Carsten Timm: Theory of superconductivity

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10.6 Ginzburg-Landau-Gor’kov theory We conclude this chapter by remarking that Lev Gor’kov managed, two years after the publication of BCS theory, to derive Ginzburg-Landau theory from BCS theory. The correspondence is perfect if the gap is sufficiantly small, i.e., T is close to T c , and the electromagnetic field varies slowly on the scale of the Pippard coherence length ξ 0 (see Sec. 5.4). These are indeed the conditions under which Ginzburg and Landau expected their theory to be valid. Gor’kov used equations of motion for electronic Green functions, which he decoupled with a mean-field-like approximation, which allowed for spatial variations of the decoupling term ∆(r). The derivation is given in Schrieffer’s book and we omit it here. Gor’kov found that in order to obtain the Ginzburg-Landau equations, he had to take as anticipated, and (using our conventions) ψ(r) = q = −2e, (10.119) m ∗ = 2m, (10.120) √ 7ζ(3) 4π √ n 0 s ∆(r) k B T c , (10.121) where n 0 s := n s 1 − T T c ∣ ∣∣∣∣T →T − c . (10.122) Recall that n s ∼ 1 − T/T c close to T c in Ginzburg-Landau theory. The spatially dependent gap is thus locally proportional to the Ginzburg-Landau “condensate wavefunction” or order parameter ψ(r). Since we have already found that London theory is a limiting case of Ginzburg-Landau theory, it is also a limiting case of BCS theory. But London theory predicts the two central properties of superconductors: Ideal conduction and flux expulsion. Thus Gor’kov’s derivation also shows that BCS theory indeed describes a superconducting state. (Historically, this has been shown by BCS before Gor’kov established the formal relationship between the various theories.) 106
11 Josephson effects Brian Josephson made two important predictions for the current flowing through a tunneling barrier between two superconductors. The results have later been extended to various other systems involving two superconducting electrodes, such as superconductor/normal-metal/superconductor heterostructures and superconducting weak links. Rather generally, for vanishing applied voltage a supercurrent I s is flowing which is related to the phase difference ∆ϕ of the two condensates by I s = I c sin ( ∆ϕ − 2π Φ 0 ∫ ) ds · A . (11.1) We will discuss the critical current I c presently. We consider the case without magnetic field so that we can choose the gauge A ≡ 0. Then the Josephson relation simplifies to I s = I c sin ∆ϕ. (11.2) It should be noted that this DC Josephson effect is an equilibrium phenomenon since no bias voltage is applied. The current thus continues to flow as long as the phase difference ∆ϕ is maintained. Secondly, Josephson predicted that in the presence of a constant bias voltage V , the phase difference would evolve according to d ∆ϕ = −2e dt V (11.3) (recall that we use the convention e > 0) so that an alternating current would flow, ( I s (t) = I c sin ∆ϕ 0 − 2e ) V t . (11.4) This is called the AC Josephson effect. The frequency ω J := 2eV (11.5) of the current is called the Josephson frequency. The AC Josephson effect relates frequencies (or times) to voltages, which makes it important for metrology. 11.1 The Josephson effects in Ginzburg-Landau theory We consider a weak link between two identical bulk superconductors. The weak link is realized by a short wire of length L ≪ ξ and cross section A made from the same material as the bulk superconductors. We choose this setup since it is the easiest to treat in Ginzburg-Landau theory since the parameters α and β are uniform, but the only property that really matters is that the phase ϕ of the order parameter ψ(r) only changes within the weak 107
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Theory of Superconductivity Carsten
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Contents 1 Introduction 5 1.1 Scope
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1 Introduction 1.1 Scope Supercondu
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2 Basic experiments In this chapter
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Superconductivity with rather high
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• In 1979, Frank Steglich et al.
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3 Bose-Einstein condensation In thi
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We note the identity g n (y) = ∞
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We find a phase transition at T c ,
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gives a reasonable approximation fo
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We now consider the force F = −eE
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5 Electrodynamics of superconductor
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Thus the supercurrent flows in the
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where φ j is the polar angle of el
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with the penetration depth √ √
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N s is the total number of particle
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6.2 Ginzburg-Landau theory for neut
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scope of this course, though. The i
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As above, we keep only terms up to
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Within the mean-field theories we h
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Including the superconducting contr
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The Gibbs free energy of the domain
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The magnetic flux Φ through this l
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Another useful quantity is the free
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We obtain e −ik yy (− 2 2m ∗
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has a simple interpretation: It is
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This is a Gaussian average since F
Page 55 and 56: Note that in two dimensions the cur
Page 57 and 58: Thus the energy is ∫ E 2 = 2E cor
Page 59 and 60: The energy of an isolated vortex-an
Page 61 and 62: Next, we have to express Z ′ in t
Page 63 and 64: quasi−long−range order free vor
Page 65 and 66: which is valid within the film and
Page 67 and 68: attraction of the magnetic monopole
Page 69 and 70: Finally, the voltage measured for a
Page 71 and 72: In this case it is useful to expand
Page 73 and 74: It will be useful to write the elec
Page 75 and 76: (the minus sign is conventional) an
Page 77 and 78: Note that summing up the more and m
Page 79 and 80: 8.4 Effective interaction between e
Page 81 and 82: Re RPA, R V eff V c ( ) RPA q 0 Re
Page 83 and 84: diagrams describing this situation.
Page 85 and 86: where |0⟩ is the vacuum state wit
Page 87 and 88: This is called the BCS gap equation
Page 89 and 90: 10 BCS theory The variational ansat
Page 91 and 92: we obtain ( ) 2ξ k |u k | |v k | e
Page 93 and 94: 1 0 2 u k v 2 k k F k We also find
Page 95 and 96: 10.2 Isotope effect How can one che
Page 97 and 98: and expanding for small ∆T and sm
Page 99 and 100: For tunneling between two normal me
Page 101 and 102: In the limit k B T → 0 this becom
Page 103 and 104: tant. The first one is (u k ′u k
Page 105: for quasiparticle creation and anni
Page 109 and 110: Ginzburg-Landau theory also gives u
Page 111 and 112: Equation (11.25) can be used to stu
Page 113 and 114: The averaged current is just Ī = I
Page 115 and 116: where = 1. In the superconductor,
Page 117 and 118: with ˜k 1 := (−k 1x , k 1y , k 1
Page 119 and 120: S N S 2∆ 0 2eV eV In particular,
Page 121 and 122: But now the right-hand side is alwa
Page 123 and 124: The fact that the gap closes at som
Page 125 and 126: where τ is the imaginary time, T
Page 127 and 128: At lower temperatures, details beco
Page 129 and 130: Also note that spin fluctuations st
Page 131 and 132: k y Γ X’ M hole−like Q 2 elect
Page 133 and 134: Thus ∆ τσ (−k) = − 1 N or,
Page 135: It is plausible that in a crystal t
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Carsten Timm: Theory of superconductivity

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