Phys 446 Solid State Physics Superconductivity is a phenomenon occurring in certain materials at Lecture 11 Nov 25 low temppyyeratures, characterized by exactly zero electrical resistance (Ch. 10) Discovered in 1911 by Heike Kamerlingh Onnes - the resistance of solid mercury abruptly disappeared at the temperature of 4.2 K normal metal superconductor SUPERCONDUCTIVITY m ρ = − zero ρ means there are some electrons with infinite τ – 2 ne τ no scattering Superconducting elements Persistent current in a superconductor • More than 20 metallic elements are superconductors • Cu, Au, Ag, Na, K and magnetically ordered metals (Fe, Ni, Co) are not This gives an upper value on the value of R: superconductors • Certain elements are superconducting at high pressures or as thin films • Highest Tc of an element is 9.3 K for Nb • There are thousands of alloys and compounds that exhibit superconductivity • The highest Tc superconductors tend to be poor conductors in the normal state • Record Tc is currently ~ 138 K (a ceramic consisting of Tl, Hg, Cu, Ba, Ca, Sr, and O) Limitations of persistent current flow Perfect diamagnetism Persistent currents will flow in a superconductor unless: Second basic property of superconductors – they expel magnetic field compltlletely w hen in supercon duc ting p hase (T<T ). 1) A sufficiently large magnetic field is applied c This phenomenon is called the Meissner effect. 2) The current exceeds a certain critical current, Ic (the Silsbee effect) 3) An AC electric field above a certain frequency is applied The transition from dissipationless to normal response occurs at ω ~ ∆/ħ, where ∆ is the energy gap normal Superconductors are poor Distinguishes the thermal conductors superconductor from an ideal but normal conductor, supercon- for which dB/dt = 0 dtiducting Thermal conductivity of lead From J.H.P. Watson and G. M Graham, Could explain the expulsion of the flux if a S/C is moved into a magnetic field → Can. J. Phys. 41, 1738 (()1963) motion of metal pp,roduces currents, which do not decay because R = 0 However, R = 0 does not explain the flux expulsion when a S/C is already in a magnetic field and is cooled from its normal to S/C state (through Tc) Magnetic properties: type I superconductors Superconductors are divided into 2 types, depending on their behavior Rough correlation between critical field and TC in a magnetic field. All superconducting elements are type I, except Nb Supercon duc tiv ity is des troye d by the presence o f some magne tic field, the critical field, Bc. Typically Bc ~ tens of mT for type I B-T phase diagram Field Bc does not need to be external: Critical current Ic causes a magnetic field Bc,,y which destroys superconductivity Penetration depth Type II superconductors Surface currents expel magnetic flux from type I superconductors • Alloys (and Nb and V) Currents actually penetrate the sample slightly (~ 100 nm) • Higher TC, BC, IC than type I Magnetic field decreases exponentially inside sample: • More applications e. g. Nb3Sn magnets • 2 critical fields: B < BC1 - just like a type I B > BC2 - normal Intermediate fields – sample has both superconducting and normal regions As T→Tc, λ increases and B penetrates deeper into the sample. At Tc, λ →∞, and the sample goes normal Isotope effect Experimental evidence for an energy gap – heat capacity • TC depends on atomic mass: Discontinuity at TC a • More general: TcM = constant • Suggests superconductivity Hg - Maxwell, Reynolds et al . (1950) is not just an electronic effect Tc Isotope effect: 1st direct evidence ofif in terac tion o f e lec trons and the lattice Exponential T-dependence Suggests that lattice vibrations play a part in the superconducting process Characteristic of thermal behavior ofthf a system whose exc itdllited levels 1 are separated from the ground M state by an energy 2∆ From Kittel-Phillips Experimental evidence for an energy gap – Theory tunneling Phenomenological: Thin insulator → • F & H. London (1935) • Ginzburg& Landau (1950) electrons tunnel Quantum: • Fröhlich (1950) • Bardeen, Cooper & Schrieffer, BCS (1957) London model • Using two fluid model of Gorter and Casimir: As T increases Assume only a fraction of electrons ns(T)/n participate in supercurrent towards Tc, threshold voltage decreases • ns(T) is the ‘density of superconducting electrons: n ~n~ n at T<<Tc, n → 0at0 at T → T ⇒ energy gap s s c decreases with • n - ns electrons exhibit normal dissipation increasing T • Current and supercurrent flow in parallel ⇒ superconducting electrons carry all current, normal current is inert and can be ignored London equations m Combining equationΒ = − ∇× j and In an electric field E, S/C electrons will accelerate without dissipation, 2 ∇ × B = µ0J nse so we can reltlate the mean ve loc ity vs tthto the curren tdt dens ity j: and using ∇×∇× B = ∇(∇ ⋅B) − (∇ ⋅∇)B = −∇2B using get 2 2 2 µ n e µ n e get ∇ B = 0 s B and ∇2 j = 0 s j 1st London equation m m In a steady state, j = const ⇒ E = 0 Electric field inside a S/C vanishes Solution (one-dimensional case): ∂Β ∂Β Maxwell’s equation: ∇× E = − ⇒ = 0 ⇒ B = const 1 2 ∂t ∂t ⎛ m ⎞ λ = ⎜ ⎟ These equations describe the magnetic fields and current densities within a ⎜ 2 ⎟ ⎝ µ0nse ⎠ perfect conductor, but they are incompatible with the Meissner effect. - the London penetration depth From the above, have London assumed that m Β = − ∇× j ∂Β m ∂j 2 = − ∇× nd n e i.e. the Meissner effect is predicted 2 (2( London eqq)uation) s ∂t nse ∂t i.e. to successfully predict the Meissner effect the constant of integration must be chosen to be zero Solution for j gives a surface current – exponentially decaying into a S/C BCS theory • Cooper calculation: solve Schrödinger eq.for 2 interacting electrons • Fröhlich (1950): e-e attraction via phonons (…. Isotope effect) in the pppresence of a Fermi sphere of non-interactinggy electrons. Only • Cooper (1956): e lec trons jus t a bove the Ferm i sur face form boun d pa irs effect of N-2 electrons - restrict k values of e-e pair to be > kF, i.e. • Most stable when center of mass is at rest and total spin = 0, outside the Fermi sphere So, +k↑ and –k↓ CiCooper pair – boson. • Attractive interaction is provided by lattice vibrations – phonons • • First electron deforms the lattice and second electron is then attracted by • A single, coherent wave function extending over entire system. the deformation (i.e. the changed positive charge distribution) Can’ t change momentum of a pair without changing all pairs • Bardeen, Cooper and Schrieffer (BCS) → extend Cooper’s theory, construct a ground state where all electrons form bound pairs • Each electron now has 2 roles: - Provide restriction on allowed wavevectors via Pauli principle - Participate in bound pair (called a Cooper pair) • Time-scales: electron motion ~ 10-16 s; lattice deformed for ~10-13 s • Electron-phonon interactions: responsible for resistance of metals In this time, first electron has traveled and superconductivity Lattice deformation attracts 2nd electron without it feeling the Coulomb • Superconductors are generally poor conductors in normal state repulsion of the 1st Summary Energy gap: a Cooper pair has a lower energy than 2 individual When a superconductor is cooled below the critical temperature (TC), electrons. BCS gives it enters a new state , in which its resistance vanishes. Superconductors expel magnetic field completely when in superconducting phase – the Meissner effect When a magnetic field higher than a certain value called the critical field (Bc) is applied to a superconductor, it reverts to a normal state BCS calculated gap T-dependence of energy gap Type I and type II superconductors are distinguished by their 1.0 behavior in a magnetic field. In a type II S/C there are 2 critical fields. ) At intermediate fields, the material has both superconducting and 00 ( 0.8 ∆ normal regions (T)/ ∆ 0.6 Electrodynamics of superconductors is described by Gap 0.4 phenomenological London equations BCS theory MgB , sample 1 0.2 2 BCS theory – microscopic mechanism for superconductivity through Reduced MgB , sample 2 2 the formation of e-e Cooper pairs via electron-phonon interaction. 0.0 0.0 0.2 0.4 0.6 0.8 1.0 Reduced Temperature, T/T A Cooper pair has a lower energy than 2 individual electrons. The c energy difference is 2∆ - energy gap..