Carsten Timm: Theory of superconductivity

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we can write the gap matrix in a compact form as ˆ∆(k) = (∆ k 1 + d(k) · σ) iσ y , (12.63) where σ is the vector of Pauli matrices. The mean-field Hamiltonian H MF is diagonalized by a Bogoliubov transformation and the gap function ˆ∆(k) is obtained selfconsistently from a gap equation in complete analogy to the singlet case discussed in Sec. 10.1, except that ˆ∆(k) is now a matrix and that we require four coefficients u k↑ , u k↓ , v k↑ , v k↓ . We do not show this here explicitly. A few remarks on the physics are in order, though. The symmetry ˆ∆(−k) = − ˆ∆ T (k) implies ⇒ (∆ −k 1 + d(−k) · σ) iσ y = −i (σ y ) T } {{ } ( ∆k 1 + d(k) · σ T ) (12.64) iσ y ∆ −k 1 + d(−k) · σ = σ y ( ∆ k 1 + d(k) · σ T ) σ y ⎛ ⎞ = ∆ k 1 + d(k) · ⎝ σy (σ x ) T σ y σ y (σ y ) T σ y ⎠ ⎛ σ y (σ z ) T σ y ⎞ σy σ x σ y = ∆ k 1 + d(k) · ⎝ −σ y σ y σ y σ y σ z σ y ⎠ = ∆ k 1 − d(k) · σ. (12.65) Thus we conclude that ∆ k is even, whereas d(k) is odd, ∆ −k = ∆ k , (12.66) d(−k) = −d(k). (12.67) Hence, the d-vector can never be constant, unlike the single gap in Sec. 10.1. Furthermore, we can expand ∆ k into even basis functions of rotations in real (and k) space, ∆ k = ∆ s ψ s (k) + ∆ dx 2 −y 2 ψ dx 2 −y 2 (k) + ∆ d3z 2 −r 2 ψ d3z 2 −r 2 (k) + ∆ dxy ψ dxy (k) + ∆ dyz ψ dyz (k) + ∆ dzx ψ dzx (k) + . . . , (12.68) and expand (the components of) d(k) into odd basis functions, d(k) = d px ψ px (k) + d py ψ py (k) + d pz ψ pz (k) + . . . (12.69) For the singlet case we have already considered the basis functions ψ s (k) = 1 for conventional superconductors, ψ s (k) = cos k x a cos k y a for the pnictides, and ψ dx 2 −y 2 (k) = cos k x a − cos k y a for the cuprates. A typical basis function for a triplet superconductor would be ψ px = sin k x a. He-3 in the so-called B phase realized at now too high pressures (see Sec. 2.2) has the d-vector d(k) = ˆx ψ px (k) + ŷ ψ py (k) + ẑ ψ pz (k), (12.70) where the basis functions are that standard expressions for p x , p y , and p z orbitals (note that there is no Brillouin zone since He-3 is a liquid), √ 3 ψ px (k) = 4π sin θ k cos ϕ k , (12.71) √ 3 ψ py (k) = 4π sin θ k sin ϕ k , (12.72) √ 3 ψ pz (k) = 4π cos θ k. (12.73) This is the so-called Balian-Werthamer state. 134
It is plausible that in a crystal the simultaneous presence of a non-vanishing k-even order parameter ∆ k and a non-vanishing k-odd order parameter d(k) would break spatial inversion symmetry. Thus in an inversionsymmetric crystal singlet and triplet superconductivity do not coexist. However, in crystals lacking inversion symmetry (noncentrosymmetric crystals), singlet and triplet pairing can be realized at the same time. Moreover, one can show that spin-orbit coupling in a noncentrosymmetric superconductor mixes singlet and triplet pairing. In this case, ∆ k and d(k) must both become non-zero simultaneously below T c . Examples for such superconductors are CePt 3 Si, CeRhSi 3 , and Y 2 C 3 . 135
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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
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8.4 Effective interaction between e
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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: Thus ∆ τσ (−k) = − 1 N or,
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Carsten Timm: Theory of superconductivity

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