Carsten Timm: Theory of superconductivity

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and −V RPA eff ≡ := + + + · · · (8.80) or As above, we can sum this up, −V RPA eff (q, iν n ) = −V eff (q, iν n ) + V eff (q, iν n ) Π 0 (q, iν n ) V eff (q, iν n ) − V eff (q, iν n ) Π 0 (q, iν n ) V eff (q, iν n ) Π 0 (q, iν n ) V eff (q, iν n ) + · · · (8.81) Veff RPA V eff (q, iν n ) (q, iν n ) = 1 + V eff (q, iν n )Π 0 (q, iν n ) = V C(q) = V C (q) = 1 + V C (q) (iν n ) 2 (iν n ) 2 − Ω 2 + (iν n ) 2 V C (q)Π 0 (q, iν n ) V C (q) 1 + V C (q)Π 0 (q, iν n ) } {{ } = VC RPA (q,iν n) = V RPA C (q, iν n ) (iν n ) 2 (iν n ) 2 −Ω 2 (iν n ) 2 (iν n ) 2 −Ω 2 Π 0 (q, iν n ) (iν n ) 2 + (iν n ) 2 V C (q)Π 0 (q, iν n ) (iν n ) 2 − Ω 2 + (iν n ) 2 V C (q)Π 0 (q, iν n ) (iν n ) 2 (iν n ) 2 − Ω 2 1+V C (q)Π 0 (q,iν n ) = VC RPA (iν n ) 2 (q, iν n ) (iν n ) 2 − ωq(iν 2 n ) (8.82) with the renormalized phonon frequency ω q (iν n ) := Ω √ 1 + VC (q)Π 0 (q, iν n ) . (8.83) To see that this is a reasonable terminology, compare V RPA eff V eff (q, iν n ) = V C (q) to the bare effective interaction (iν n ) 2 (iν n ) 2 − Ω 2 . (8.84) Evidently, screening leads to the replacements V C → VC RPA and Ω → ω q . For small momenta and frequencies, we have Π 0 → N(E F ), the density of states at E F . In this limit we thus obtain ω q ∼ = Ω √ 1 + 4π e2 q N(E 2 F ) = Ω √ 1 + κ2 s √ κ 2 s q 2 ∼ = Ω q 2 = Ω κ s q. (8.85) Due to screening we thus find an acoustic dispersion of jellium phonons. This is of course much more realistic than an optical Einstein mode. Beyond the low-frequency limit it is important that Π 0 and thus ω q obtains a sizable imaginary part. It smears out the pole in the retarded interaction Veff RPA (q, ν) or rather moves it away from the real-frequency axis—the lattice vibrations are now damped. The real part of the retarded interaction is sketched here for fixed q: 80
Re RPA, R V eff V c ( ) RPA q 0 Re ω q ν Note that • the interaction still vanishes in the static limit ν → 0, • the interaction is attractive for 0 < ν < Re ω q , where Re ω q ∼ q. It is important that the static interaction is not attractive but zero. Hence, we do not expect static bound states of two electrons. To obtain analytical results, it is necessary to simplify the interaction. The main property required for superconductivity is that the interaction is attractive for frequencies below some typical phonon frequency. The typical phonon frequency is the material specific Debye frequency ω D . We write the effective RPA interaction in terms of the incoming and transferred momenta and frequencies, V RPA eff = V RPA eff (k, iω n ; k ′ , iω ′ n; q, iν n ). (8.86) k − q,iω n − iν n q, iν n k’ + q, iω’ n + iν n k,iω n k’ , i ’ ω n We then approximate the interaction (very crudely) by a constant −V 0 < 0 if both incoming frequencies are smaller than ω D and by zero otherwise, { Veff RPA −V 0 for |iω n | , |iω ′ ≈ n| < ω D , (8.87) 0 otherwise. 81
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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
Page 9 and 10:
Superconductivity with rather high
Page 11 and 12:
• In 1979, Frank Steglich et al.
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3 Bose-Einstein condensation In thi
Page 15 and 16:
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
Page 21 and 22:
We now consider the force F = −eE
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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 and 46: The magnetic flux Φ through this l
Page 47 and 48: Another useful quantity is the free
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: 8.4 Effective interaction between e
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
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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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