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```Lecture 21: Superconductors Superconductivity: The electrical resistance of some solids disappears completely at sufficiently low temperature. Look to the next figure: Resistivity of superconductors versus temperature TC: critical temperature <10‐23 Ω ∙ cm compared to 𝜌 of 10‐9 Ω ∙cm in Cu. ‐Sharp decrease in 𝜌 at TC (critical temperature) 1 𝜌
‐Metallic elements: TC: Hg 4.15K W 0.01 K Nb 9.3K ‐Metal oxides (Ceramics): High temperature semiconductors YBaCuO7 HgCuo (contain CuO2 or NiO2). ‐Cu, Ag, Au, … or Fe, Ni, Co are not superconductors The Two Characteristics of Superconductivity: • The absence of resistance to current flow • Exclusion of magnetic flux (Meissner effect) Meissner Effect As a superconducting substance is cooled below its critical temperature in the presence of applied magnetic field, it expels all magnetic flux from its interior. 2 Ill explain in more detail This leads to a very interesting phenomena (magnetic levitation) 3 A permanent magnet floating over a superconducting surface ‐magnetic train Let’s go back to the previous figure: 4 Now, if the external magnetic field increases beyond a certain value (the critical field HC) the material ceases to be superconductor and becomes normal. ‐If the temperature increases beyond TC, the material becomes normal. Thus, there is a critical temperature, critical magnetic field, and critical current. 5 ‐Critical temperature, critical magnetic field, critical current Let’s look again to (H‐T) curve: HC: critical magnetic field Above HC, the superconductivity is destroyed 𝑇!
𝐻! = 𝐻! 1 − ! 𝑇!
A: normal state Magnetic field=zero 6 Path: ABCD Cooling down:( A⟶B) B: superconductor H: still zero C: Turn in magnetic field, changing flux creates electric field that sets up a current that opposes magnetic field (Lenz law). Since there is no resistivity, eddy currents don’t decay but create magnetic field that cancels the applied field. Superconductors act as perfect diamagnet We can summarize: Now, let’s go through the path ADCB: i.e. apply a field to a normal state, then cool down. What happened? 7 It expelled the magnetic field. We call the two cases Meissner effect: generated field exactly cancels out the applied field in superconductor What is Behind Superconductivity Before I give some explanation for the phenomena, let me remind you what we have so far: − Classical theory of metallic conduction treated electrons as a gas of independent particles. − Quantum Mechanics: treated electrons as independent particles, and took into account the wave nature of electrons and exclusion principle. ∗ Fermi distribution ∗ Quantization of energy level and band theory of solids Independent particles failed to explain superconductivity. We need many‐body effects There was an obvious experimental evidence for the need of many‐body effects. Experimental Evidence: Isotope effect: The critical temperature of a superconductor depends on the total mass of the nucleus 1
𝑇! ∝
𝑀
if we use the isotope, (this means we add more neutrons) the critical temperature decreases. This 8 suggests that the lattice vibrations play a role in superconductivity. We must take into account: • electron‐phonon interaction • electron‐electron interaction BCS Theory (Bardeen, Cooper, Schrieffer, 1957) takes that into account. Although thermal vibrations are behind the resistivity, in some materials and below a certain temperature, the lattice vibrations play a role as an intermediate between two electrons. It results in an attractive force between the two electrons. 9 The first electron deforms the lattice. The second electron is attracted by the deformation (region of increased positive charge density). The first electron has emitted a phonon. A second electron passing by the region will absorb the phonon. The two electrons have exchanged momentum with each other. Attractive interaction exceeds repulsive interaction. The two electrons are weakly bound and form Cooper pair. Cooper pairs are responsible for superconductivity. 10 11 ‐What are the conditions for superconductivity? 1. 2. 3. 4. 12 • According to BCS theory, the binding energy of a cooper pair at absolute zero is about 3kbTC , where kb is Boltzmann constant, TC is critical temperature • Cooper pairs do not obey Fermi‐Dirac statistics. They are bosons. • BCS theory presently fails to explain superconductivity of high temperature super conductors Summary of Superconductor Properties: • They behave as (no DC resistors) • They behave as a perfect dimagnet and experience “Meissner” effect • A band gap was implied by the very fact that the resistance is precisely zero. If charge carriers can move through a crystal lattice without interacting at all, it must be because their energies are quantized such that they do not have any available energy levels within reach of the energies of interaction with the lattice. • Energy gap 13 The lower energy of superconductivity electrons are separated from the normal state by energy gap about EF Classification of Superconductors: Type I (soft) Type II (hard) ‐Type I superconductors: most superconductive elements except Nb 14 Superconductivity is destroyed by applying a small magnetic field HC ‐Type II superconductors: The elimination of superconductivity state is gradually 15 mixed (vortex) state Less than Hc 1→ superconductor Above HC 2 → normal It has There are intermediate fields, where material has both normal and superconductor regions 16 Type II includes: ‐ Nb ‐ All alloys Note: Type II have higher TC, higher HC and higher IC 17 !
𝜙! = ℎ = 2.07 𝑥 10!! 𝑔𝑎𝑢𝑠𝑠 − 𝑐𝑚! known as a !!
fluxoid or flux quantum. 18 19 20 ```
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