Superconductivity

Dept of Phys M.C. Chang Superconductivity • Introduction • Thermal properties • specific heat, entropy, free energy • Magnetic properties • critical field, critical current, Meissner effect, type II SC • London theory of the Meissner effect • penetration length, coherence length, surface energy • Microscopic (BCS) theory • Cooper pair, BCS ground state • Flux quantization • Quantum tunneling • single particle tunneling, DC/AC Josephson effect • SQUID

Discovery of superconductivity Au  Hg T 1913 A brief history of low temperature (Ref: 絕對零度的探索) • 1800 Charles and Gay-Lusac (from P-T relationship) proposed that the lowest temperature is -273 C (= 0 K) • 1877 Cailletet and Pictet liquified Oxygen (-183 C or 90 K) • soon after, Nitrogen (77 K) is liquified • 1898 Dewar liquified Hydrogen (20 K) • 1908 Onnes liquified Helium (4.2 K) • 1911 Onnes measured the resistance of metal at such a low T. To remove residual resistance, he chose mercury. Near 4 K, the resistance drops to 0! R R

Is the resistivity very small or really zero? : Persistent current (Onnes) Switch 1 • open S2 and close S1: there is current in SC coil. • close S2 and open S1: the current in SC coil remains the same for several hours. • similar experiment years later detected no decay of current for 2 years! • Such a current can be a powerful source of magnetic field (however, see later discussion on critical current). Liquid He Switch 2 SC coil compass

0.03K 1.14K 1.09K 0.39K 5.38K 0.88K 0.0003K 7.77K 3.72K 3.40K 0.92K 0.51K 0.56K 9.50K 0.55K 4.88K 0.12K 4.48K 0.01K 1.4K 0.14K 2.39K 7.19K 0.66K 4.15K 0.20K 0.60K 1.37K 1.4K In the form of nanostructure (type II) Tc's given are for bulk, except for Palladium, which has been irradiated with He+ ions, Chromium as a thin film, and Platinum as a compacted powder http://superconductors.org/Type1.htm

HgBa2Ca2Cu3O9 (under pressure) 160 140 HgBa2Ca2Cu3O9 TlBaCaCuO 120 BiCaSrCuO Applications of superconductor 100 • powerful magnet • MRI, the deceased SSC... • magnetic levitation • speed train • SQUID (超導量子干涉儀) • detect tiny magnetic field • quantum bits • lossless powerline (IF there is room temperature SC) YBa2Cu3O7 Superconducting transition temperature (K) 80 60 40 (LaBa)CuO Nb3Ge Nb3Sn NbN 20 NbC Nb Pb Hg V3Si 1910 1930 1950 1970 1990 Superconductivity in alloys and oxides Liquid Nitrogen temperature (77K) Bednorz Muller 1987 From Cywinski’s lecture note

Introduction • Thermal properties • specific heat, entropy, free energy • Magnetic properties • critical field, critical current, Meissner effect, type II SC • London theory of the Meissner effect • penetration length, coherence length, surface energy • Microscopic (BCS) theory • Cooper pair, BCS ground state • Flux quantization • Quantum tunneling • single particle tunneling, DC/AC Josephson effect • SQUID

Thermal properties of SC: specific heat Critical temperature The exponential dependence with T is called “activation” behavior and implies the existence of an energy gap above Fermi surface ~ 0.1-1 meV (10-4~-5 EF)

Temperature dependence of  (obtained from Tunneling) Connection between energy gap and Tc Universal behavior of (T) ‘s scale with different Tc’s 2(0) ~ 3.5 kBTc

Entropy and free energy of SC state Al Less entropy in SC state: more ordering Al FN-FS = Condensation energy  10-8 eV per electron! 2nd order phase transition

EM wave absorption 2 suggests excitations created in “e-h” pairs More evidences of energy gap • Electron tunneling (discussed later)

Magnetic property of the superconductor Superconductivity is destroyed by a strong magnetic field Hc for metal is of the order of 0.1 Tesla or less All curves can be collapsed onto a similar curve after re-scaling Temperature dependence of Hc(T) normal sc

Hi Radius, a Current i Magnetic field Critical currents (no applied field) Actual situation is more complicated! (London, 1937; see Tinkham, p.34) so The critical current density of a long thin wire is therefore (thinner wire has larger Jc) jc~108A/cm2 for Hc=500 Oe, a=500 A Jc has a similar temperature dependence as Hc, and Tc is similarly lowered as J increases (R = a at j = jc) From Cywinski’s lecture note

Meissner effect(Meissner and Ochsenfeld, 1933) normal sc Lenz law not only dB/dt=0 but also B=0! Active exclusion that violates Faraday’s law! Perfect diamagnetism. different same

Meissner effect for a hollow cylinder Apply a field, then lower below Tc: There are surface currents on both inside and outside. no field inside the ring. Remove the field: surface current on the outside disappears; surface current on the inside persists Magnetic flux is trapped! Q: what if we reduce T first, apply a field, then remove the field? (Alan Portis, Sec 8.7)

2003 Superconducting alloy: partial exclusion and remains superconducting at high B (1935) (intermediate/mixed/vortex/Shubnikov state) pure In HC2 is of the order of 10~100 Tesla (called hard, or type II, superconductor)

B=H+4M Hc2 Comparison between type I and type II superconductors Lead + (A) 0%, (B) 2.08%, (C) 8.23%, (D) 20.4% Indium Areas below the curves (=condensation energy) remain the same! Condensation energy (for type I)

nn Carrier density Tc T London theory of the Meissner effect(Two-fluid model) (Fritz London and Heinz London, 1934) Superfluid density ns =  Normal fluid density nn ns like free charges They assumed where London proposed It can be shown that =0 for simply connected sample (See Schrieffer)

Penetration length L Outside the SC, B=B(x) z (expulsion of magnetic field) Temperature dependence of L tin also decays Predicted L(0)=340 A, measured 510 A Higher T, smaller nS

ns surface superconductor x  Coherence length 0(Pippard, 1939) In fact, ns cannot be uniform near a surface. The length it takes for ns to drop from full value to 0 is called 0 0 ~ 1 m >>  for type I SC Microscopically it’s related to the range of the Cooper pair The pair wave function (with range 0) is a superposition of one-electron states with energies within of EF(A+M, p.742) Therefore Therefore, the spatial range of the variation of nS

Penetration depth, correlation length, and surface energy: Surface energy is positive: Type I superconductivity Surface energy is negative: Type II superconductivity For 0 >  For 0 <  From Cywinski’s lecture note • smaller , cost more energy to expel the magnetic field • smaller 0, get more “negative” condensation energy • When 0 >> (type I), there is a net positive surface energy. (difficult to create an interface) • When 0 << (type II), the surface energy is negative. Interface may spontaneously appear.

H 0 Hc1 Hc2 -M Vortex state of type II superconductor (Abrikosov, 1957) 2003 • the magnetic flux  in a vortex is always quantized (discussed later) • the vortices repel each other slightly • the vortices prefer to form a triangular lattice (Abrikosov lattice) • the vortices can move and dissipate energy (unless pinned by impurity) From Cywinski’s lecture note

Estimation of Hc1 and Hc2 Near Hc1, there begins with a single vortex with flux quantum 0, therefore Near Hc2, vortex are as closely packed as the coherence length allows, therefore Typical values, for Nb3Sn, 0  34 A, L  1600 A

mercury Microscopic picture of the SC state? • Metal X can (cannot) superconduct because its atoms can (cannot) superconduct? Neither Au nor Bi is superconductor, but alloy Au2Bi is! White tin can, grey tin cannot! (the only difference is lattice structure) • good normal conductors (Cu, Ag, Au) are bad superconductor, • bad normal conductors are good superconductors, why? • Why is the superconducting gap so small? • Failed attempts: polaron, CDW... • Isotope effect (1950): It is found that Tc =const  M   1/2 for different materials

2003 • 1956 Cooper: attractive interaction between electrons (with the help of crystal vibrations) near the FS forms a bound state • 1957 Bardeen, Cooper, Schrieffer: BCS theory 1972 • Microscopic wave function for the condensation of Cooper pairs Ref: 1972 Nobel lectures by Bardeen, Cooper, and Schrieffer Brief history of the theories of superconductors • 1935 London: superconductivity is a quantum phenomenon on a macroscopic scale. There is a “rigid” (due to the energy gap) superconducting wave function  • 1950 • Frohlich: electron-phonon interaction maybe crucial • Reynolds et al, Maxwell: isotope effect • Ginzburg-Landau theory: Scan be varied in space. Suggested the connection and wrote down the eq. for (r) (App. I) Difficulty: the condensation energy is 108 eV per electron!

+++ p e p1 -p2 P D (a) p1 P=0 p1 p2 p2 P (c) (b) Dynamic electron-lattice interaction Effective attractive interaction between 2 electrons  0.1 m Phase space argument (more phase space available, stronger interaction): Momentum is conserved during phonon exchange p1+p2=p’1+p’2=P The # of energy-reducing phonon exchange processes is max for P=0p1 = -p2

Cooper pair (Cooper, 1956) 2 electrons with opposite momenta (p,-p) can form a bound state with binding energy (the spin is opposite by Pauli principle) • Fraction of electrons involved  kTc/EF  10-4 • Average spacing between condensate electrons  10 nm • Therefore, within the volume occupied by the Cooper pair, there are approximately (0.1 m/10 nm)3  103 other pairs. • These pairs (similar to bosons) are highly correlated and form a macroscopic condensate state BCS with (non-perturbative result!) 2(0) ~ 3.5 kBTc Schafroth (1951): Meisner effect cannot be obtained in any finite order of perturbation. Migdal (1958): no energy gap from the perturbation theory. • Electrons within kTC of the FS have their energy lowered by the order of kTC in the condensation (therefore 10-8 eV per electron).

Distribution function and DOS below Tc BCS ground state (Schrieffer, 1957. Ref: 李正中固體理論) (phase coherent state) Density of states of quasi-particles D(E) ~ O(1) meV

Flux quantization in a superconducting ring (F. London 1948 with a factor of 2 error, Byers and Yang, also Brenig, 1961) Current density operator In the presence of B field Not simply connected London eq. with Inside a ring 0 the flux of the Earth's magnetic field through a human red blood cell (~ 7 microns)

Single particle tunneling (Giaever, 1960) SIN dI/dV Phonon structure 20-30 A thick SIS Ref: Giaever’s 1973 Nobel prize lecture For T>0 (Tinkham, p.77) 1973

Josephson effect(predicted by Josephson, 1962.For related debate, see “The true genius”, by L. Hoddeson) 1) DC effect:there is a DC current through SIS in the absence of voltage 1973

2) AC Josephson effect apply a DC voltage, then there is a rf current oscillation • An AC supercurrent of Cooper pairs with freq. =2eV/h, a weak microwave is generated. •  can be measuredvery accurately, so tiny V as small as 10-15 V can be detected. • Also, since V can be measured with accuracy about 1 part in 1010, so 2e/h can be measured accurately. • JJ-based voltage standard (1990): 1 V  the voltage that produces a frequency of 483,597.9 GHz (exact). • advantage: independent of material, lab, time (similar to the quantum Hall standard)

3) DC+AC: apply a DC+rf voltage, there is a DC current Shapiro steps (1963) given I, measure V Another way of providing a voltage standard NIST 1 Volt standard using 3020 JJs connected in series Microwave in

For junction with finite thickness SQUID (Superconducting QUantum Interference Device) The current of a SQUID with area 1 cm2 could change from max to min by a tiny H=10-7 gauss!

SuperConducting Magnet Non-destructive testing MCG, magnetocardiography MEG, magnetoencephlography

Super-sentitive photon detector Transition edge sensor semiconductor detector superconductor detector 科學人,2006年12月

Superconductivity