Lecture 3. Phenomenological theory of the EM properties of superconductors *

Transcription

1 Phys. 598SC Fa 2011 Prof. A. J. Leggett Lecture 3. Phenomenoogica theory of the EM properties of superconductors * 1. London theory (F. and H. London, 1935) [Recap on significance of Meissner effect] Consider first T=0, assume a eectrons behave in superconducting way. Eqn. of motion in norma meta woud be = = Experimentay σ, so 1/τ 0. Thus eqn. of motion of eectrons in superconductor is = Maxwe = = = i.e. + =0 + =. (in time) So far, nothing new above simpy a consequence of infinite conductivity. [in particuar, Φ+(m/ne 2 ) J d = const. (Lippmann s rue)] But: Meissner shows B=0 in interior of superconductor. So, Londons postuate that the const.=0, i.e. + = - London eqn. (*) Combine with Maxwe eqn. H = J + D/ t zero if t independent situation = * Refs: F. London, Superfuids, Tinkham ch. 1, Rickayzen. Note, historicay this materia is a pre-bcs.

2 PHYS 598 SC Lecture 3: Phenomenoogica theory of the EM properties of superconductors 2 or since B 0, (and J 0 in time-independent situation) = = with = ~10 London penetration depth (de Haas-Lorentz) Note λ L is just HF skin depth in N phase, but now has quite different significance: e.g. infinite fat-pate geometry: V S = = = into page B Screening currents on surface, B screened out in o(λ L ). Note: at surface of a superconductor occupying an infinite haf-space, =0, i.e. magnetic fied is parae to surface. Proof by reduction ad absurdum: if, 0 i.e. B z 0 just inside superconductor, then from div B = 0 and transation invariance in parae direction, B z 0 infinitey far into superconductor. But then by London eqn. J x / y and/or J y / x 0, vioating condition of TI to surface. For a finite geometry, this argument suggests that B is approximatey parae to surface provided a dimensions are à λ L (e.g. macroscopic sphere). For sampes with one or more dimensions d λ L, situation more compicated: e.g. for infinite thin pate, dd λ L, effective no. of eectrons (n) reduced by factor ~ d/λ where λ is effective 2D penetration depth. Thus, λ ~. Note that for a 2D sab the current does not fow principay around boundaries but through buk! Finite T: n s (T) of e - s superconducting, n n (T) n-n s (T) norma. At dc, norma e - s don t contribute formua same except λ = 0 Assume n s n at T=0, and 0 at T T c, then λ L (T) as T T c. If we make defaut assumption n s (T)~T c - T for T T c, then λ L (T)~ (T c -T) -1/2. (Approximate empirica reation: ~ 01 / / ) Experimenta measurement of λ: inductance of cavity, cooid suspensions Josephson effect Note that generay it is easier to measure changes in λ with some parameter (e.g. T) than absoute vaue.

3 PHYS 598 SC Lecture 3: Phenomenoogica theory of the EM properties of superconductors 3 2 Impications of London eqn. Since B A, the London equation (*) can be rewritten + = 0 i.e. + = ψ In simpy-connected sampe, in region beyond λl where J = 0, A = ψ A (= B) = 0 A = ongitudina & can be gauged away. Hence in any simpy-connected arge sampe, can write for a r, = But in ongitudina case we know A can induce no J system knows difference between L and T forms of A even in imit q 0. Perturbation theory: in presence of A. p i p i -ea i (r) H = + 2 But current = 0 = 0 take F.T.: = 0 For L case, f-sum rue ensures δjk/δak = 0 as above (no response to purey ongitudina static vector potentia). For T case, get London resut if we assume that for some reason matrix eements of J k 0 with k but (reevant) energy eves stray finite ( rigidity, gap). Cf. atomic diamagnetism (Bohr-van Leeuwen theorem)

4 PHYS 598 SC Lecture 3: Phenomenoogica theory of the EM properties of superconductors 4 In mutipy connected case (e.g. ring) cannot necessariy infer A = 0 in midde of ring possibiity of trapped fux. (but no statement about what vaues possibe, for now) Anaogy between Meissner diamagnetism and HF effect in superfuid 4 He: If we pace a norma iquid (incuding 4 He above T λ is an annuar container and rotate the container sowy < ħ/, the iquid rotates with the container. If now in the case of 4 He we coo through T λ and on down towards T = 0, the iquid comes out of equiibrium with the container and as T 0, is (approximatey) at rest in the ab frame. This is the Hess-Fairbank (HF) effect (or noncassica rotationa inertia, NCRI). To see the correspondence with Meissner diamagnetism, consider the Hamitonian formuation of the probem in the rotating frame. (indicate variabes in this frame by primes). For a singe partice the canonica momentum is + =, and the (canonica) Hamitonian is, = + 2 = centrifuga term and = / Compare case of eectricay charged system, viewed from ab. frame but in presence of EM vector potentia A(r):,= +V 2 j = (p-e,a(r))/m except for centrifuga term, exact correspondence between EM system viewed from ab. frame & neutra system viewed from rotating frame, with ea(r) m(ω r), or for constant fied B such that =, ω eb/2m. In particuar, nonzero EM current in ab. frame nonzero neutra-atom current in rotating frame. [Can generaize straightforwardy to many-body case provided V(r' i r' j ) = V(r i r j )] 3 Pippard modification. There are two obvious probems with the London theory: (1) it does not expain the possibiity or nature of type-ii superconductivity. (2) The actua vaue of the experimenta penetration depth, as measured e.g. from inductance experiments, is often consideraby greater than the London vaue

5 PHYS 598 SC Lecture 3: Phenomenoogica theory of the EM properties of superconductors 5 / and moreover is very sensitive to aoying, even though thermodynamic properties itte affected. Pippard hypothesis (discussed in much more detai ater, in context of BCS theory): J(r) is a nonoca function of A(r), i.e. * [pure materia for now] ~, (t) where range of K is of order some ength ξo Pippard coherence ength. If, then essentiay reduces to London theory provided = /. What if t? Suppose actua penetration depth is ~λ. Then the contribution to the RHS of (t) is ~ Ar n s (T)e 2 /m (λ/ξo = Ar λ/ξo. Thus, ~. /, which can be àλl. In a dirty materia (mfp á λpure) then Pippard supposed reduction woud be by a factor /ξo rather than (λ/ξo). Thus, (/ ) ~ ( /) / So in Pippard approach, ξo is essentiay the range of nonocaity (in a pure meta) of eectromagnetic effects. It turns out (from the experiments on λ(t)) that ξo is ony weaky sensitive to temp, and in particuar does not diverge for T Tc.: in hindsight, wi interpret ξo as essentiay radius of Cooper pairs. Definition of London and Pippard imits: note aways in London imit for (a) sufficient dirt, and (b) for T T c (λl, ξo ~ finite) (crucia for vaidity of GL approach). [Sti no expanation of type-ii ] 4 GL theory: type-ii superconductivity Suppose we appy an externa fied of the order of the thermodynamic critica fied H c to the sampe. Let s consider the possibiity that it punches hoes (vortices) through, with a norma core (since H = 0 in buk S) and circuating currents around the core. Is this energeticay advantageous? [for a more quantitative cacuation, see Tinkham section 4.3., which foows the historica arguments more cosey]. Consider first for definiteness T=0. * Specific choice: ( )~ exp +, (Chambers)

6 PHYS 598 SC Lecture 3: Phenomenoogica theory of the EM properties of superconductors 6 First, what is the gain in energy? Essentiay, we expect that the fied punches through over a region of dimension ~λ, so the gain per unit ength of vortex ine is ~. On the other hand we need to form a norma core. Let s assume that the bending energy to go from S to N over a distance L is K/L 2 per unit vo., and define a ength ξ so that K = Econd ξ 2, where Econd is the combination energy. Then the tota bending energy per unit area is independent of L and of order Econdξ 2 ~ (df of Hc!), whie the oss of buk condensation energy is ~ Econd L 2 : thus, for L dξ this term is of the same order as the bending energy, and the tota energy/unit ength of the vortex ine is given by ~ ( ) a,b~1. Thus, for ξ à λ the energy is positive and it is not advantageous to introduce vortex ines, but for λ à ξ it becomes advantageous. [At finite T, E G (T) but argument otherwise the same]. Note, so far, no specification of strength of vortex. These considerations made more quantitative by phenomenoogica theory of GL (1950). Introduce compex order parameter (wave function) ψ(r) and postuate foowingexpression for free energy density (after Landau & Lifshitz): () = Ñ () In this expression m* and e* are at present stage unknown, though it seems reasonabe to guess they are ~ eectron mass and charge. The coefficients α, β are given by α(t) = α o (T T c ) β (T) = β o (~ ind. of T) Thus for a uniform state, potentia ooks ike: The eectric current is defined as F/ A(r) and hence () = 2 () +.. just as for a singe partice described by a Schrödinger wave function ψ. In the case where ψ is constant in space, () = : thus, we can tentativey write =.

7 PHYS 598 SC Lecture 3: Phenomenoogica theory of the EM properties of superconductors 7 Note GL impicity assume a oca response. (vaid for T T c, at east) If ψ is taken to be the equiibrium OP, it is given for T < T c by ψ 2 = α /β and thus T c T; thus λ L (T c T) -1/2 as observed. The GL free energy defines another characteristic ength which is independent of e *, namey () = ( /2. Since ~, aso ~. () is GL coherence (correation) ength: do not confuse with! The ratio of to is independent of T for T T c and is usuay denoted κ: from the above = 2 1 where β can be derived from the experimenta vaues of H c (T) and λ(t) (see Tinkham 4.1). Actuay in BCS theory we have in the cean imit ~ 01 / ~ 1 / so κ is actuay ~ 0/ (0.96 times this, in cean imit). It foows from a detaied anaysis (cf.. 10) that the formation of vortices is favorabe when > 1 2. thus this is the discriminant between type-i and type-ii superconductivity. For a cean superconductor, the type-i type-ii distinction is essentiay the same as Pippard-London. For a dirty superconductor, ~ (0)/. 5. The reevance of Bose condensation Consider simpe neutra system of noninteracting partices (statistics so for unspecified) in narrow annuar geometry, radius R. Container rotates at anguar veocity ω. = = + transverse drops out Expectation vaue of anguar momentum: = ħ.

8 PHYS 598 SC Lecture 3: Phenomenoogica theory of the EM properties of superconductors 8 norma state: cassica, ~ exp /.. (Fermi, Bose..) in a cases smoothy varying function: consider cassica case (or F, B in nondegenerate regime) for simpicity. = / ħ ~ exp ħ ħ / 2 exponentia function smooth for / ; since smooth, exp / exp Introduce new variabes, /ħ so as to compete square, then = ħ + ħ.. 2 / ħ /2 = ħ ie iquid rotates exacty with cyinder (to ħ / 1. Now consider Bose system beow T c : norma component described by = (exp 1, but, so define condensate no. (~N). These must a pie into the owest singe-partice state, i.e. one with minimum vaue of. Thus, = + ħ = nearest integer to / ħ/. 6a A possibe definition of the order parameter for a BEC system: Φ No = ft in genera case Definition of characteristic veocity: in Schrödinger singe-partice case ρrt = ψrt 2

9 PHYS 598 SC Lecture 3: Phenomenoogica theory of the EM properties of superconductors 9 if introduce ψrt Λ(rt) exp i ϕ(rt), then One can define veocity by = ħ 2 =Λ, = ħ Λ / = ħ quantum object, but not terriby usefu physicay, because subject to arge fuctuations. In BEC case, try defining. Ψ(rt) = Λ(rt)exp i ϕ(rt) () ħ () (or this, doesn t matter) superfuid veocity satisfies: (a) cur vs = 0 (b) " vs d = nh/m (Onsager-Feynman) Note these conditions are not satisfied by hydrodynamic veocity of norma fuid () () ()/ () Thus, vs is quantum object, but not subject to onger fuctuations because made up of contributions of No~N partices. Charged system: p p-ea) so = ħ ( /ħ) " = ħ ( Φ/(h/)) and in particuar if vs = 0 (e.g. in interior of thick ring) then Φ = nh/e (London, with e = actua eectron charge)