Carsten Timm: Theory of superconductivity

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as it is approached. Consequently, the superconductivity caused by them is strongest and has the highest T c right above the QCP. Compare the previous argument: There, the spin fluctuations are assumed to be strongest close to the (finite-temperature) antiferromagnetic phase and one needs to envoke small n s and vortex fluctuations to argue why the maximum T c is not close to the antiferromagnetic phase. 12.3 Pnictides The iron pnictide superconductors are a more heterogeneous group than the cuprates. However, for most of them the phase diagram is roughly similar to the one of a typical cuprate in that superconductivity emerges at finite doping in the vicinity of an antiferromagnetic phase. This antiferromagnetic phase is metallic, not a Mott insulator, though, suggesting that interactions are generally weaker in the pnictides. T 140 K CeFeAsO 1−xF x tetragonal orthorombic AFM superconductor 0 0.05 doping x The crystal structure is quasi-two-dimensional, though probably less so than in the cuprates. The common structural motif is an iron-pnictogen, in particular Fe 2+ As 3− , layer with Fe 2+ forming a square lattice and As 3− sitting alternatingly above and below the Fe 2+ plaquettes. Fe 2+ As 3− While the correct unit cell contains two As and two Fe ions, the glide-mirror symmetry with respect to the Fe plane allows to formulate two-dimensional models using a single-iron unit cell. The fact that different unit cells are used leads to some confusion in the field. Note that since the single-iron unit cell is half as large as the two-iron unit cell, the corresponding single-iron Brillouin zone is twice as large as the two-iron Brillouin zone. A look at band-structure calculations or angular resolved photoemission data shows that the Fermi surface of pnictides is much more complicated than the one of cuprates. The k z = 0 cut typically shows five Fermi pockets. 130
k y Γ X’ M hole−like Q 2 electron−like Q 1 X k x single−iron Brillouin zone The (probably outer) hole pocket at Γ and the electron pockets at X and X ′ are well nested with nesting vectors Q 1 = (π/a, 0) and Q 2 = (0, π/a), respectively. Not surprisingly, the spin susceptibility is peaked at Q 1 and Q 2 in the paramagnetic phase and in the antiferromagnetic phase the system orders antiferromagnetically at either of three vectors. Incidentally, the antiferromagnet emerges through the formation and condensation of electron-hole pairs (excitons), described by a BCS-type theory. The same excitonic instability is for example responsible for the magnetism of chromium. Assuming that the exchange of spin fulctuations in the paramagnetic phase is the main pairing interaction, the gap equation ∆ k = − 1 ∑ 1 − n F (E k ′) ∆ k ′ (12.37) N 2E k ′ k ′ V kk ′ with V kk ′ large and positive for k − k ′ = ±Q 1 or ±Q 2 , favors a gap function ∆ k changing sign between k and k + Q 1 and between k and k + Q 2 , see Sec. 12.1. This is most easily accomodated by a nodeless gap changing sign between the electron and hole pockets: k y 131 k x
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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
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Note that in two dimensions the cur
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Thus the energy is ∫ E 2 = 2E cor
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The energy of an isolated vortex-an
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Next, we have to express Z ′ in t
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quasi−long−range order free vor
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which is valid within the film and
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attraction of the magnetic monopole
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Finally, the voltage measured for a
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In this case it is useful to expand
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It will be useful to write the elec
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(the minus sign is conventional) an
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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: Also note that spin fluctuations st
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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