Superconductivity:As Macroscopic Phenomena

SUJEET KUMAR CHOUDHARY(10MS54)

SANMOY MANDAL(10MS38)

31ST OCTOBER 2013

Introduction•The property of zero electrical resistance in some substance at very low absolute temperature.

•How it is different with normal conductor?

Historical Overview •Heike Kamerlingh Onnes (1911), Nobel Prize [1913]

•Meissner and Ochsenfeld (1933)

•Fritz and Heinz London (1935)

•John Bardeen, Leon N. Cooper, J. Robert Schriffer (1957) , Nobel Prize [1972]

Cooper Pair•Due to the interaction of the electrons with the vibration of atoms in the lattice there is small neteffective attraction between the electrons. The result is that electrons form together.

•A movement of C-P when a supercurrent is flowing is considered as a movement of center ofmass of two electrons creating C-P.

•All the cooper pair are in the same quantum sate with the same energy.

Cooper Pair•Cooper pair have a spatial extension of order 𝜉0. This is the length over which electrons are saidto be coherent.

•Fermi sea is unstable against the formation of cooper pair.

•An energy gaps opens up between the ground and higher excited state.

•Hence, super current ones formed persist.

Cooper Pair Animation

Courtesy : http://www.youtube.com/watch?v=wbAFvKMRlSI

BCS Ground State•In BCS theory wave function is written as:

|𝜓𝐺 =

𝒌

(𝑢𝒌 + 𝑣𝒌𝑐𝒌↑∗𝑐−𝒌↓∗)| 𝜓0

•Evidently, this | 𝜓𝐺 can be expressed as sum

| 𝜓𝐺 =

𝑁

𝜆𝑁| 𝜓𝑁

BCS Ground State

|𝜓𝜙 =

𝒌

( 𝑢𝒌 + 𝑣𝒌 𝑒𝑖𝜙𝑐𝒌↑∗𝑐−𝒌↓∗)| 𝜓0

We can project out the ground state into definite particle state, by the following equation

| 𝜓𝑁 =1

2𝜋𝜆𝑁

0

2𝜋

𝑑𝜙𝑒−𝑖𝑁𝜙| 𝜓𝜙

Now, acting on the | 𝜓𝜙 by − 𝑖𝜕

𝜕𝜙and integrating by parts we get, N| 𝜓𝑁 .

Conjugacy Relationship

•Particle number and phase are related by

𝑁 ↔ − 𝑖𝜕

𝜕𝜙

•As long as we regard the particle number as continuous variable.

•Hence, here we get a relationship:

Δ𝑁Δ𝜙 ≳ 1

London Equation•We can write our wave function as:

Ψ 𝒓 = 𝜌 𝒓 𝑒𝑖𝜃(𝒓)

•We get the equation for current density if we plug the above wave function into probabilitycurrent:

𝐉 =ℏ

𝑚(𝛁𝜃 −𝑞

ℏ𝑨)𝜌

London Equation•Taking divergence of the above equation

𝜵. 𝑱 =ℏ

𝑚(𝛁𝟐𝜃 −

𝑞

ℏ𝛁. 𝑨)𝜌

•Using and we get

𝛁𝟐𝜃 = 0

•So 𝜃 is constant inside the superconductor,

• Since 𝜌 inside the superconductor is also constant.

•From above argument we get London and London Equation:

𝑱 = − 𝜌𝑞

𝑚𝑨

Meissner Effect•Meissner effect is a expulsion of magnetic field from superconductor during its transition to thesuperconducting state.

•The vector potential is related to the current density by

𝜵2𝑨 = −1

휀0𝑐2𝑱

•From London and London equation we get:

𝜵2𝑨 = 𝜆2𝑨

where 𝜆2= 𝜌𝑞

0𝑚𝑐2

Meissner Effect•Solution to the above equation has the form of

𝑨 = 𝑨0𝑒𝝀𝒓

𝑩 = 𝝀 × 𝑨0𝑒𝝀𝒓

with 𝝀 = 𝜆

Meissner Effect

𝑭𝒊𝒈: Magnetic Field as a function of 𝑟

Picture courtesy: The Feynman lectures on physics. Volume 3 (1965)

Meissner Effect

Courtesy: http://www.youtube.com/watch?v=Xts42tFYRRg&list=PLK0Bh1ZBTbAaHM3lcsbXGgX9kAVz6zVBI&index=3

:http://www.youtube.com/watch?v=Xts42tFYRRg&index=2&list=PLK0Bh1ZBTbAaHM3lcsbXGgX9kAVz6zVBI

Aharonov-Bohm Effect•Quantum mechanical phenomenon where a charged particle is affected by an electromagneticfield (E, B).

•To demonstrate the phenomenon we will deal with toroidal ring like physical object and introducethe quantization in the process.

Flux Quantization•From Maxwell-Faraday equation (Faraday’s Law of Induction), we have

𝜕Φ

𝜕t= E.dl

•But, for a superconducting substance, R.H.S. goes to zero.

•Again from earlier discussed equation = 0 inside a superconductor. That leads to:

ℏ𝛁𝜃 = 𝑞𝑨

𝛁𝜃. 𝒅𝒔 =𝑞

ℏΦ

•Since the L.H.S. is not zero and that leads to non-trivial solution of the case.

Quantization of Flux

𝛁𝜃. 𝒅𝒔 =𝑞

ℏΦ

Φ = 2n𝜋ℏ

𝒒

•The trapped flux must always be an integer times 2𝜋ℏ

𝒒of where 𝑞 = 2𝑒 i.e. charge of a cooper

pair.

Josephson Junction•Two superconducting materials are connected by a thin layer of insulator is called Josephson Junction.

Picture courtesy: Nature 474 589-597(30 June 2011) doi:10.1038/nature10122

Salient Features of Josephson Junction•Thin insulator is needed to produce quantum tunneling of having supercurrent.

•At a very large no. of particles(1022), hence the relative uncertainty in phase becomes low andwe can treat phase as a semiclassical physically observable quantity.

•For Josephson Junction phase degree of freedom is quite important.

Guiding Equations•If the Josephson Junction is placed within a voltage gap V, then the basic equations will be as follows

𝑖ℏ𝜕𝜓1

𝜕𝑡=𝑞𝑉

2𝜓1+ 𝐶𝜓2

𝑖ℏ𝜕𝜓2

𝜕𝑡=−𝑞𝑉

2𝜓2+ 𝐶𝜓1

• C is the characteristic of the josephson junction that determines the hoping or flipflop amplitude of the two level system.

•Now one can use following ansatz: 𝜓i = 𝜌𝑖exp(i𝜃𝑖)

where 𝜃2− 𝜃1 = 𝛿

Guiding Equation•Decoupling 𝜌𝑖 and from each and other and subtracting one another one will arrive at the following equation

𝜕𝜌1

𝜕𝑡= −𝜕𝜌2

𝜕𝑡=2𝐶

ℏ𝜌1𝜌2 sin 𝛿

𝜕𝛿

𝜕𝑡=𝑞𝑉

2ℏ

Time Evolution Of 𝜌𝑖•In the guiding equations we have not considered the currents due to external supply of energy.

•That is why we see that the charge density in both sides is changing. If we consider these we see that the junction current is given by,

J=J0𝑠𝑖𝑛 𝛿 where J0= 2𝐶

ℏ𝜌1𝜌2

and the charge density remain unchanged in both the region.

Time Evolution of 𝛿• We have,

𝜕𝛿

𝜕𝑡=𝑞𝑉

2ℏ

𝛿𝑡 = 𝛿0+𝑞

2ℏ 𝑉(𝑡)𝑑𝑡

DC Josephson Junction•For 𝑉 𝑡 = 𝑉0we have,

J = J0sin( 𝛿0+𝑞

ℏV0t)

• DC Josephson junction the junction current is oscillatory and it is very rapid and we do not have net current.

•No DC current leads to oscillatory J = J0sin( 𝛿0) which is featuring characteristic of this junction.

Josephson current in Magnetic Field •We have the following equation:

𝐼 = 𝐼0(𝛿0 + 2𝑞

ℏ𝑨. 𝒅𝒔)

•Here we introduce how the phase factor changes due to Pierls substitution.

•Result: Gauge Invariant !

AC Josephson Junction•For 𝑉 𝑡 = 𝑉0 + 𝑣𝑠𝑖𝑛(𝜔𝑡)where 𝑣 ≪ 𝑉0then

𝛿𝑡 = 𝛿0+𝑞

ℏV0t +𝑞

ℏ

v

𝜔𝑠𝑖𝑛𝜔𝑡

•Then, J = J0𝑠𝑖𝑛( 𝛿0 +𝑞

ℏV0t) +

𝑞

ℏ

v𝜔𝑠𝑖𝑛 𝜔𝑡cos(𝛿0 +

𝑞

ℏV0t) , using Taylor series expansion for small

shift from initial position.

•The first term goes to zero on average over time on the other hand the second term contributes

significantly for 𝜔=𝑞

ℏV0 and gives resonance.

Quantum Interference amongst phasesPath1

Path 2

• Path 1 picks up phase 𝛿1 = 𝛿0+𝑒𝜙

ℏand,

• Path 2 picks up phase 𝛿2 = 𝛿0−𝑒𝜙

ℏ

• Hence phase difference between the paths is 𝛿2− 𝛿1 = 2𝑒𝜙

ℏ

Picture courtesy: The Feynman lectures on physics. Volume 3 (1965)

DC SQUID•The current passing through will have following expression:

𝐼 = 𝐼0 sin 𝛿𝑎 + sin 𝛿𝑏

= 2𝐼0 sin 2𝛿0 cos(2𝑒𝜙

ℏ)

•Here current maxima occurs at 𝜙 = 𝑛𝜋ℏ

𝑞𝑒= 𝑛𝜙0.

•And one can create magnetometer sensitive up to order of 10−7 𝐺𝑎𝑢𝑠𝑠 .

•Current flux relationship is nonlinear.

Flux and current nonlinear relationship

Picture courtesy: The Feynman lectures on physics. Volume 3 (1965)

Rf SQUID•Aharonov-Bohm Effect and Flux Quantization revisited

𝛿 = 2𝜋𝜙

𝜙0

•Rf SQUID double well.

Fig. Rf SQUID

Picture courtesy: Google Images

Practical Realization

Japanese levitating train has

superconducting magnets

onboard

SQUID measurement of

neuro-magnetic signals

Superconducting power cable installed in Denmark

Medical MRI Scanner

Picture courtesy: Google Images

What lied ahead...•High 𝑇𝐶 superconductor. ( Ex. – Materials like ceramics)

•High performance Smart grid, Electrical Power transmission, Transformer, Power storagedevices , Electric Motors , Magnetic levitation devices ,Fault current limiters.

• Nanoscopic materials such as Buckyballs, Nanotube and Superconducting MagneticRefrigeration.

References•Phillip W Phillips, Advanced Solid State Physics.

•Michael Tinkham , Introduction to Superconductivity.

•Feynman Lectures in Physics ,Volume 3

•Bardeen Cooper Schreifer, Physical Review, 108,1175[1957]

•Cooper, Physical Review, 104,1189[1956]

Thank You