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Superconductivity Electrical resistance r 𝑇c … critical temperature maα Tc const. pure metal a is a material constant 𝑇c metal with impurities 0.1 K (isotopic shift of the critical temperature) 𝑇 1 Superconductivity Heike Kamerlingh Onnes 1913 Nobel prize in physics The superconductivity was discovered in 1911 by Heike Kamerlingh Onnes at the Leiden University. At 4.2 K (-296°C), he observed a disappearance of resistivity in mercury. His experiments were made possible by the condensation of helium (1908). 2 Superconductivity Superconducting elements Al Cd Ga Hg In Ir La Mo Nb Os Pb Re T [K] 1.19 0.56 1.09 4.00 3.40 0.14 5.00 0.92 9.13 0.65 7.19 1.70 Ru Sn Ta Tc Th Ti Tl U V Zn Zr T [K] 0.49 3.72 4.48 8.22 1.37 0.39 2.39 0.68 5.30 0.87 0.55 3 Isotopic Shift maα Tc const. Material T [K] Zn Cd Sn Hg Pb Tl 0.87 0.56 3.72 4.00 7.19 2.39 a 0.45±0.05 0.32±0.07 0.47±0.02 0.50±0.03 0.49±0.02 0.61±0.10 Material T [K] a Ru Os Mo Nb3Sn Mo3Ir Zr 0.49 0.65 0.92 18 0.00±0.05 0.15±0.05 0.33 0.08±0.02 0.33±0.03 0.00±0.05 0.55 4 Superconductivity Temperature dependence of the critical magnetic field Superconductor in a magnetic field Hc T2 H c H 0 1 2 Tc normal state B µ0 ( H M ); 0 4 10 7 Vs /( Am) Superconductor: Meissner effect B 0 H M M H 1 superconducting state Tc T Otherwise: -10-6 5 Meissner-Ochsenfeld effect 6 Magnetic levitation train 7 Superconductor in a magnetic field External field: Inner field: Magnetization: Be µ0 H e B µ0 ( H e M ) 0 M H e Be Be2 W MdB BdB 2 0 0 Be Work per unit of volume (magnetization direction of a superconductor is opposite to the magnetic field direction) Energy of a superconductor within an magnetic field is higher than without an magnetic field This is caused by the “superconducting” electrons 8 Transition between normal and superconducting state Thermodynamic consideration G U TS Be2 G U TS 2 Be2 1 Tc U G S 2 Be 2U G Tc S 𝐺 … Gibbs free energy 𝑈 … enthalpy 𝑇 … temperature 𝑆 … entropy 𝐵e … external magnetic field 𝑇 < 𝑇c: 𝑈 (and 𝑆) small for SC state, therefore the SC state is stable 𝑇 > 𝑇c: 𝑆 bigger in normal state (less order), therefore the normal state is stable 𝐵 > 0: free Gibbs energy is smaller, if 𝑆 is bigger (normal state) 9 Superconductivity Material 𝑇 [K] NbC 14 NbN 16 Nb3Al 18 Nb3Ge 23 Nb3Sn 18 SiV3 17 La2-xBaxCuO4 30 MgB2 40 YBa2Cu3O7-d 110 S.L. Bud’ko and P.C. Canfield: Temperature-dependent Hc2 anisotropy in MgB2 as inferred from measurements on polycrystals, Phys. Rev. B 65 (2002) 212501. 10 Crystal structures of La2-xBaxCuO4 and YBa2Cu3O7-x La2-xBaxCuO4 Space group: Bmab Lattice parameters: a = 5.33915(9) Å b = 5.35882(9) Å c = 13.2414(2) Å YBa2Cu3O7-x Space group: Pmmm Lattice parameters: a = 3.856(2) Å b = 3.870(2) Å c = 11.666(3) Å ab a/2 < c/3 < a a b c/3 11 Superconductivity Type I superconductors Type II superconductors • Transition to normal state after exceeding 𝐻c • Superconductivity decreases gradually between 𝐻c1 und 𝐻c2 • Transition to normal state after exceeding 𝐻c2 −𝑀 −𝑀 superconducting normal state 𝐻c 𝐻 𝐻c1 𝐻c 𝐻c2 𝐻 12 Theories of Superconductivity Superelectrons : • No scattering • Entropy of the system is zero (the system is perfectly ordered) • Large coherence length 13 London Theory (Meissner Effect) 1 A; rot A B London: j 2 µ0 λL Ohm: j E London: Maxwell: Meissner effect: 1 rot j B 2 µ0 λL E rot B µ0 j µ0 0 rot B µ0 j (static conditions) t Solution: rot rot B grad div B B µ0 rot j 0 x B B x B0 exp B 2 λL λL B 𝜆L … London penetration depth x 14 Consequences of the London Theory 𝜆L describes the penetration depth of the magnetic field into a material. Inside the material at a distance 𝜆L to the surface the intensity of the magnetic field falls to 1/e of its original value. An external magnetic field 𝐵e penetrates completely homogeneous a thin layer, if the thickness is much smaller than 𝜆L . In such a layer, the Meissner effect is not complete. The induced field (in the material) is smaller than 𝐵e , therefore the critical magnetic field, which is oriented parallel to the thin layers is very high. 15 Coherence Length The distance in which the width of the energy gap, in a spatial variable magnetic field, doesn’t change essentially. London: 1 j r Ar 2 µ0 λL rot Ar Br 16 BCS Theory of Superconductivity J. Bardeen, L.N. Cooper and J.R. Schrieffer, Phys. Rev. 106 (1957) 162. J. Bardeen, L.N. Cooper and J.R. Schrieffer, Phys. Rev. 108 (1957) 1175. 1. Interactions between electrons can cause a ground state, which is separated from the electronically excited states by an energy gap. However: there are also superconductors without an energy gap! 𝐸 𝐸 17 BCS Theory of Superconductivity 2. The energy gap is caused by the interaction between electrons via lattice vibrations (phonons). One electron distort the crystal lattice, another electron “sees” this and assimilate his energy to this state in a way, which reduces the own energy. That’s how the interaction between electrons via lattice vibrations work. 18 BCS Theory of Superconductivity 3. The BCS theory delivers the London penetration depth for the magnetic field and the coherence length. Thereby the Meissner effect is explained. London: 1 j r Ar ; rot Ar Br 2 µ0 λL Meissner: B x B 2 B x B0 exp λL λL Coherence length: 0 vF πE g 19