Downloaded 30 times
![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]](https://image.slidesharecdn.com/superconductivityasmacroscopicphenomenatermpaperpresentation-140405131649-phpapp02/85/Superconductivity-as-macroscopic-phenomena-term-paper-presentation-3-320.jpg)
![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]](https://image.slidesharecdn.com/superconductivityasmacroscopicphenomenatermpaperpresentation-140405131649-phpapp02/85/Superconductivity-as-macroscopic-phenomena-term-paper-presentation-34-320.jpg)
More Related Content
Similar to Superconductivity(as macroscopic phenomena) term paper presentation
Superconductivity(as macroscopic phenomena) term paper presentation
- 1. Superconductivity: As Macroscopic Phenomena SUJEETKUMAR CHOUDHARY(10MS54) SANMOY MANDAL(10MS38) 31ST OCTOBER 2013
- 2. Introduction •The property ofzero electrical resistance in some substance at very low absolute temperature. •How it is different with normal conductor?
- 3. Historical Overview •Heike KamerlinghOnnes (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]
- 4. Cooper Pair •Due tothe interaction of the electrons with the vibration of atoms in the lattice there is small net effective 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 of mass of two electrons creating C-P. •All the cooper pair are in the same quantum sate with the same energy.
- 5. Cooper Pair •Cooper pairhave a spatial extension of order 𝜉0. This is the length over which electrons are said to 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.
- 6. Cooper Pair Animation Courtesy: http://www.youtube.com/watch?v=wbAFvKMRlSI
- 7. BCS Ground State •InBCS theory wave function is written as: |𝜓 𝐺 = 𝒌 (𝑢 𝒌 + 𝑣 𝒌 𝑐 𝒌↑ ∗ 𝑐−𝒌↓ ∗)| 𝜓0 •Evidently, this | 𝜓 𝐺 can be expressed as sum | 𝜓 𝐺 = 𝑁 𝜆 𝑁| 𝜓 𝑁
- 8. BCS Ground State |𝜓𝜙 = 𝒌 ( 𝑢 𝒌 + 𝑣 𝒌 𝑒 𝑖𝜙 𝑐 𝒌↑ ∗ 𝑐−𝒌↓ ∗ )| 𝜓0 We can project out the ground state into definite particle state, by the following equation | 𝜓 𝑁 = 1 2𝜋𝜆 𝑁 0 2𝜋 𝑑𝜙𝑒−𝑖𝑁𝜙 | 𝜓 𝜙 Now, acting on the | 𝜓 𝜙 by − 𝑖 𝜕 𝜕𝜙 and integrating by parts we get, N| 𝜓 𝑁 .
- 9. Conjugacy Relationship •Particle numberand 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
- 10. London Equation •We canwrite our wave function as: Ψ 𝒓 = 𝜌 𝒓 𝑒 𝑖𝜃(𝒓) •We get the equation for current density if we plug the above wave function into probability current: 𝐉 = ℏ 𝑚 (𝛁𝜃 − 𝑞 ℏ 𝑨)𝜌
- 11. London Equation •Taking divergenceof the above equation 𝜵. 𝑱 = ℏ 𝑚 (𝛁 𝟐 𝜃 − 𝑞 ℏ 𝛁. 𝑨)𝜌 •Using and we get 𝛁 𝟐 𝜃 = 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 = − 𝜌 𝑞 𝑚 𝑨
- 12. Meissner Effect •Meissner effectis a expulsion of magnetic field from superconductor during its transition to the superconducting state. •The vector potential is related to the current density by 𝜵2 𝑨 = − 1 𝜀0 𝑐2 𝑱 •From London and London equation we get: 𝜵2 𝑨 = 𝜆2 𝑨 where 𝜆2= 𝜌 𝑞 𝜀0 𝑚𝑐2
- 13. Meissner Effect •Solution tothe above equation has the form of 𝑨 = 𝑨0 𝑒 𝝀𝒓 𝑩 = 𝝀 × 𝑨0 𝑒 𝝀𝒓 with 𝝀 = 𝜆
- 14. Meissner Effect 𝑭𝒊𝒈: MagneticField as a function of 𝑟 Picture courtesy: The Feynman lectures on physics. Volume 3 (1965)
- 15.
- 16. Aharonov-Bohm Effect •Quantum mechanicalphenomenon where a charged particle is affected by an electromagnetic field (E, B). •To demonstrate the phenomenon we will deal with toroidal ring like physical object and introduce the quantization in the process.
- 17. Flux Quantization •From Maxwell-Faradayequation (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 ℏ Φ •Since the L.H.S. is not zero and that leads to non-trivial solution of the case.
- 18. Quantization of Flux 𝛁𝜃.𝒅𝒔 = 𝑞 ℏ Φ Φ = 2n𝜋 ℏ 𝒒 •The trapped flux must always be an integer times 2𝜋 ℏ 𝒒 of where 𝑞 = 2𝑒 i.e. charge of a cooper pair.
- 19. Josephson Junction •Two superconductingmaterials 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
- 20. Salient Features ofJosephson 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 and we can treat phase as a semiclassical physically observable quantity. •For Josephson Junction phase degree of freedom is quite important.
- 21. Guiding Equations •If theJosephson Junction is placed within a voltage gap V, then the basic equations will be as follows 𝑖ℏ 𝜕𝜓1 𝜕𝑡 = 𝑞𝑉 2 𝜓1 + 𝐶𝜓2 𝑖ℏ 𝜕𝜓2 𝜕𝑡 = −𝑞𝑉 2 𝜓2 + 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
- 22. Guiding Equation •Decoupling 𝜌𝑖and from each and other and subtracting one another one will arrive at the following equation 𝜕𝜌1 𝜕𝑡 = − 𝜕𝜌2 𝜕𝑡 = 2𝐶 ℏ 𝜌1 𝜌2 sin 𝛿 𝜕𝛿 𝜕𝑡 = 𝑞𝑉 2ℏ
- 23. Time Evolution Of𝜌𝑖 •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=J0 𝑠𝑖𝑛 𝛿 where J0= 2𝐶 ℏ 𝜌1 𝜌2 and the charge density remain unchanged in both the region.
- 24. Time Evolution of𝛿 • We have, 𝜕𝛿 𝜕𝑡 = 𝑞𝑉 2ℏ 𝛿𝑡 = 𝛿0 + 𝑞 2ℏ 𝑉(𝑡)𝑑𝑡
- 25. DC Josephson Junction •For𝑉 𝑡 = 𝑉0we have, J = J0sin( 𝛿0 + 𝑞 ℏ V0t) • 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.
- 26. Josephson current inMagnetic Field •We have the following equation: 𝐼 = 𝐼0(𝛿0 + 2𝑞 ℏ 𝑨. 𝒅𝒔) •Here we introduce how the phase factor changes due to Pierls substitution. •Result: Gauge Invariant !
- 27. AC Josephson Junction •For𝑉 𝑡 = 𝑉0 + 𝑣𝑠𝑖𝑛(𝜔𝑡)where 𝑣 ≪ 𝑉0then 𝛿𝑡 = 𝛿0 + 𝑞 ℏ V0t + 𝑞 ℏ v 𝜔 𝑠𝑖𝑛 𝜔𝑡 •Then, J = J0 𝑠𝑖𝑛( 𝛿0 + 𝑞 ℏ V0t) + 𝑞 ℏ v 𝜔 𝑠𝑖𝑛 𝜔𝑡cos(𝛿0 + 𝑞 ℏ V0t) , 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 ℏ V0 and gives resonance.
- 28. Quantum Interference amongstphases Path1 Path 2 • Path 1 picks up phase 𝛿1 = 𝛿0 + 𝑒𝜙 ℏ and, • Path 2 picks up phase 𝛿2 = 𝛿0 − 𝑒𝜙 ℏ • Hence phase difference between the paths is 𝛿2 − 𝛿1 = 2 𝑒𝜙 ℏ Picture courtesy: The Feynman lectures on physics. Volume 3 (1965)
- 29. DC SQUID •The currentpassing through will have following expression: 𝐼 = 𝐼0 sin 𝛿 𝑎 + sin 𝛿 𝑏 = 2𝐼0 sin 2𝛿0 cos( 2𝑒𝜙 ℏ ) •Here current maxima occurs at 𝜙 = 𝑛𝜋 ℏ 𝑞 𝑒 = 𝑛𝜙0. •And one can create magnetometer sensitive up to order of 10−7 𝐺𝑎𝑢𝑠𝑠 . •Current flux relationship is nonlinear.
- 30. Flux and currentnonlinear relationship Picture courtesy: The Feynman lectures on physics. Volume 3 (1965)
- 31. Rf SQUID •Aharonov-Bohm Effectand Flux Quantization revisited 𝛿 = 2𝜋 𝜙 𝜙0 •Rf SQUID double well. Fig. Rf SQUID Picture courtesy: Google Images
- 32. Practical Realization Japanese levitatingtrain has superconducting magnets onboard SQUID measurement of neuro-magnetic signals Superconducting power cable installed in Denmark Medical MRI Scanner Picture courtesy: Google Images
- 33. What lied ahead... •High𝑇𝐶 superconductor. ( Ex. – Materials like ceramics) •High performance Smart grid, Electrical Power transmission, Transformer, Power storage devices , Electric Motors , Magnetic levitation devices ,Fault current limiters. • Nanoscopic materials such as Buckyballs, Nanotube and Superconducting Magnetic Refrigeration.
- 34. 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]
- 35.