SHANGHAI JIAO TONG UNIVERSITY LECTURE

Transcription

1 Lecture 4 SHANGHAI JIAO TONG UNIVERSITY LECTURE Anthony J. Leggett Department of Physics University of Illinois at Urbana-Champaign, USA and Director, Center for Complex Physics Shanghai Jiao Tong University

2 Ginzburg Landau (GL) theory of superconductivity SJTU 4.1 Formulated 1950 (pre BCS): even in 017 mostly adequate for engineering applications of superconductivity. (SC) historically, takes its origins in general Landau-Lifshitz theory (1936) of nd order phase transitions, hence quantitative validity confined to T close to T c. Here, I will start by arguing with hindsight for its qualitative validity at T = 0, and only later generalize to T 0 and in particular T T c. Recall: could explain 3 major characteristics of SC state (persistent currents, Meissner effect, vanishing Peltier coefficient) by scenario in which fermionic pairs form effective bosons, and these undergo BEC. Suppose that N fermions form N/ bosons, which are then condensed into the same particle state, Neglect for now relative wave function and df. COM wave function of condensed pairs φ r COM coordinate

3 SJTU 4. What is energy E o (at T = 0) of pairs as function(al) of φ (r)? At first sight, for a single pair, E o φ r = E b න φ r dr + ħ m න binding energy of pair ie ħ A(r) φ r electrons involved! so, convenient to define order parameter (OP) Ψ r N φ r α T=0 E b then energy of N/ pairs is E o Ψ r = const. + න dr α T=0 Ψ r + ħ m ie A r ħ Ψ r However, by itself this will not generate stability of supercurrents (lecture 3). To do so, must add (e.g.) term in Ψ 4 Adding also EM field term: Ε 0 Ψ r = න dr α Τ=0 Ψ r + 1 β Τ=0 Ψ r 4 + ħ m + 1 μ 0 ie A r ħ Ψ 1 A r r Standard form of GL (free) energy (at T=0)

4 Notes: 1. Normalization of Ψ r is conventional and arbitrary. Rescaling SJTU 4.3 Ψ 1 r qψ r, α Τ=0 q α Τ=0, β Τ=0 q 4 β Τ=0, ħ Τm ħ Τmq leaves Ε 0 unchanged. eq. If A r = 0 and Ψ r = constant Ψ Τ=0, eq Ψ Τ=0 = α Τ=0 Τβ 1Τ Τ=0 then value is and energy is Ε eq Τ=0 = α Τ=0 Τ β Τ=0 3. Minimization with respect to A r δe o ΤδA r = 0 yields Maxwell s equation Η = j r provided that we identify j r = e m Ψ r iħ eα Ψ r + c. c. 4. Minimization with respect to Ψ r δe 0 ΤδΨ r = 0 yields α Τ=0 Ψ r + β Τ=0 Ψ r Note that for A r = 0 this defines a characteristic length ξ Τ=0 ħ mα Τ=0 1 A second characteristic length follows if we put Ψ r = constant Ψ 0 and compare terms in A and Α. λ Τ=0 m/e 1Τ μ o Ψ 0 or since with our normalization Ψ 0 = Ν/V, Ψ r ħ m ie ħ A r Ψ r = 0 λ Τ=0 m/μ 0 e n 1Τ ( London penetration depth at Τ = 0) electron density

5 SJTU 4.4 Generalization to Τ 0: Free energy All we need do is to let Ε 0 Ϝ Τ and let the coefficients α 0, β 0 be Τ dependent: Ϝ Ψ r : Τ = Ϝ 0 Τ + න drf r, Τ F r, Τ α Τ Ψ r + 1 β Τ Ψ r 4 + ħ m ie ħ Α r Ψ r +μ 0 1 A r standard form of GL free energy density Note that there is now a contribution to α Τ (and possibly also β Τ ) from the entropy term ΤS Ψ r. (condensate itself carries no entropy, but electrons liberated from it do!)

6 Since Ψ r describes S state, should 0 at some T c, then α Τ should change sign there. Most natural choice (corresponding to E b ~ constant, S Ψ r ~ constant constant Ψ ): α Τ = α 0 T c T β Τ = β 0 For Τ 0, all the above results go through with α Τ=0 α Τ, β T=0 β Τ. In particular in limit Τ T c from below: Ψ eq Τ = α Τ /β Τ F eq Τ F 0 Τ = ξ Τ α Τ λ Τ Ψ eq Τ Ϝ * Τ < Τ c α > 0 Τ > Τ c α Τ < 0 Ϝ 1Τ = 1Τ α0 /β 0 Τ c Τ 1Τ 0 Ψ α Τ Τβ Τ = α 0 Τβ 0 Τ c Τ 1Τ Τ c Τ 1Τ 1 Τ c Τ Τ 1 SJTU 4.5 Ψ 0 Mexican-hat potential so ratio κ λ/ξ is independent of Τ in limit Τ Τ c. From F eq Τ F 0 Τ Τ c Τ, entropy S has no discontinuity at Τ c, but sp. ht. C v ΤdS/dΤ has discontinuity Τ c α 0 Τβ 0 second order phase transition at T c. C v S N Τ c Τ

7 Some simple applications of the GL theory A. Zero magnetic field A = 0 1. Recovery of OP near wall (etc.) Suppose boundary condition at z = 0 is that Ψ must 0 (e.g. wall ferromagnetic). Then must solve GL eq Ψ = 0 ξ Τ z SJTU 4.6 Ψ = Ψ α Τ Ψ z + β 0 Ψ z Ψ z ħ m Ψ z = 0 subject to boundary conditions Ψ z = 0 = 0 Ψ z = Ψ α Τ Τβ 1Τ 0 Solution: Ψ z = Ψ tanh Τ z ξ Τ ξ Τ ħ Τ 1Τ mα Τ Τ c Τ 1Τ Note: Ψ does not have to vanish at boundary with vacuum or insulator! [except on scale ~k F 1 where N state wave functions do too] Thus, no objection to SC occurring in grains of dimension ξ Τ. However, contact with N metal tends to suppress SC.

8 . Current carrying state in thin wire SJTU 4.7 If d λ Τ, can neglect A to first approximation d By symmetry, j Ψ = Ψ exp iφ, Ψ = constant, j = eħ m Ψ φ F = α Τ Ψ + β 0 Ψ 4 + ħ m Ψ φ and we need to minimize this with respect to Ψ for fixed φ. Result: Ψ = ħ α Τ m φ β 0 Τ 1 Superfluid velocity v s = ħ m φ Ψ 1 ξ Τ ) φ Τ 1 Ψ 0 for φ = ξ 1 Τ (condensation energy changes sign). However, j is nonmonotonic function of φ, with maximum at point when φ = 1 3 ξ 1 Τ. (at which point Ψ = Τ3 Ψ ). Thus, critical current j c given by j c = eħ m α Τ 1 3 β 0 3 ξ 1 Τ α 1 T ξ 1 Τ 1 ΤΤΤ 3Τ c

9 B. Behavior in magnetic field SJTU 4.8 Because of the Meissner effect, any superconductor will completely expel a weak magnetic field. If we consider just the competition between the resulting state and the Nstate, we have (per unit volume) ΔE magn = + 1 μ 0Η ΔE cond = α Τ /β 0 S vs. N and so ( thermodynamic ) critical field Η c is given by Η c Τ = α Τ /μ 0 β 0 Τ 1 1 Τ/Τ c However, in general this works only for type-i superconductors in form of long cylinder parallel to field. More generally: (a) in type-i superconductors, intermediate state forms with interleaved macroscopic regions of N and S. (b) in type-ii superconductors, magnetic field punches through sample in the form of vortex lines.

10 Isolated vortex line SJTU 4.9 Consider λ Τ ξ T ( extreme type-ii ) Magnetic field turns region ~ξ Τ normal, and punches through there. Field is screened out of bulk on scale λ Τ ξ Τ λ Τ Ν ξ Τ At distances λ Τ, no current flows: however since j φ e ħ Α φ and Α may be individually non zero, with φ = e ħ Α. But φ must be single-valued mod. π, hence ර φ dl = nπ ර Α dl Φ = nφ 0 trapped flux h/e superconducting flux quantum n = 0 is trivial (no vortex!) and n is unstable, so restrict consideration to n = ± 1. Since field extends over λ Τ into bulk metal, field at core Η 0 ~Φ 0 /λ Τ.

11 Ψ SJTU 4.10 Energetics of single vortex line (per unit length) circ. currents Ψ (1) intrinsic energies: ξ Τ r λ Τ (a) field energy ~ 1 μ 0Η 0 λ ~φ 0 / μ o λ (b) (minus) condensation energy ~ 1 μ oη c ξ (c) flow energy: v s ~ ħ m 1 r, for r λ Τ, so c ~ Ψ ħ m λ rdr න r ~ Ψ ħ m ξ ln λ/ξ but λ Τ = μ 0 Ψ e /m 1Τ, so c ~ Φ 0 /μ 0 λ Τ ln λ/ξ dominant over (a) and (b) for λ Τ ξ Τ Thus intrinsic energy per unit length of vortex lines for λ ξ is Ε 0 Τ ~ Φ 0 / μ o λ Τ ln λ/ξ.

12 SJTU 4.11 () On the other hand, the extrinsic energy saving due to admission of the external field is ~μ 0 Η ext Η 0 ~Η ext Φ 0 /λ. Hence the condition for it to be energetically advantageous to admit a single vortex line is roughly Η ext ~ Φ 0 /λ Τ lnκ κ λ Τ /ξ Τ f Τ This defines the lower critical field Η c1. What is the maximum field the superconductor can tolerate before switching to the N phase? Roughly, defined by the point at which the vortex cores (area ~ξ Τ ) start to overlap. Since for near-complete penetration we must have nφ 0 ~Η ext, this gives the condition number of vortices/unit area Η ext ~ Φ 0 /ξ Τ This defines the upper critical field Η c Note that for λ Τ ξ Τ, we have Η c1 Η c and would expect type-i behavior.

13 SJTU 4.1 Results of more quantitative treatment: (a) condition for type-ii behavior is κ λ Τ /ξ Τ > 1Τ (b) in extreme type-ii limit κ 1, Η c1 Τ = Φ 0 /4πλ Τ lnκ (c) in same limit, Η c Τ = Φ 0 /πξ Τ. Note that quite generally we have up to logarithmic factors Η c1 Τ Η c Τ ~Η c Τ thermodynamic critical field hence if Η c Τ held fixed, Η c1 varies inversely to Η C Application: dirty superconductors Alloying does not change Ε cond Τ, hence Η c Τ, much. However, it drastically increases λ Τ ( ξ Τ is also increased, but less). Hence κ is increased, and many elements which on type-i when pure became type-ii when alloyed. As alloying increases, Η c increases while Η c1 decreases.

14 Summary of lecture 4 SJTU 4.13 Ginzburg-Landau theory is special case of Landau- Lifshitz theory of nd order phase transitions: quantitatively valid only near T c. Introduces order parameter ( macroscopic wave function )Ψ r which couples to vector potential A r with charge e: F Ψ r = const. α Ψ r + 1 β Ψ r 4 + const. ie ħ A r Ψ r + 1 μ o 1 A r characteristic lengths: healing length ξ T penetration depth λ T both T c T 1/ for T T c. Magnetic behavior depends on ratio κ = λ T /ξ T T Tc For κ > 1/, ( type-i ) field completely expelled up to a thermodynamic critical field H c T, at which point turns fully normal. For κ < 1/, ( type-ii ) field starts to penetrate at H c in form of vortices, continues to do so up to H c where turns completely normal.