Carsten Timm: Theory of superconductivity

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vortex core. For a vortex along the z-axis we choose cylindrical coordinates ϱ, φ, z. We then have to solve the Ginzburg-Landau equations with the boundary conditions From symmetry, we have and we can choose where ψ(ϱ = 0) = 0, (6.95) |ψ(ϱ → ∞)| = ψ 0 , (6.96) B(ϱ → ∞) = 0. (6.97) ψ(r) = |ψ| (ϱ) e iϕ = |ψ| (ϱ) e −iφ , (6.98) v s (r) = ˆφ v s (ϱ), (6.99) B(r) = ẑB(ϱ) (6.100) A(r) = ˆφA(ϱ) (6.101) B(ϱ) = 1 ϱ ∂ ϱA(ϱ). (6.102) ∂ϱ Choosing a circular integration path of radius r centered at the vortex core, the enclosed fluxoid is Φ ′ (r) = Φ(r) − mc ∮ ds · v s = 2πrA(r) − mc e e 2πr v s(r) = ! Φ 0 ⇒ ∂S A(r) − mc e v s(r) = Φ 0 2πr . (6.103) This relation follows from fluxoid quantization and thus ultimately from the second Ginzburg-Landau equation. To obtain j(r), ψ(r), and A(r), one also has to solve the first Ginzburg-Landau equation and Ampère’s Law ( 1 4m i ∇ + 2e ) 2 c A ψ + α ψ + β |ψ| 2 ψ = 0 (6.104) j = c ∇ × B. (6.105) 4π This cannot be done analytically because of the non-linear term β |ψ| 2 ψ. For small distances ϱ from the core, one can drop this term, thereby linearizing the Ginzburg-Landau equation. The solution is still complicated, the result is that |ψ| increases linearly in ϱ. Numerical integration of the full equations gives the results sketched here for a cut through the vortex core: ~ λ ψ( r) B( r) 0 ~ ξ x 46
Another useful quantity is the free energy per unit length of a vortex line (its line tension). An analytical expression can be obtained in the strong type-II case of κ ≫ 1. We only give the result here: The vortex line tension is ( ) 2 Φ0 ϵ v ≈ ln κ 4πλ = H2 c 8π 4πξ2 ln κ. (6.106) We can now calculate the field for which the first vortex enters the superconductor, the so-called lower critical field H c1 . This happens when the Gibbs free energy for a superconductor without vortices (Meißner phase) equals the Gibbs free energy in the presence of a single vortex. We assume the sample to have cross section A and thickness L, parallel to the applied field. B L A Then we have the condition Thus The line tension in Eq. (6.106) can also be written as 0 = G one vortex − G no vortex = ✚✚F s + Lϵ v − 1 ∫ d 3 r H · B −✚✚F s 4π = Lϵ v − H ∫ c1 d 3 r B(r) 4π = Lϵ v − H c1L 4π Φ 0. (6.107) H c1 = 4πϵ v Φ 0 . (6.108) ϵ v ≈ Φ 0 H c ξ √ ln κ = Φ 0 H c ln κ √ 4πλ 2 4π 2 κ (6.109) so that H c1 = H c √ 2 ln κ κ . (6.110) Recall that this expression only holds for κ ≫ 1. H c is the thermodynamic critical field derived above. In a type-II superconductor, nothing interesting happens at H = H c . The Abrikosov vortex lattice We have considered the structure of an isolated vortex line. How does a finite magnetic flux penetrate a type-II superconductor? Based on Ginzburg-Landau theory, A. A. Abrikosov proposed in 1957 that flux should enter as a periodic lattice of parallel vortex lines carrying a single flux quantum each. He proposed a square lattice, which was due to a small mistake. The lowest-free-energy state is actually a triangular lattice. 47
Page 1 and 2: Theory of Superconductivity Carsten
Page 3 and 4: Contents 1 Introduction 5 1.1 Scope
Page 5 and 6: 1 Introduction 1.1 Scope Supercondu
Page 7 and 8: 2 Basic experiments In this chapter
Page 9 and 10: Superconductivity with rather high
Page 11 and 12: • In 1979, Frank Steglich et al.
Page 13 and 14: 3 Bose-Einstein condensation In thi
Page 15 and 16: We note the identity g n (y) = ∞
Page 17 and 18: We find a phase transition at T c ,
Page 19 and 20: gives a reasonable approximation fo
Page 21 and 22: We now consider the force F = −eE
Page 23 and 24: 5 Electrodynamics of superconductor
Page 25 and 26: Thus the supercurrent flows in the
Page 27 and 28: where φ j is the polar angle of el
Page 29 and 30: with the penetration depth √ √
Page 31 and 32: N s is the total number of particle
Page 33 and 34: 6.2 Ginzburg-Landau theory for neut
Page 35 and 36: scope of this course, though. The i
Page 37 and 38: As above, we keep only terms up to
Page 39 and 40: Within the mean-field theories we h
Page 41 and 42: Including the superconducting contr
Page 43 and 44: The Gibbs free energy of the domain
Page 45: The magnetic flux Φ through this l
Page 49 and 50: We obtain e −ik yy (− 2 2m ∗
Page 51 and 52: has a simple interpretation: It is
Page 53 and 54: 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 and 106:
for quasiparticle creation and anni
Page 107 and 108:
11 Josephson effects Brian Josephso
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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