in a Cooper pair were too close, such as a couple of
atomic spacing apart; the electrostatic (coulomb) repulsion will be much
larger than the attraction from the lattice deformation and so they will repel
each other. Thus there will be no superconductivity. A current flowing in the
superconductor just shifts the total moment slightly from zero so that, on
average, one electron in a cooper pair has a slightly larger momentum
magnitude that its pair. They do, however, still travel in opposite directions.
The interaction between a Cooper pair is transient. Each electron in the pair
goes on to form a Cooper pair with other electrons, and this process
continues with the newly formed Cooper pair so that each electron goes on
to form a Cooper pair with other electrons. The end result is that each
electron in the solid is attracted to every other electron forming a large
network of interactions. Causing just one of these electrons to collide and
scatter from atoms in the lattice means the whole network of electrons must
be made to collide into the lattice, which is energetically too costly. The
collective behavior of all the electrons in the solid prevents any further
collisions with the lattice. Nature prefers situations that spend a minimum of
energy. In this case, the minimum energy situation is to have no collisions
with the lattice. A small amount of energy is needed to destroy the
superconducting state and make it normal. This energy is called the energy
gap. Although a classical description of Cooper pairs has been given here,
the formal treatment from the BCS theory is quantum mechanical. The
electrons have wave-like behavior and are described by a wave function that
extends throughout the solid and overlaps with other electron wave
functions. As a result, the whole network of electrons behaves line one wave
function so that their collective motion is coherent. In addition to having a
linear momentum, each electron behaves as if it is spinning. This property,
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surprisingly, is called spin. This does not mean that the electron is actually
spinning, but behaves as though it is spinning. The requirement from the
BCS theory is that spins of a Cooper pair be in opposite directions. Note that
the explanation and pictorial representation of a Cooper pair presented here
comes directly from BCS theory. However, current HSC textbooks tend to
distort this picture with unphysical situations such as the Cooper pair being
within one or two atomic spacing and traveling in the same direction – each
of these situations is false.
High – Tc superconductors:
It has long been a dream of scientists working in the field of
superconductivity to find a material that becomes a superconductor at room
temperature. A discovery of this type will revolutionize every aspect of
modern day technology such as power transmission and storage,
communication, transport and even the type of computers we make. All of
these advances will be faster, cheaper and more energy efficient. This has
not been achieved to date. However, in 1986 a class of materials was
discovered by Bednorz and Müller that led to superconductors that we use
today on a bench-top with liquid nitrogen to cool them. Not surprisingly,
Bednorz and Müller received the Nobel Prize in 1987 (the fastest-ever
recognition by the Nobel committee). The material we mostly use on bench-
tops is Yttrium – Barium – Copper Oxide, or YBa2Cu3O7, otherwise known as
the 1-2-3 superconductor, and are classified as high temperature (Tc)
superconductors.
The critical temperature of some high-Tc superconductors is given in
Figure 2. Critical temperatures as high as 135 K have been achieved. Whilst
this is not room temperature, it has made experiments on superconductivity
accessible to more people since these need only be cooled by liquid nitrogen
(with a boiling point of liquid nitrogen is 77 K), which is cheap and readily
available. This is in contrast to the expensive and bulky equipment that used
liquid helium for cooling the traditional types of superconductors. Moreover,
the superconductors are calculated to have an upper critical magnetic field,
Bc2, of about 200 Tesla – huge! The crystal lattice structure of YBa2Cu3O7 is
shown in Figure 9. Unlike traditional superconductors, conduction mostly
occurs in the planes containing the copper oxide. It has been found that the
critical temperature is very sensitive to the average number of oxygen
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atoms present, which can vary. For this reason the formula for 1-2-3
superconductor is sometimes given as YBa2Cu3O7-δ where δ is a number
between 0 and 1. The nominal distance between cooper pairs (coherence
length) in these superconductors can be as short as one or two atomic
spacing’s. As a result, the coulomb repulsion force will generally dominate at
these distances causing electrons to be repelled rather than coupled. For this
reason, it is widely accepted that Cooper pairs, in these materials, are not
caused by a lattice deformation, but may be associated with the type of
magnetism present (known as antiferromagnetic) in the copper oxide layers.
So high–Tc superconductors cannot be explained by the BCS theory since
that mainly deals with a lattice deformation mediating the coupling of
electron pairs. The research continues into the actual mechanism
responsible for superconductivity in these materials.
Applications of superconductors:
The first large scale commercial application of superconductivity was in
magnetic resonance imaging (MRI). This is a non-intrusive medical imaging
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technique that creates a two-dimensional picture of say tumors and other
abnormalities within the body or brain. This requires a person to be placed
inside a large and uniform electromagnet with a high magnetic field.
Although normal electromagnets can be used for this purpose, because of
resistance they would dissipate a great deal of heat and have large power
requirements. Superconducting magnets on the other hand have almost no
power requirements apart from operating the cooling. Once electrical current
flows in the superconducting wire, the power supply can be switched off
because the wires can be formed into a loop and the current will persist
indefinitely as long as the temperature is kept below the transition
temperature of the superconductor.
Superconductors can also be used to make a device known as a
superconducting quantum interference device (SQUID). This is incredibly
sensitive to small magnetic fields so that it can detect the magnetic fields
from the heart (10-10 Tesla) and even the brain (10-13 Tesla). For
comparison, the Earth’s magnetic field is about 10-4 Tesla. As a result,
SQUIDs are used in non-intrusive medical diagnostics on the brain.
The traditional use of superconductors has been in scientific
research where high magnetic field electromagnets are required. The cost of
keeping the superconductor cool are much smaller than the cost of operating
normal electromagnets, which dissipate heat and have high power
requirements. One such application of powerful electromagnets is in high
energy physics where beams of protons and other particles are accelerated
to almost light speeds and collided with each other so that more
fundamental particles are produced. It is expected that this research will
answer fundamental questions such as those about the origin of the mass of
particles that make up the Universe.
Levitating trains have been built that use powerful electromagnets
made from superconductors. The superconducting electromagnets are
mounted on the train. Normal electromagnets, on a guide way beneath the
train, repel (or attract) the superconducting electromagnets to levitate the
train while pulling it forwards.
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A use of large and powerful superconducting electromagnets is in a
possible future energy source known as nuclear fusion. When two light nuclei
combine to form a heavier nucleus, the process is called nuclear fusion. This
results in the release of large amounts of energy without any harmful waste.
Two isotopes of hydrogen, deuterium and tritium, will fuse to release energy
and helium. Deuterium is available in ordinary water and tritium can be
made during the nuclear fusion reactions from another abundantly available
element – lithium. For this reason it is called clean nuclear energy. For this
reaction to occur, the deuterium and tritium gases must be heated to
millions of degrees so that they become fully ionized. As a result, they must
be confined in space so that they do not escape while being heated. Powerful
and large electromagnets made from superconductors are capable of
confining these energetic ions. An international fusion energy project, known
as the International Thermonuclear Experimental Reactor (ITER) is currently
being built in the south of France that will use large superconducting
magnets and is due for completion in 2017. It is expected that this will
demonstrate energy production using nuclear fusion.
Types of superconductor:
Depending upon on the magnetic properties,
superconductors are divided into two types.
Type I superconductors: (Soft superconductor):
Type-I superconductor acts as a perfect diamagnetic material
and obeys Meissnar effect.
Type I superconductors expel the magnetic field totally, but if the
field is too big, the superconductivity is destroyed.
As value of magnetic field (H) increases, magnetization of
superconductor also increases. Above critical magnetic field
(H c ) it turns into normal state.
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This is reversible process. When value of applied field
decreases, material expels magnetic field lines and retains
to superconducting state. value of H
Type-I superconductor allows magnetic field lines to
penetrate at lower value of c Therefore, it is also known as
“Soft superconductor”. Therefore, it is also known as “Soft
superconductor”.
Mostly pure elements like Aluminum (Hc = 0.0105 Tesla),
Zinc (Hc = 0.0054) etc. are examples of soft
superconductors.
Type-II Superconductor: (Hard superconductors):
Type-II superconductors do not obey perfect Meissner effect.
As value of magnetic field field (H) increases,
magnetisation of superconductor increases. Upto H c1
Above H c1 it shows perfect superconducting behavior.
force of external magnetic field lines increases and it
starts penetrating superconducting material. At H c2
completely.
The state between H c1 and H c2 material losts its
superconductivity is known as “Vortex state” or “Mixed
state”.
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As to destroy superconductivity of type As to destroy
superconductivity of type-- II superconductor is difficult
than type II superconductor is difficult than type
superconductor due to its high value of H c , it is known
as “Hard superconductor”.
Mostly alloys and ceramics like NbN (Hc = 8 x 10^ 6
Tesla),Babi 3 (Hc = 59 x 10^3) are examples of hard
superconductors.
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