Electrical resistance Superconductivity Electrical resistance 𝑇 r pure metal metal with impurities 0.1 K 𝑇c 𝑇c … critical temperature a is a material constant (isotopic shift of the critical temperature)

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).

Superconductivity Superconducting elements T [K] T [K] Al 1.19 Ru 0.49 Cd 0.56 Ga 1.09 Hg 4.00 In 3.40 Ir 0.14 La 5.00 Mo 0.92 Nb 9.13 Os 0.65 Pb 7.19 Re 1.70 T [K] Ru 0.49 Sn 3.72 Ta 4.48 Tc 8.22 Th 1.37 Ti 0.39 Tl 2.39 U 0.68 V 5.30 Zn 0.87 Zr 0.55

Isotopic Shift Material T [K] a Zn 0.87 0.45±0.05 Cd 0.56 0.32±0.07 Sn 3.72 0.47±0.02 Hg 4.00 0.50±0.03 Pb 7.19 0.49±0.02 Tl 2.39 0.61±0.10 Material T [K] a Ru 0.49 0.00±0.05 Os 0.65 0.15±0.05 Mo 0.92 0.33 Nb3Sn 18 0.08±0.02 Mo3Ir 0.33±0.03 Zr 0.55 0.00±0.05

Superconductor in a magnetic field Superconductivity Superconductor in a magnetic field Temperature dependence of the critical magnetic field Hc normal state Superconductor: Meissner effect superconducting state Otherwise:   -10-6 Tc T

Meissner-Ochsenfeld effect

Magnetic levitation train

Superconductor in a magnetic field External field: Inner field: Magnetization: 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

Transition between normal and superconducting state Thermodynamic consideration 𝐺 … Gibbs free energy 𝑈 … enthalpy 𝑇 … temperature 𝑆 … entropy 𝐵 e … 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)

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.

Crystal structures of La2-xBaxCuO4 and YBa2Cu3O7-x Space group: Bmab Lattice parameters: a = 5.33915(9) Å b = 5.35882(9) Å c = 13.2414(2) Å a  b a/2 < c/3 < a YBa2Cu3O7-x Space group: Pmmm Lattice parameters: a = 3.856(2) Å b = 3.870(2) Å c = 11.666(3) Å a  b  c/3

Superconductivity Type I superconductors Type II superconductors Transition to normal state after exceeding 𝐻c Type II superconductors Superconductivity decreases gradually between 𝐻c1 und 𝐻c2 Transition to normal state after exceeding 𝐻c2 −𝑀 −𝑀 superconducting normal state 𝐻c 𝐻c1 𝐻c 𝐻c2 𝐻 𝐻

Theories of Superconductivity Superelectrons : No scattering Entropy of the system is zero (the system is perfectly ordered) Large coherence length

London Theory (Meissner Effect) Ohm: London: London: (static conditions) Maxwell: Solution: Meissner effect: B 𝜆 L … London penetration depth x

Consequences of the London Theory 𝜆 L describes the penetration depth of the magnetic field into a material. Inside the material at a distance 𝜆 L to the surface the intensity of the magnetic field falls to 1/e of its original value. An external magnetic field 𝐵 e penetrates completely homogeneous a thin layer, if the thickness is much smaller than 𝜆 L . In such a layer, the Meissner effect is not complete. The induced field (in the material) is smaller than 𝐵 e , therefore the critical magnetic field, which is oriented parallel to the thin layers is very high.

Coherence Length The distance in which the width of the energy gap, in a spatial variable magnetic field, doesn’t change essentially. London:

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

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.

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: Meissner: Coherence length: