Superconductivity Introduction Thermal properties Magnetic properties London theory of the Meissner effect Microscopic (BCS) theory Flux quantization Quantum tunneling Dept of Phys M.C. Chang
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 -73 C (= 0 K) G. Amontons 1700 1877 Cailletet and Pictet liquified Oxygen (-183 C or 90 K) soon after, Nitrogen (77 K) is liquified 1898 Dewar liquified Hydrogen (0 K) 1908 Onnes liquified Helium (4. 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. ρ Au Hg ρ R ρ R 1913 T
0.03K 1.14K 0.39K 5.38K 0.88K 1.09K 0.55K 9.50K 0.9K 7.77K 0.51K 0.0003K 0.56K 3.40K 3.7K 4.88K 0.1K 4.48K 0.01K 1.4K 0.66K 0.14K 4.15K.39K 7.19K 1.37K 1.4K 0.0K 0.60K 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
Superconductivity in alloys and oxides Applications of superconductor Superconducting transition temperature (K) 160 140 10 100 80 60 40 0 Hg Pb Liquid Nitrogen temperature (77K) NbC Nb NbN Nb 3 Sn V 3 Si HgBa Ca Cu 3 O 9 (under pressure) HgBa Ca Cu 3 O 9 TlBaCaCuO BiCaSrCuO YBa Cu 3 O 7 (LaBa)CuO Nb 3 Ge powerful magnet MRI, LHC... magnetic levitation SQUID ( 超導量子干涉儀 ) detect tiny magnetic field quantum bits lossless powerline 1910 1930 1950 1970 1990 From Cywinski s lecture note Bednorz Muller 1987
Introduction Thermal properties Magnetic properties London theory of the Meissner effect Microscopic (BCS) theory Flux quantization Quantum tunneling
Thermal properties of SC: specific heat For different superconductors, C S C C N N ~ 143. at T C C el S e Δ(0) The exponential dependence with T is called activation behavior and implies the existence of an energy gap above Fermi surface. k B T Δ ~ 0.1-1 mev (10-4~-5 E F )
Connection between energy gap and T c Temperature dependence of Δ (obtained from Tunneling) Universal behavior of Δ(T) Δ s scale with different T c s Δ(0) ~ 3.5 k B T c 1/ Δ( T) T = 1.74 1 for T T Δ(0) TC C
Entropy Al S C = T T H Less entropy in SC state: more ordering free energy F N -F S = Condensation energy ~10-8 ev per electron! Al F S = T H C S F = = T T H T nd order phase transition H
More evidences of energy gap Electron tunneling EM wave absorption Δ suggests excitations created in e-h pairs Δ ν = = h 480 GHz (microwave)
Magnetic property of the superconductor 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) All curves can be collapsed onto a similar curve after re-scaling. { } = { ( ) } H ( T ) H 1 T T c o c normal sc
Critical currents (no applied field) Radius, a Current H i Magnetic field From Cywinski s lecture note H 4π d = i c so ca ic = H c The critical current density of a long thin wire is therefore chc jc = πa (thinner wire has larger J c ) j c ~10 8 A/cm for H c =500 Oe, a=500 A J c has a similar temperature dependence as H c, and T c is similarly lowered as J increases. Cross-section through a niobium tin cable Phys World, Apr 011
Meissner effect (Meissner and Ochsenfeld, 1933) A SC is more than a perfect conductor Lenz law not only db/dt=0 but also B=0! Perfect diamagnetism different same
Superconducting alloy: type II SC partial exclusion and remains superconducting at high B (1935) (also called intermediate/mixed/vortex/shubnikov state) STM image NbSe, 1T, 1.8K pure In H C is of the order of 10~100 Tesla (called hard, or type II, superconductor)
Comparison between type I and type II superconductors B=H+4πM H c Lead + (A) 0%, (B).08%, (C) 8.3%, (D) 0.4% Indium Areas below the curves (=condensation energy) remain the same! Condensation energy (for type I) ( H is B in Kittel) a df = M dh 1 For a SC, dfs = HdH 4π H FS( H) FS(0) = 8π F ( H ) = F ( H ) N c S c F ( H ) = F (0) for nonmagnetic material N c N Hc Δ F = FN(0) FS(0) = 8π (Magnetic energy density)
Introduction Thermal properties Magnetic properties London theory of the Meissner effect Microscopic (BCS) theory Flux quantization Quantum tunneling
London theory of the Meissner effect (Fritz London and Heinz London, 1934) Carrier density n s n n T c T Two-fluid model: n + n = n= constant s n Superfluid density n s σ = ne n τ Normal fluid density n n σ n = m Assume dj s (1) = dt m () J = σ E where J J n = en v = en v s s s n n n nse E n like free charges London proposed 4π use B = J s and c v = v v ( ) ( ) 1 B Eq.(1) + E = c t d ne s B ( J s ) = dt mc t ne s J s = B mc ne s J S = A + φ mc 4 π ne s B B= B mc λ L It can be shown that ψ=0 for simply connected sample (See Schrieffer)
Penetration length λ L Outside the SC, B=B(x) z db L λ dx = B Bx = Be x/ λl ( ) 0 (expulsion of magnetic field) λ L mc = 170 A if n = 10 /cm S 4π ne S 3 3 Temperature dependence of λ L tin 4π B= Js c c db cb0 Jsy = = 4π dx 4πλ L e x/ λ L also decays Predicted λ L (0)=340 A, measured 510 A Higher T, smaller n S ( ) λ(0) λ T = 4 1/ ( T T ) 1 / C
Coherence length ξ 0 (Pippard, 1939) In fact, n s cannot remain uniform near a surface. The length it takes for n s to drop from full value to 0 is called ξ 0 Microscopically it s related to the range of the Cooper pair. The pair wave function (with range ξ 0 ) is a superposition of one-electron states with energies within Δ of E F (A+M, p.74). n s surface ξ 0 superconductor x Energy uncertainty of a Cooper pair pδp m Δ Therefore, the spatial range of the variation of n S vf vf ξ0 = from BCS theory Δp Δ πδ ξ 0 ~ 1 μm >> λ for type I SC
Penetration depth, correlation length, and surface energy Type I superconductivity Type II superconductivity ξ 0 > λ, surface energy is positive ξ 0 < λ, surface energy is negative smaller λ, cost more energy to expel the magnetic field. When ξ 0 >> λ (type I), there is a net positive surface energy. Difficult to create an interface. smaller ξ 0, get more negative condensation energy. When ξ 0 << λ (type II), the surface energy is negative. Interface may spontaneously appear. From Cywinski s lecture note
Vortex state of type II superconductor (Abrikosov, 1957) i sc Normal core the magnetic flux φ in a vortex is always quantized (discussed later). the vortices repel each other slightly. the vortices prefer to form a triangular lattice (Abrikosov lattice). 003 the vortices can move and dissipate energy (unless pinned by impurity Flux pinning) 0 H c1 H c H -M From Cywinski s lecture note
Estimation of Hc 1 and Hc (type II) Near H c1, there begins with a single vortex with flux quantum φ 0, therefore πλ H φ φ H πλ 0 c1 0 c1 Near H c, vortex are as closely packed as the coherence length allows, therefore Nπξ H Nφ H φ 0 0 c 0 c πξ0 Therefore, H H c λ ξ c1 0 Typical values, for Nb 3 Sn, ξ 0 ~ 34 A, λ L ~ 1600 A
Origin of superconductivity? Metal X can (cannot) superconduct because its atoms can (cannot) superconduct? Neither Au nor Bi is superconductor, but alloy Au 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? What leads to the superconducting gap? Failed attempts: polaron, CDW... Isotope effect (1950): It is found that T c =const M -α α~ 1/ for different materials lattice vibration? mercury
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 Ψ. 1950 Frohlich: electron-phonon interaction maybe crucial. 003 Reynolds et al, Maxwell: isotope effect Ginzburg-Landau theory: ρ S can be varied in space. Suggested the connection ρ ( r S ) = ψ ( r ) and wrote down the eq. for order parameter Ψ(r) (App. I) 1956 Cooper pair: attractive interaction between electrons (with the help of crystal vibrations) near the FS forms a bound state. 1957 Bardeen, Cooper, Schrieffer: BCS theory 197 Microscopic wave function for the condensation of Cooper pairs. Ref: 197 Nobel lectures by Bardeen, Cooper, and Schrieffer
Dynamic electron-lattice interaction Cooper pair +++ e Effective attractive interaction between electrons (sometimes called overscreening) (range of a Cooper pair; coherence length) ~ 1 μm
Cooper pair, and BCS prediction electrons with opposite momenta (p,-p ) can form a bound state with binding energy (the spin is opposite by Pauli principle) Δ( 0) = 1 D( E ) V ω int D e F, see App. H Fraction of electrons involved ~ kt c /E F ~ 10-4 Δ(0) ~ 3.5 k B T c Average spacing between condensate electrons ~ 10 nm Therefore, within the volume occupied by the Cooper pair, there are approximately (1μm/10 nm) 3 ~ 10 6 other pairs. These pairs (similar to bosons) are highly correlated and form a macroscopic condensate state with (BCS result) kt ω = 113. ω B C D 500 K, D( E ) V 1/3 D 3 Tc 500e = e 1 D( EF ) V int F 5 K int (~upper limit of T c )
Energy gap and Density of states D(E) ~ O(1) mev Electrons within kt C of the FS have their energy lowered by the order of kt C during the condensation. On the average, energy difference (due to SC transition) per electron is T 1 0.1 10 C 8 kt B C mev ev 4 TF 10
Families of superconductors Cuperate (iron-based) T.C. Ozawa 008 Conventional BCS Heavy fermion F. Steglich 1978 wiki
Introduction Thermal properties Magnetic properties London theory of the Meissner effect Microscopic (BCS) theory Flux quantization Quantum tunneling (Josephson effect, SQUID)
Flux quantization in a superconducting ring (F. London 1948 with a factor of error, Byers and Yang, also Brenig, 1961) q Current density operator j = ψ * ψ ψ ψ *, q= e m i i SC, in the presence of B let ψ= ψ e iφ and assume ψ vary slowly with r * * * * q q q j = ψ * A ψ + ψ A ψ * * m i c i c q * m * = e = m e e then j = φ + A ψ m mc Inside a ring j d = 0 c c Ad = φ d = Δφ e e hc hc flux Φ= n = nφ 0, φ0 = e e φ 0 ~ the flux of the Earth's magnetic field through a human red blood cell (~ 7 microns) London eq. with n s = ψ 7 10 gauss-cm
Single particle tunneling (Giaever, 1960) SIN di/dv 0-30 A thick SIS For T>0 (Tinkham, p.77) Ref: Giaever s 1973 Nobel prize lecture
Josephson effect (Cooper pair tunneling) Josephson, 196 1) DC effect: There is a DC current through SIS in the absence of voltage. ψ ψ θ 1 = e i ψ = ψ e i 1 1 θ 1973 0 iθ1 K( x+ d/) iθ+ K( x d/) ( ) ψ = n / e + e S iens Kd i( θ1 θ) i( θ θ1) j = Ke ( e + e ) m = j0 sin δ0 Kd j en Ke / m, δ θ θ Δ /e S 0 1 Josephson tunneling Giaever tunneling
) AC Josephson effect Apply a DC voltage, then there is a rf current oscillation. ψ = 1 ˆ = N N 1 N ψ N e e i( E E ) t/ iμt/ θ () t = μt/ + θ ( i = 1,) i i i μ μ = ev 1 ev ev δ = t+ δ0 j = j0sin t+ δ0 (see Kittel, p.90 for an alternative derivation) An AC supercurrent of Cooper pairs with freq. ν=ev/h, a weak microwave is generated. ν can be measured very accurately, so tiny ΔV as small as 10-15 V can be detected. Also, since V can be measured with accuracy about 1 part in 10 10, so e/h can be measured accurately. JJ-based voltage standard (1990): 1 V the voltage that produces ν=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, then there is a DC current V = V0 + υ cosωt e υ j = j0sin V0t+ sinωt + δ0 ω n eυ ev0 = j0 ( 1) Jn sin t nωt + δ0 n ω ω there is DC current at V0 = n e Another way of providing a voltage standard Shapiro steps (1963) given I, measure V
SQUID (Superconducting QUantum Interference Device) j = j0sinδa + j0sinδb δa δb δa + δb = j0 cos sin c Similar to A d= d e θ e We now have Ad θb 1 θa 1 c = C1 e Ad = θa θb c C e φ δa δb = Ad π c = φ π φ jmax = j0 cos φ0 C 0 The current of a SQUID with area 1 cm could change from max to min by a tiny ΔH=10-7 gauss! For junction with finite thickness
SuperConducting Magnet Non-destructive testing MCG, magnetocardiography MEG, magnetoencephlography
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