solid state physics lecture

1. UEEP2024 Solid State Physics
Topic 6 Superconductivity

2. Superconductivity
• The electrical resistivity of many metals and
alloys drops suddenly to zero when the
specimen is cooled to a sufficiently low
temperature. This phenomenon is called
superconductivity
• Superconductor is not a normal conductor
with zero electrical resistivity

3. Superconductivity
• Two basic properties of superconductivity
– 1. Zero resistivity
– 2. Perfect diamagnetism
• Missing any one of these two properties will
make the superconducting phase
thermodynamically unstable. Hence it needs
both properties to prove a material is a
superconductor.

4. Superconductivity
• Zero resistivity
– Below a certain temperature, the critical temperature Tc
(property of the superconductor), resistivity of a
superconductor will become exactly zero. The first
superconductor was mercury, discovered by Onnes in 1911.

5. • Perfect diamagnetism
– A superconductor expels magnetic field completely when
it is in superconducting phase (T<Tc). This phenomenon
was discovered by Meissner (and Ochsenfeld) in 1933, so it
is called the Meisner effect.
Superconductivity
T<Tc
T>Tc

6. Superconductivity
• A sufficiently strong magnetic field will destroy
superconductivity. A superconducting state will become
normal when H>Hc(T)

7. Superconductivity
Material Tc (K) Material Tc (K)
Al 1.20 Nb3Sn 18.05
Hg 4.15 Nb3Al 17.5
Mo 0.92 V3Ga 16.5
Nb 9.26 La3In 10.4
Pb 7.19 LaBaCuO 35
Ta 4.48 YBCO 90
Ti 0.39 Tl2Ba2Ca2Cu3O10 125
Zn 0.88 Hg12Tl3Ba30Ca30
Cu45O125
138

8. Superconductivity
Superconductors can be classified in accordance with several
criteria that depend on our interest in their physical properties,
on the understanding we have about them, on how expensive is
cooling them or on the material they are made of.
By their physical properties
-type I superconductors
-type II superconductors
By the understanting we have about them
-Conventional superconductors
-Unconventional superconductors

9. Superconductivity
By their critical temperature
-Low temperature superconductors
(those whose critical temperature is below 77K )
-High temperature superconductors
(those whose critical temperature is above 77K )
By material
-Pure element
(Al, Pb, Mo, Hg, Ti, Zn……….)
-Alloy
(Nb3Al, NbN, Ti2Co ……)
-Ceramics
(YBa2Cu3O7, MgB2……….. )

10. Superconductivity
• Type I superconductor
Magnetization (M) versus applied magnetic field (H) for type I superconductor
exhibits a complete Meissner effect (perfect diamagnetism). Above the critical
field Hc the specimen is a normal conductor. The value of Hc are always too low
for type I superconductors to have any useful technical application in coils for
superconducting magnets

11. Superconductivity
• Type II superconductor
Type II superconductor has two critical fields, Hc1 and Hc2, being a perfect
superconductor under the lower critical field (Hc1) and leaving completely the
superconducting state above the upper critical field (Hc2). Between the lower critical
field Hc1 and the upper critical field Hc2 the flux will penetrate the specimen (B ≠ 0)
and the Meissner effect is said to be incomplete. In this region (between Hc1 and
Hc2), the superconductor is said to be in the mixed state or Vortex state. The value of
Hc2 is much higher than the value of the critical field Hc calculated from the
thermodynamics of the transition.

12. Superconductivity
Conventional superconductors: those that can be fully explained
with the BCS theory or related theories.
Unconventional superconductors: those that failed to be explained
using such theories.
This criterion is important, as the BCS theory is explaining the
properties of conventional superconductors since 1957, but on the
other hand there have been no satisfactory theory to explain fully
unconventional superconductors. In most of cases type I
superconductors are conventional, but there are several exceptions
as niobium, which is both conventional and type II.

13. Superconductivity
As described in the BSC theory developed by John Bardeen, John
Schrieffer and Leon Cooper, the Cooper pair state is responsible for
superconductivity.
Cooper pair :
A Cooper pair is the name given to electrons that are bound
together at low temperatures in a certain manner first described
in 1956 by American physicist Leon Cooper. Cooper showed that
an arbitrarily small attraction between electrons in a metal can
cause a paired state of electrons to have a lower energy than the
Fermi energy, which implies that the pair is bound. Cooper pair is
a composite boson as its total spin is integer (0 or 1).

14. Superconductivity
How the Cooper pair is formed in a superconductor?
An electron in a metal normally behaves as a free particle. The
electron is repelled from other electrons due to their negative
charge, but it also attracts the positive ions that make up the rigid
lattice of the metal. This attraction can distort the positively
charged ion lattice in such a way as to attract other electrons
(electron–phonon interactions). At long distances this attraction
between electrons due to the displaced ions can overcome the
electrons' repulsion due to their negative charge, and cause them
to pair-up.

15. Superconductivity
• BCS theory predicts that at absolute zero the
band gap is given by E=3.53kTc.
When we considers the more realistic state consisting of many electrons forming
pairs (Cooper pair fluid) we find that, due to quantum mechanics, the energy
spectrum of the Cooper pair fluid possesses an energy gap ΔE, meaning that all
excitations of the system must possess some minimum amount of energy. This
gap to excitations ΔE leads to superconductivity. If ΔE is larger than the thermal
energy of the lattice, given by kT, the fluid will not be scattered by the lattice
since this small excitations (scattering of electrons) are forbidden
• the Cooper pair fluid in a superconductor can
flow without energy dissipations

16. Example
Calculate the energy gap for superconducting
tin and the minimum frequency that can be
adsorbed given the critical temperature of the
tin is 3.72 K.

17. Solution
• E= 3.53kT= 3.53×1.38×10-23×3.72
= 1.81×10-22J.
• Frequency of light absorbed
• Note this frequency is 1000 times lower than that
of visible light and corresponds to the microwave
region of the electromagnetic spectrum.
.1073.2
1063.6
1081.1 11
34
22
Hz
h
E





 



18. Superconductivity
• Normal conductor
In a normal conductor, an electrical current may be visualized as a
fluid of electrons moving across a heavy ionic lattice. The
electrons are constantly colliding with the ions in the lattice, and
during each collision some of the energy carried by the current is
absorbed by the lattice and converted into heat, which is
essentially the vibrational kinetic energy of the lattice ions. As a
result, the energy carried by the current is constantly being
dissipated. This is the phenomenon of electrical resistance.

19. Superconductivity
Material Tc (K) ΔE(eV) ΔE/ (kTc )
Al 1.20 3.4 x 10-4 3.3
Hg 4.15 16.5 x 10-4 4.6
Mo 0.92 2.7 x 10-4 3.4
Nb 9.26 30.5 x 10-4 3.8
Pb 7.19 27.3 x 10-4 4.38
Ta 4.48 14 x 10-4 3.6
Ti 0.39 7.35 x 10-4 3.57
Zn 0.88 2.4 x 10-4 3.2
ΔE ~ 4kTc

20. Superconductivity
(Isotope effect)
• It has been observed that the critical temperature of
superconductors varies with isotopic mass
.constTM C 
The experimental results within each series of isotopes may be a relation of
the form :
Original BCS model :  = 0.5
Material  Material 
Zn
Cd
Pb
~0.45
~0.32
~0.49
Hg
Mo
Sn
~0.5
~0.33
~0.47

21. Superconductivity
(Isotope effect)
KTM C 0If
Absence of lattice
vibrations
No superconductivity
The importance of lattice vibrations suggests that an electron-
phonon interaction is responsible for superconductivity. There
is no other reason for the superconducting transition
temperature to depend on the number of neutrons in the
nucleus.

22. Superconductivity
• The conditions for the Cooper pairs formation, in
numbers large enough to allow superconductivity
– Low T (limited random thermal phonons)
– The interaction between an electron and a phonon are
strong
– The number of electrons in states lying just below the Fermi
energy be large
– The two electrons have “antiparallel” spins
– The two electrons of a pair have linear momenta of equal
magnitude but opposite direction
Cooper pairs are weakly bound, they are constantly breaking up
and then reforming. Also, they are large (~mm)

23. Josephson effect
• The Josephson effect is the phenomenon of current
flow across two weakly coupled superconductors,
separated by a very thin insulating barrier. This
arrangement—two superconductors linked by a
non-conducting barrier—is known as a Josephson
junction; the current that crosses the barrier is the
Josephson current.
The Josephson junction

24. Josephson effect
The Josephson junction
Al2O3 layer
Al
Glass slide
Indium
contact

25. Josephson effect
the current-voltage relation for a Josephson junction

26. • The DC Josephson effect -- A DC current flows across the
junction in the absence of any electric or magnetic field
• The AC Josephson effect -- A DC voltage applied across the
junction causes RF current oscillations across the junction.
Further, an RF voltage applied with the DC voltage can then
cause a DC current across the junction.
• Macroscopic long-range quantum interference -- A DC
magnetic field applied through a superconducting circuit
containing two junctions causes the maximum supercurrent
to show interference effects as a function of magnetic field
intensity.
Josephson effect

27. Josephson effect
 
te
tV




2

   )(sin tItI c 
The basic equations governing the dynamics of the Josephson
effect are
(superconducting phase evolution equation)
where V(t) and I(t) are the voltage and current across the
Josephson junction, (t) is the "phase difference" across the
junction, and Ic is a constant, the critical current of the junction.
(Josephson or weak-link current-phase relation)
magnetic flux quantum
(1)
(2)

28. Josephson effect
• The Josephson effect has found wide usage, for
example in the following areas:
1. SQUIDs, or superconducting quantum interference devices
2. In precision metrology, the Josephson effect provides an
exactly reproducible conversion between frequency and
voltage. Since the frequency is already defined precisely
and practically by the caesium standard, the Josephson
effect is used, for most practical purposes, to give the
definition of a volt
3. superconducting single-electron transistor
4. Josephson junctions are integral in Superconducting
quantum computing as qubits such as in a Flux qubit or
others schemes where the phase and charge act as the
Conjugate variables
5. Superconducting Tunnel Junction Detectors (STJs)

29. Josephson effect
Y1 -- The probability amplitude of electron pairs in region 1
Y2 -- The probability amplitude of electron pairs in region 2
Region 1 Region 2
K

30. Josephson effect
1
2
2
1
, Y

Y
Y

Y
K
t
iK
t
i 
With coupling by the weak link K and applying the time-
dependent Schrodinger equation
If a DC voltage V is applied across the junction, an electron
pair experiences a potential difference 2eV, (1) can be
rewritten as
12
2
21
1
, YY

Y
YY

Y
KeV
t
iKeV
t
i 
(1)
(2)

31. Josephson effect
2
2
21
2
1 , nn YYSince
Thus, (2) can be rewritten as
1222
2111
12
2
2
2
2
21
1
1
1
1
2
1
2
1




iiii
iiii
enKeneV
t
eni
t
n
e
n
i
enKeneV
t
eni
t
n
e
n
i






























(3)
We can write )exp(,)exp( 222111  inin YY

32. Josephson effect
From (3) we can get
)(
212
2
2
2
)(
211
1
1
1
21
12
2
1
2
1

























i
i
ennKeVn
t
in
t
n
i
ennKeVn
t
in
t
n
i


(4)
Equate the real parts of (4)
)cos(
)cos(
21212
2
2
12211
1
1












nnKeVn
t
n
nnKeVn
t
n

 (5)
(6)

33. Josephson effect
Equate the imginary parts of (4)
)sin(
2
)sin(
2
2121
2
1221
1








nnK
t
n
nnK
t
n


(7)
(8)
From (7) and (8) and let (t)=2-1
))(sin(
4 2121
t
nneK
dt
dn
e
dt
dn
eJ 

 (9)

34. Josephson effect
(10)
It is more common to use current than current density. We can
rewrite (9)
   )(sin tItI c 
Josephson super
current
Phase difference across the
junction
DC Josephson
super current
Constant phase difference across
the junction

35. Josephson effect

eV
t
t 2)(



From (5) and (6) and let (t)=2-1

















2
1
1
2
12
))(cos(
2
)(
n
n
n
n
t
KeV
t
t
tt



(11)
If n1=n2
 
te
tV




2

or
AC Josephson
super current
Oscillating phase difference across
the junction
(12)

36. Josephson effect






eVt
II c
2
)0(sin 
By integration of (12) that with a DC voltage across the junction

eVt
t
2
)0()(  (13)
The superconducting current is given by (10) with (13) for the
phase
The current oscillates with frequency
(14)

eV2


37. Josephson effect
Josephson junction A
Josephson junction B
1
2

38. Josephson effect
BA   0


e
AB
2
0 
Let the phase difference between points 1 and 2 taken on a
path through junction A(B) be A(B)
the phase difference around a
closed circuit which encompassed a
total magnetic flux 
Thus


ee
oAoB  ,

39. Josephson effect





























e
I
ee
I
III
oc
ooc
BAtotal
cossin2
sinsin


The total current
The current varies with  and has maxima when
integerss
e
 ,


40. Superconducting magnets
• Superconducting magnet is an application of superconductor.
• Conceptually the simplest, and perhaps most obvious, application is to use
the supercurrent to generate an intense magnetic field.
• For long solenoid, the magnetic field produced in a solenoid is B = µonI
where n is the number per unit length and I is the current in the wire.
• What is the largest magnetic field that can be generated with this
arrangement?
• The answer is related to the current-carrying capacity of the wire, which in
turn is determined by the amount of heat generated by the current.
• For most copper wire , current density is about 400 A cm-2, and
corresponds to maximum magnetic field of few millitesla.
• By placing an iron core within the solenoid, this magnetic field can be
enhanced by a factor about a thousand to produce a maximum field of
about 2T.
• Main disadvantage of this arrangement is that the iron core is extremely
heavy and cumbersome.

41. Superconducting magnets
• Similar structure using a superconducting wire.
• It can achieve current density of 107 A cm-2, it can
produced very large magnetic field.
• An additional advantage is that no iron core is required.
• This dramatically reduces the size and weight of
electromagnet and open up a whole new range of
applications.
• Type I superconductors are usually unsuitable as they
have critical fields of only a fraction of Tesla, but type II
superconductors it is possible to achieve upper critical
fields up to 50T.

42. Example
• An electromagnet is formed by winding a
solenoid with 150 turns per metre using
copper wire of diameter 3 mm. If the
maximum current density for copper is 400 A
cm-2, determine the largest magnetic field that
can be produced with this solenoid.

43. Solution
• Maximum current
• Maximum magnetic field
   
A
mAmrJI
3.28
105.110400
23242
maxmax

 

   
T
AmHmnIB o
3
116
max
1034.5
3.281501026.1



 m

44. High-temperature Superconductors
• The discovery of new superconductors with increasingly
higher critical temperature has been slow and erratic.
• By the mid 1980s the highest recorded critical temperature
was 23K for the alloy Nb3Ge, and it was generally believed
that the maximum possible critical temperature for any
material was in the region of 40K.
• All this changed in 1986 when Alex Muller and Georg Bednorn
at IBM Zurich made a dramatic breakthrough. They discovered
a material with a critical temperature of 35K.
• The result was important not only because it represented a
significant increase in the maximum critical temperature but
also because the material lanthanum-barium-copper oxide
(LBCO), belongs to a new class of superconducting materials.

45. High-temperature Superconductors
• Up to the time of discovery, most known superconducting
materials were metals.
• Only a few groups in the world were investigating the
superconducting properties of oxides and only a handful of
superconducting oxides were known.
• Over the next couple of years there was a dramatic
increase in research into these materials, and the results
were stunning.
• In 1987 another oxide, yttrium-barium-copper-oxide(YBCO)
was found to superconducting at temperature up to 90K,
and the following year the highest critical temperature had
been pushed up to 125K for a related material, thallium-
barium-calcium-copper-oxide (TBCCO).

46. High-temperature Superconductors
• So room temperature superconducting is not yet
a reality, and is unlikely to be unless other new
family of superconducting material is discovered.
• However, the critical temperature of these new
oxide superconductors comfortably exceeds
another significant temperature- the boiling point
of liquid nitrogen (at 77K).
• This mean that the superconducting state can be
achieved using liquid nitrogen, which is far
cheaper and easier to handle than liquid helium.

47. High-temperature Superconductors
• The high critical temperature is not the only
unusual property of these oxide superconductors.
• They also exhibit extreme type II behavior.
• Estimates of the upper critical field vary widely,
but even the most conservative calculation
suggest that the material will remain
superconducting in magnetic fields of 250T.
• There are still some practical difficulties to be
overcome before high temperature
superconductors can be used commercially.