Carsten Timm: Theory of superconductivity

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V V int r barrier x There is a finite barrier for vortex-antivortex unbinding at a separation r barrier determined from ∂V/∂r = 0. This gives where K barrier := K(r barrier ). The barrier height is For small currents we have r barrier ≫ r 0 and thus The rate at which free vortices are generated is r barrier = 2π k BT K barrier 2F Magnus , (7.117) ∆E := V (r barrier ) − V (r 0 ) ∼ = V (r barrier ) The recombination rate <strong>of</strong> two vortices to form a pair is since two vortices must meet. In the stationary state we have and thus a resistance <strong>of</strong> Since the Magnus force is F Magnus ∝ I we have = 2π k B T K barrier ln r barrier − 2F Magnus r barrier r ( 0 = 2π k B T K barrier ln r ) barrier − 1 . (7.118) r 0 ∆E ∼ = 2π k B T K(l → ∞) ln r barrier . (7.119) } {{ } r 0 ≡ K R gen ∝ e −β∆E ∼ = ( rbarrier r 0 ) −2πK . (7.120) R gen = R rec ⇒ n v ∝ √ R gen ∝ R ∝ n v ∝ R rec ∝ n 2 v, (7.121) ( rbarrier r 0 ) −πK (7.122) ( rbarrier r 0 ) −πK . (7.123) r barrier ∝ 1 F Magnus ∝ 1 I (7.124) so that ( ) −πK 1 R ∝ = I πK . (7.125) I 68
Finally, the voltage measured for a current I is V = RI ∝ I πK I = I 1+πK , (7.126) where K ≡ K(l → ∞) is the renormalized stiffness. Since K = 2 π + ∆K ∼ = 2 π ( 1 + √ 2b √ ) T c − T (7.127) we find for the exponent for T T c . 1 + πK ∼ = 3 + 2 √ √ } {{ 2b } Tc − T (7.128) = const 1+ πK 3 2 1 ohmic T c T Above T c we have K = 0 and thus ohmic resistance, V ∝ I, as expected. Below T c , the voltage is sub-ohmic, i.e., the voltage is finite for finite current but rises more slowly than linearly for small currents. This behavior has been observed for thin superconducting films. 69
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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
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) = ∞
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 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: attraction of the magnetic monopole
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
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S N S 2∆ 0 2eV eV In particular,
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But now the right-hand side is alwa
Page 123 and 124:
The fact that the gap closes at som
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where τ is the imaginary time, T
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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,
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It is plausible that in a crystal t
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Carsten Timm: Theory of superconductivity

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