Carsten Timm: Theory of superconductivity

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the second term, ∫ ∫ d 2 r j ¯D i,j r≤|r i−r j|
Next, we have to express Z ′ in terms of the new length scale r ′ := r + dr. This is only relevant in expressions not already linear in dr. This applies to, on the one hand, 1 r 2 = 1 (r ′ − dr) 2 = 1 + 2 dr r ′ (r ′ ) 2 , (7.62) and, on the other, ⎛ ⎞ ⎛ ⎞ exp ⎝− 1 ∑ 2πK N i N j ln r⎠ = exp ⎝− 1 ∑ 2πK N i N j ln(r ′ − dr) ⎠ 2 2 i≠j i≠j ⎛ ⎞ ⎛ = exp ⎝− 1 ∑ 2πK N i N j ln r ′ ⎠ exp ⎝ 1 ∑ 2 2 i≠j i≠j ⎛ ⎞ = exp ⎝− 1 ∑ ( 2πK N i N j ln r ′ ⎠ 2 neglecting terms of order N 0 compared to N. The renormalized partition function is finally [ ( ) ] 2 ( ) 2N [ y Z ′ = exp 2π (r ′ ) 2 r ′ 1 y dr V (N!) 2 (r ′ ) 2 1 + (2 − πK) dr ] 2N r ′ N } {{ } ∫ × D ′ 1 d 2 r 1 · · · =: Z pair ∑ ∫ D ′ 2N i≠j ⎡ d 2 r 2N exp ⎣− 1 ∑ 2 i≠j ( −2πK + 8π 4 y 2 K 2 dr r ′ ) ⎞ dr 2πK N i N j ⎠ r ′ 1 − πK dr r ′ ) 2N , (7.63) ⎤ N i N j ln |r i − r j | ⎦ r ′ . (7.64) Here, Z pair is the partition function of the small pair we have integrated out. It is irrelevant for the renormalization of y and K. Apart from this factor, Z ′ is identical to Z if we set [ y(r ′ ) = 1 + (2 − πK(r)) dr ] r ′ y(r) (7.65) Introducing the logarithmic length scale we obtain the Kosterlitz RG flow equations K(r ′ ) = K(r) − 4π 3 y 2 K 2 (r) dr r ′ . (7.66) l := ln r r 0 ⇒ dl = dr r , (7.67) The initial conditions are dy dl dK dl = (2 − πK) y, (7.68) = −4π 3 y 2 K 2 . (7.69) i.e., the parameters assume their “bare” values at r = r 0 (l = 0). y(l = 0) = y 0 = e −E core/k B T , (7.70) K(l = 0) = K 0 = − 1 k B T 2γ α β , (7.71) 61
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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.
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 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: The energy of an isolated vortex-an
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
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Equation (11.25) can be used to stu
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The averaged current is just Ī = I
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where = 1. In the superconductor,
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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
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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
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Also note that spin fluctuations st
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k y Γ X’ M hole−like Q 2 elect
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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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