Superconductivity M.C. Chang Dept of Phys 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

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!  T Au 1913 Hg Discovery of superconductivity RR RR

Liquid He Switch 2 SC coil 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). compass Is the resistivity very small or really zero? : Persistent current (Onnes) Switch 1

1.14K 3.72K 7.19K 3.40K 1.09K 2.39K 4.15K 0.39K5.38K 0.55K 0.12K 9.50K 4.48K 0.92K 7.77K 0.01K1.4K 0.66K 0.14K 0.88K 0.56K0.51K 0.03K 1.37K1.4K 4.88K 0.20K0.60K K 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 In the form of nanostructure (type II)

Superconductivity in alloys and oxides From Cywinski’s lecture note Superconducting transition temperature (K) Hg Pb Nb NbC NbN V 3 Si Nb 3 Sn Nb 3 Ge (LaBa)CuO YBa 2 Cu 3 O 7 BiCaSrCuO TlBaCaCuO HgBa 2 Ca 2 Cu 3 O 9 (under pressure) HgBa 2 Ca 2 Cu 3 O 9 (under pressure) Liquid Nitrogen temperature (77K) Bednorz Muller 1987 Applications of superconductor 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)

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  ~ meV (10 -4~-5 E F )

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

Entropy and free energy of SC state Less entropy in SC state: more ordering 2nd order phase transition F N -F S = Condensation energy  eV per electron! Al

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

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

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

Critical currents (no applied field) Current i Radius, a Magnetic field HiHi The critical current density of a long thin wire is therefore j c ~10 8 A/cm 2 for H c =500 Oe, a=500 A so J c has a similar temperature dependence as H c, and T c is similarly lowered as J increases From Cywinski’s lecture note Actual situation is more complicated! (London, 1937; see Tinkham, p.34) (R = a at j = j c ) (thinner wire has larger Jc)

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

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)

The Hirsch paradox (cond-mat/ ) Induction Expulsion e lattice e lattice? Proposed experiment ?

pure In Superconducting alloy: partial exclusion and remains superconducting at high B (1935) (intermediate/mixed/vortex/Shubnikov state) H C2 is of the order of 10~100 Tesla (called hard, or type II, superconductor) 2003

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! H c2 B=H+4  M Condensation energy (for type I)

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

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

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

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

But in fact, it is nonzero only within. So If A y (x) is a constant within a thickness  0 from the surface, then it reduces to the “local” form For type-I SC, we need to use (see Tinkham, App.3 for details) local in q-space

Penetration depth, correlation length, and surface 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. smaller, cost more energy to expel the magnetic field smaller  0, get more “negative” condensation energy For  0 > Surface energy is positive: Type I superconductivity For  0 < Surface energy is negative: Type II superconductivity From Cywinski’s lecture note

Vortex state of type II superconductor (Abrikosov, 1957) 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) H c2 H c1 H 0 -M-M From Cywinski’s lecture note 2003

Estimation of Hc 1 and Hc 2 Near H c1, there begins with a single vortex with flux quantum  0, therefore Near H c2, vortex are as closely packed as the coherence length allows, therefore Typical values, for Nb 3 Sn,  0  34 A, L  1600 A

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

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 Au 2 Bi 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... It is found that T c =const  M     1/2 for different materials Isotope effect (1950):

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

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 p 1 +p 2 =p’ 1 +p’ 2 =P The # of energy-reducing phonon exchange processes is max for P=0  p 1 = -p e P p1p1 p2p2 (c) p2p2 p1p1 P=0 (b) (a) P p1p1 -p 2 pp DD

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  kT c /E F  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  10 3 other pairs. These pairs (similar to bosons) are highly correlated and form a macroscopic condensate state  BCS with (non-perturbative result!) Electrons within kT C of the FS have their energy lowered by the order of kT C in the condensation (therefore eV per electron). Schafroth (1951): Meisner effect cannot be obtained in any finite order of perturbation. Migdal (1958): no energy gap from the perturbation theory. 2  (0) ~ 3.5 k B T c

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

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

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 London eq. with Inside a ring  0  the flux of the Earth's magnetic field through a human red blood cell (~ 7 microns) Not simply connected

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

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

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 measured very accurately, so tiny  V as small as V can be detected. Also, since V can be measured with accuracy about 1 part in 10 10, 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)

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

SQUID (Superconducting QUantum Interference Device) The current of a SQUID with area 1 cm 2 could change from max to min by a tiny  H=10 -7 gauss! For junction with finite thickness

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

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

Sci Am. Aug, 1994 磁力梯度儀

rf SQUID RSJ (resistively shunted junction) model A realistic JJ = an ideal JJ  potential + a resistive shunt  damping + capacitative shunt  mass