Superconductor

Group 2
B112035010 郭峻甫
B112035012 陳科佑
B112035018 范紘瑋
 What is superconductor?
Superconductors have zero electrical resistivity
when it’s below a critical temperature (𝑇𝐶 ).

When a material is in the superconducting state,

current can flow without attenuation for more than
a year.

File and Mills used precision nuclear magnetic

resonance methods (NMR) to measure the magnetic
field with the supercurrent, and they concluded that
the decay time of the supercurrent can be over
100,000 years. The relationship between the resistance
(vertical axis) and temperature of mercury
(horizontal axis).
 Occurrence of superconductivity
Superconductivity occurs in many metallic elements of the periodic system
or in alloys, intermetallic compounds, and doped semiconductors.

Different types of materials has unique requirements to achieve the

superconducting state, such as specific temperatures, magnetic fields, and
pressure.

Such as 90 K for the compound YBa2Cu3O7 to below 0.001 K for the

element Rh, and Si has a superconducting form at 165 kbar, with Tc 8.3 K.
 Meissner effect
When a bulk superconductor is placed in a weak magnetic field, it will
represent to be diamagnet, which has zero magnetic induction in interior.

When the temperature T is below the critical temperature 𝑇𝑐 (T< 𝑇𝑐 ), the

magnetic flux originally present is ejected (such as the figure).

T > 𝑇𝑐 T < 𝑇𝑐
 We can limit ourselves to long thin axes parallel to 𝐵𝑎 , and the demagnetizing
field contribution to B will be negligible:
4π factor stems from the way magnetic
fields were originally defined in CGS,
ensuring consistency across magnetic
field equations.
𝑀 1
𝐵 = 𝐵𝑎 + 4𝜋𝑀 =0≫ =− [𝐶𝐺𝑆]
𝐵𝑎 4𝜋
൞ 𝑀 1
𝐵 = 𝐵𝑎 + 𝜇0 𝑀 =0≫ =− = 𝜀0 𝑐 2 [𝑆𝐼]
𝐵𝑎 𝜇0

Which B is the total magnetic field inside the

superconductor, 𝐵𝑎 is the applied external
magnetic field, and M is the magnetization of
the material. Temperature dependence of magnetic susceptibility
(magnetization M/H) of the Ba214 compound.
 ※ Why the magnetic properties cannot be accounted for by the assumption
that a superconductor is a normal conductor with zero electrical resistivity ?

From Ohm’s law:

𝐸 = 𝜌Ԧ𝑗
which 𝐸 is the electric field, 𝜌 is resistivity, and 𝑗Ԧ is the current density.
→ If 𝜌 = 0, with the finite value of 𝑗Ԧ, 𝐸 will still be 0.
𝜕𝐵
→ From Maxwell equation, 𝐸 = 0 only shows = 0, not 𝐵 = 0.
𝜕𝑡

Meissner effect → perfect diamagnetism ⇔ essential property of superconducting state

 What is “perfect conductor” ?
Perfect conductor ≠ superconductor

Perfect conductor ⇔ conductors with electrons have infinite mean free path

→ When observe closely, a perfect conductor placed in a magnetic field cannot

maintain a permanent eddy current shield. Instead, the magnetic field gradually
penetrates to the conductor at a slow rate, reaching about 1 cm depth over the
course of 1 hour.
 Destruction of Superconductivity by Magnetic Fields

Sufficiently strong magnetic field → Destroy superconductivity

𝐻𝐶 : Critical magnetic field, which is the maximum magnetic

field strength that a superconducting material can withstand
before losing its superconducting properties.

1. 𝐿𝑜𝑤 𝑡𝑒𝑚𝑝𝑒𝑟𝑎𝑡𝑢𝑟𝑒
To become superconducting state: ቊ
2. 𝐿𝑜𝑤 𝑎𝑝𝑝𝑙𝑖𝑒𝑑 𝑓𝑖𝑒𝑙𝑑
⇔
Threshold curves separate normal state & superconducting state.
 Type I and type II superconductor

(a) Type I superconductor, (b) Type II superconductor,

the “ perfect“ Meissner effect. which is the “ incomplete“ Meissner effect.
Type I superconductors Type II superconductors
Usually pure specimens Tend to be alloy or transition metal (with
high resistance in normal state)
𝐻𝑐 is too low ⇔ hard to application
Superconducting electrical property
Show exact Meissner effect
(until 𝐻𝑐2 )
𝐻𝑐1 < H < 𝐻𝑐2 → flux density B ≠ 0 →
Meissner effect is incomplete.
𝐻𝑐2 > 100𝐻𝑐
(calculated by thermodynamics)
 Heat capacity and entropy

1. When a material (Aluminum in figure)

turns from normal state to the
superconducting state, it will be more
ordered, and its entropy will decrease.

2. The superconducting state is more

ordered due to the formation of
Cooper pairs (pairs of electrons bound
together in a specific quantum state).

We will see in further pages!

The superconducting state has lower entropy is because the

electrons are more ordered here than in the normal state.
The change of entropy is small, which means only small fraction participate in the
transition, so the free energies of normal and superconducting states are different,
too.

This is the free energy as a function of temperature for aluminum in

the superconducting state and in the normal state.
(a) The comparison of capacity between (b) The electronic part 𝐶𝑒𝑠 of the heat capacity
normal and superconducting state, the in the superconducting state of Gallium versus
capacity will decrease sharply when it the ratio of Tc and T, and it represent the
reach critical temperature. exponential dependence on 1/T.
 Energy gap
The energy gap of superconductor and insulators is different:

Insulators: Its energy gap is caused by the electron-lattice interaction.

Superconductors: Its energy gap forms as the result of Cooper pairing, which is the electron -
electron interaction that orders the electrons in k space compared to the Fermi gas of electrons.

Energy gap at the Fermi level

in the superconducting state.

Electrons in excited states above the gap will behave as

normal electrons in rf fields, they will cause resistance, too.
 Isotope Effect

𝑀𝛼 𝑇𝑐 = 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡

It has been observed 𝑇𝑐 varies with isotopic mass M.This

figure is about the isotope effect of mercury .
Theoretical survey
Thermodynamics of the Superconducting Transition

reversible
Normal state Superconducting state
similar

reversible
Liquid Vapor

We try to apply Thermodynamics to the transition use 𝐻𝐶 and T

• The superconducting state of type I of
superconductor will disappear directly.
• For type II superconductors, 𝐻𝑐 is understood to
be the thermodynamic critical field related to the
stabilization free energy. In coexistence the free energy densities are
equal 𝐹𝑁 𝑇, 𝐵𝑎𝑐 = 𝐹𝑆 𝑇, 𝐵𝑎𝑐
 𝐵𝑎
𝑊 = − න 𝑀 ∙ 𝑑𝐵𝑎
0

𝑑𝐹 = −𝑀 ∙ 𝑑𝐵𝑎

1 Now consider a normal nonmagnetic metal. If we neglect

𝑑𝐹𝑆 = 𝐵 𝑑𝐵 [𝐶𝐺𝑆]
4𝜋 𝑎 𝑎 the small susceptibility of a metal in the normal state, then
1 M and the energy of the normal metal is independent of
𝑑𝐹𝑆 = 𝐵𝑎 𝑑𝐵𝑎 [𝑆𝐼]
𝜇0 field. At the critical field we have 𝐹𝑁 𝐵𝑎𝑐 = 𝐹𝑁 0

The increase in the free energy density of the superconductor is

𝐵𝑎 2 𝐵𝑎 2
𝐹𝑆 𝐵𝑎 − 𝐹𝑆 0 = [𝐶𝐺𝑆] 𝐹𝑁 𝐵𝑎𝑐 = 𝐹𝑆 𝐵𝑎𝑐 = 𝐹𝑆 0 + [𝐶𝐺𝑆]
8𝜋 8𝜋
𝐵𝑎 2 𝐵𝑎 2
𝐹𝑆 𝐵𝑎 − 𝐹𝑆 0 = [𝑆𝐼] 𝐹𝑁 𝐵𝑎𝑐 = 𝐹𝑆 𝐵𝑎𝑐 = 𝐹𝑆 0 + [𝑆𝐼]
2𝜇 0 2𝜇 0
 𝐵𝑎 2
𝐹𝑁 𝐵𝑎𝑐 = 𝐹𝑆 𝐵𝑎𝑐 = 𝐹𝑆 0 + [𝐶𝐺𝑆]
8𝜋
𝐵𝑎 2
𝐹𝑁 𝐵𝑎𝑐 = 𝐹𝑆 𝐵𝑎𝑐 = 𝐹𝑆 0 + [𝑆𝐼]
2𝜇 0

𝐵𝑎 2
𝐹𝑆 𝐵𝑎 − 𝐹𝑆 0 = [𝐶𝐺𝑆]
8𝜋
𝐵𝑎 2
𝐹𝑆 𝐵𝑎 − 𝐹𝑆 0 = [𝑆𝐼]
2𝜇 0

𝐵𝑎𝑐 2
∆F = 𝐹N (0) − 𝐹𝑆 0 =
8𝜋
 London equation
𝑀 1
𝐵 = 𝐵𝑎 + 4𝜋𝑀 = 0 ≫ =− [𝐶𝐺𝑆]
𝐵𝑎 4𝜋
Meissner effect give that 𝑀 1
𝐵 = 𝐵𝑎 + 𝜇0 𝑀 = 0 ≫ = − = 𝜀0 𝑐 2 [𝑆𝐼]
𝐵𝑎 𝜇0

1
𝑴
− [𝐶𝐺𝑆]
4𝜋
magnetic susceptibility 𝝌 = , superconductor 𝝌 = ൞ 1
𝑯 − = 𝜀0 𝑐 2 [𝑆𝐼]
𝜇0

Ohm’s law give that 𝒋 = 𝝈𝑬

𝑐
𝑗=− 𝐴 [𝐶𝐺𝑆]
4𝜋𝜆2𝐿
We know that 𝑩 = 𝛁 × 𝑨 and we postulate that 𝒋 ∝ 𝑨 → ൞ 𝑐
𝑗= − 2𝐴 [𝑆𝐼]
𝜇0 𝜆𝐿

𝑐
𝛁×𝑗 =− 𝐵 [𝐶𝐺𝑆]
4𝜋𝜆2𝐿
We express it by curl both side → ൞ 𝑐
𝛁×𝑗 = − 2𝐵 [𝑆𝐼]
𝜇0 𝜆𝐿
 𝑐
Gauge freedom 𝛁×𝑗 =− 𝐵 [𝐶𝐺𝑆]
4𝜋𝜆2𝐿
We know that magnectic B will not be affected by distribution of A. 𝑐
𝛁×𝑗 =− 𝐵 [𝑆𝐼]
𝜇0 𝜆2𝐿
And Curl and divergence do not affect each other.

London gauge 𝛁∙𝑨=0 →𝛁∙𝒋=0 B B

𝐴𝑛 = 0 when no current flow through the surface 𝑗𝑛 = 0

4𝜋
𝛁×𝐵 = 𝑗 [𝐶𝐺𝑆]
Ampère's law ቐ 𝑐
𝛁 × 𝐵 = 𝜇0 𝑗 [𝑆𝐼] T > 𝑇𝑐 T < 𝑇𝑐

We curl both side 𝛁 × 𝛁 × 𝐵 = 𝛁× 𝛁× 𝛁×𝑨 = 𝛁 × (𝛁 𝛁 ∙ 𝑨 − 𝛁 𝟐 A) =−𝛁 𝟐 B

4𝜋 curl formula
𝛁 × 𝛁 × 𝐵 = −𝛁 𝟐 B= 𝛁 × 𝑗 [𝐶𝐺𝑆]
→൞ 𝑐
𝛁 × 𝛁 × 𝐵 = −𝛁 𝟐 B = 𝜇0 𝛁 × 𝑗 [𝑆𝐼]

𝐵
→ −𝛁𝟐 B= Obviously, B(r)= B0 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡 𝑖𝑠 𝑛𝑜𝑡 𝑎 𝑠𝑜𝑙𝑢𝑡𝑖𝑜𝑛 𝑖𝑓 𝐵0 ≠ 0
𝜆2𝐿
The formula also confirm that if B=0 , j=0.
 London penetration depth 𝟐
𝐵
−𝛁 B= 2
𝜆𝐿
London equation actually tell as magnctic B can’t
uniformly exist in superconductor . Instead, it decays boundary
exponentially with distance from the outer boundary to
the inside just like the figure.

If B(0) is the field at the boundary,then the

field inside is :
−𝑥
𝐵 𝑥 = 𝐵 0 exp( )
𝜆𝐿
We can see that 𝜆𝐿 measure the depth of
penetration of magnetic field .

𝑐 𝑚𝑐 2
𝛁 × 𝑗 = − 4𝜋𝜆2 𝐵 [𝐶𝐺𝑆] 𝜆𝐿 = [𝐶𝐺𝑆]
𝑛𝑞2 4𝜋𝑛𝑞2
Compare ൞ 𝑐
𝐿
with 𝛁 × 𝑗 = − 𝑚𝑐 𝐵 , we get
𝛁 × 𝑗 = − 𝜇 𝜆2 𝐵 [𝑆𝐼] 𝜀0 𝑚𝑐 2
0 𝐿 𝜆𝐿 = [𝑆𝐼]
We will explain this later 𝑛𝑞2
Modulation of superconducting electron concentration

Spatial changes in the state of any electronic system require additional kinetic energy.
2 2
Schrödinger equation without time is Η ෡ = − ℏ 𝑑 + 𝑉(𝑥)
෡ Ψ = 𝐸Ψ , Η
2𝑚 𝑑𝑥
Coherence of Cooper pairs leads to periodic changes in probability density.
𝑖𝑓 𝑒𝑙𝑒𝑐𝑡𝑟𝑜𝑛𝑠 𝑎𝑟𝑒 𝑖𝑛𝑑𝑒𝑝𝑒𝑛𝑑𝑒𝑛𝑡 : Ψ ∗ Ψ = 𝒆𝒊𝒌𝒙 𝒆−𝒊𝒌𝒙 = −1
else if 𝑒𝑙𝑒𝑐𝑡𝑟𝑜𝑛𝑠 𝑎𝑟𝑒 interfere with each other : 𝛗∗ 𝛗 = 1+cos(qx) the equation is modulated by wave number q
1 1
𝛗= (𝒆𝒊(𝒌+𝒒)𝒙+ 𝒆𝒊𝒌𝒙), 𝛗∗ = (𝒆−𝒊(𝒌+𝒒)𝒙+ 𝒆−𝒊𝒌𝒙)
2 2
ℏ2 𝑘 2
The kinectic energy of the wave Ψ is
2𝑚
ℏ2 𝑑2 𝟏 ℏ2 ℏ2 ℏ2
The kinectic energy of the wave 𝛗 is ‫ ∗𝛗 𝑥𝑑 ׬‬− 2𝑚 𝑑𝑥 𝛗 = 𝟐 2𝑚
𝒌+𝒒 𝟐
+ 𝒌𝟐 ≅ 2𝑚 𝒌𝟐 + 2𝑚 k𝒒
we neglect 𝒒𝟐 for 𝒒 ≪ 𝒌
 Coherence Length
The coherence length represents the distance scale at which the electron concentration in
the superconducting state does not change drastically in space.
ℏ2 𝑘 2
The kinectic energy of the wave Ψ is 2𝑚
ℏ2 ℏ2
The kinectic energy of the wave 𝛗 ≅ 𝒌𝟐+ k𝒒
2𝑚 2𝑚
ℏ2
The increase of energy require to modulate is 2𝑚 k𝒒

If this increase exceeds the energy gap 𝐸𝑔 , superconductivity

will be destroyed.So the critical value 𝑞0 of the modulation 1
𝑞∝
wave number is given by 𝜆
ℏ2
𝑘 𝑞 = 𝐸𝑔
2𝑚 𝐹 0 𝜉∝𝜆
1
we define an intrinsic Coherence Length 𝜉 0 =
𝑞0
ℏ2 𝑘 𝑝 ℏ𝑘𝐹
𝜉 0 = 2𝑚𝐸𝐹 , electron velocity at fermi surface 𝑣𝐹 = 𝑚 =
𝑔 𝑚
ℏ𝑣 2ℏ𝑣𝐹
𝜉 0 = 2𝐸𝐹 ,which is similar to result from BCS 𝜉 0 =
𝑔 𝜋
 BCS Theory of Superconductivity
In 1957, Bardeen, Cooper, and Schrieffer proposed a detailed microscopic
theory, now known as the BCS theory.
In a superconductor, electrons don’t behave as individual particles as they
do in a normal conductor. Instead, they form pairs called Cooper pairs.
 When an electron moves through the lattice, it attracts the positive charges at
neighboring lattice points, causing a local distortion in the lattice and creating a
region of high positive charge.
≫ ≫ Attract another electron with an opposite spin, forming a Cooper pair with a
certain binding energy.
 Specification of the Cooper pairs

Opposite Spins and wavevectors

≫ ≫ Cooper pairs are formed by two electrons with opposite spins, one spin-up and
one spin-down and opposite wavevectors, +k and -k. This means that the total spin
of the pair is zero.

Lower energy state

≫ ≫ When electrons pair up, they occupy a lower energy state than they would as
independent electrons. This helps minimizing the total energy of the electrons in the
superconductor.
 The standard for the transition temperature of an element or alloy involves the electron density of
orbitals D (𝜖F ) of one spin at the Fermi level and the electron-lattice interaction U, which can be
estimated from the electrical resistivity. For UD(𝜖F ) <<1 the BCS theory predicts :
Tc = 1.14 𝜃exp[-1/UD(𝜖F )] ,
where 𝜃 is the Debye temperature and U is an attractive interaction. The result for Tc is satisfied at
least qualitatively by the experimental data. There is an interesting apparent paradox: the higher the
resistivity at room temperature the higher is U, and thus the more likely it is that the metal will be a
superconductor when cooled.

Magnetic flux through a superconducting ring is quantized and the effective unit of charge is 2e
rather than e. The BCS ground state involves pairs of electrons; thus, flux quantization in terms of
the pair charge 2e is a consequence of the theory.
 BCS Ground State

The formation of Cooper pairs leads to the

creation of an energy gap 𝐸𝑔 around the
Fermi energy 𝜖𝐹. This gap is the energy
required to break a Cooper pair and create
unpaired electron excitations in the system.

(b) BCS Ground State

(a) Noninteracting Fermi Gas
(Superconducting State)
 Cooper Pair and Quantum state
Bose-Einstein statistics
From the BCS theory we know that cooper pair follow Bose-
1
Einstein statistics.So quantum states are same for all cooper pair < 𝑛 >=
in superconductor . 𝑒 (𝜀−𝜇)/𝑘𝐵 𝑇 − 1

We assume the pair concentration 𝐧 = 𝛗∗ 𝛗= constant

𝛗= 𝒏𝒆𝒊𝜽(𝒓) ,𝛗∗ = 𝒏𝒆−𝒊𝜽(𝒓)

𝜽 is phase.Current and magnetic fields affect the motion of charged

particles, and this motion is reflected as a phase change in the wave
function.
 pair concentration 𝐧 = 𝛗∗ 𝛗
𝛗= 𝒏𝒆𝒊𝜽(𝒓) ,𝛗∗ = 𝒏𝒆−𝒊𝜽(𝒓)

London equation
In the electromagnetic field, we can use Hamilton mechanics to describe the motion of
electrons.
𝟏 𝒒
𝑯= (p− 𝑨) 𝟐 +q𝝓
𝟐𝒎 𝒄
𝟏 𝒒 𝟏 𝒒
v= (p− 𝑨) = (−iℏ𝛁− 𝑨)
𝒎 𝒄 𝒎 𝒄
𝒏 𝒒
𝒑𝒂𝒓𝒕𝒊𝒄𝒍𝒆 𝒇𝒍𝒖𝒙 𝒏𝒗 = 𝛗∗ 𝒗𝛗= = (ℏ𝛁𝜽− 𝑨)
𝒎 𝒄
𝒏𝒒 𝒒
Electric current density 𝐣 = 𝐪𝒏𝒗 = 𝒒𝛗∗ 𝒗𝛗= = (ℏ𝛁𝜽− 𝑨)
𝒎 𝒄

𝒏𝒒𝟐
London equation 𝛁 × 𝐣 = − 𝑩
𝒎𝒄
Superconducting Ring and Flux Quantization
Because it is a ring,when the current goes around a
circle and returns to the origin, its quantum state
should be the same as the initial one. The phase
change should be an integral multiple of 2𝜋.

‫ 𝟐𝜽 = 𝒍𝒅 ∙ 𝜽 𝛁 𝑐ׯ‬- 𝜽𝟏 = 2𝜋s , s is an integer

The Meissner effect tells us that B and j are zero in the
interion.
𝒏𝒒 𝒒
𝐣= (ℏ𝛁𝜽− 𝑨)=0
𝒎 𝒄
𝒒
ℏ𝛁𝜽= 𝑨
𝒄
By Stoke’s theorem

‫(𝚽 = 𝝈𝒅 ∙ 𝑩 𝑆׭ = 𝝈𝒅 ∙ 𝐴 × 𝛁 𝑆׭ =𝒍𝒅 ∙ 𝐴 𝑐ׯ‬magnetic flux)

ℏ𝒄 2𝜋ℏ𝒄
we know 𝐴 = 𝛁𝜽 ,we get 𝚽 = ( ) s , s is an integer
𝒒 𝒒
 Duration of Persistent Current
Persistent current: In a superconducting ring, there is a stable current that can last for a
long time. This current will generate magnetic flux, and the magnetic flux will be "trapped"
in the ring and will not leak easily.
Calculation of probability of leakage:
𝑃 = (𝑎𝑡𝑡𝑒𝑚𝑝𝑡 𝑓𝑟𝑒𝑞𝑢𝑛𝑐𝑦)(𝑎𝑐𝑡𝑖𝑣𝑎𝑡𝑖𝑜𝑛 𝑏𝑎𝑟𝑟𝑖𝑒𝑟 𝑓𝑎𝑐𝑡𝑜𝑟)
𝐸𝑔
𝒂𝒕𝒕𝒆𝒎𝒑𝒕 𝒇𝒓𝒆𝒒𝒖𝒏𝒄𝒚: Represents the 𝑓𝑟𝑒𝑞𝑢𝑛𝑐𝑦 leaks may occur 𝑓 =
ℏ
∆F
𝒂𝒄𝒕𝒊𝒗𝒂𝒕𝒊𝒐𝒏 𝒃𝒂𝒓𝒓𝒊𝒆𝒓 𝒇𝒂𝒄𝒕𝒐𝒓: According to Boltzmann distribution exp(− )
𝑘𝐵 𝑇

∆F=(minimum volume)(excess free energy density of normal state)

minimum volume: R𝜉 2 , 𝜉 is the coherence length , R is the wire thickness
𝐻𝑐 2
excess free energy density of normal state:
8𝜋
𝐵𝑎𝑐 2
2 𝐻𝑐 2 ∆F = 𝐹N (0) − 𝐹𝑆 0 =
𝐸𝑔 R𝜉 8𝜋
𝑃= exp(− ) 8𝜋
ℏ 𝑘𝐵 𝑇
 Magnetic field behavior of Type II superconductor
There’s a huge difference in magnetic field behavior between Type I and Type II
superconductors, it’s depended on the mean free path, which can see the relationship
between the coherence length 𝜉 and the penetration depth 𝜆.
Characteristic distance over
which an external magnetic field
decays inside a superconductor.

If 𝜉 > 𝜆, the superconductor is typically

Type I, as in most pure metals.

If 𝜉 < 𝜆, indicating a short coherence

length and large penetration depth, the
superconductor is Type II.
 Magnetic field penetration for different
superconductors Repel the magnetic flux

Part of it will penetrate

The amount of the magnetic flux

is repelled will affect the free
energy of a superconductors,
which will affect directly for the
difficulty that the external
magnetic field to destroy the
superconductivity.

(a) Thin Film Superconductor (b) Bulk Superconductor in the

mixed state
 Behavior of the magnetic field 𝐵(𝑥) and the
superconducting order parameter (energy gap)
Type I superconductors:
Penetration depth 𝜆 < coherence length 𝜉
≫ ≫ The magnetic field 𝐵(𝑥) decays quickly as it
penetrates the superconductor, because it will expel
it immediately.

≫ ≫ The superconducting order parameter changes

more slowly over a longer distance 𝜉, which order
parameter represents the energy gap for Cooper pairs,
the strength of the superconductivity. This will cause
it sensitive to magnetic fields, resulting in a complete
loss of superconductivity beyond a critical field.
The left on the x-axis is moving deeper
into the superconducting region.
 Type II superconductors:
Penetration depth 𝜆 > coherence length 𝜉
≫ ≫ The magnetic field decays more gradually inside
the superconductor, spreading over a larger distance 𝜆.
This gradual decay allows magnetic flux to penetrate
deeper into the superconductor.

≫ ≫ The order parameter changes over a much shorter

distance 𝜉, and it’s more sharply which enabling these
superconductors to maintain superconductivity in 𝐵𝑎 : Applied (external) magnetic field.
higher magnetic fields by forming the mixed state with 𝐵𝑏 : Internal magnetic field in the
vortices. superconductor, which results from the
interaction between the superconductor
and the applied field.
DC Josephson effect
Macroscopic Quantum Interference