Carsten Timm: Theory of superconductivity

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This gives a contribution to the field energy of ∫ ∫ d 3 r B2 8π = Φ2 0 32π 3 d 3 r 1 ∫∞ r 4 = Φ2 0 32π 3 4π dr r 2 = Φ2 0 1 8π 2 , (7.103) λ ⊥ λ ⊥ which is finite for an infinity film. The value of the lower cutoff does not matter for this. The cutoff has to be present, since the monopole field is not a valid approximation for small r. We next obtain the current from Ampère’s law in integral form: d ∆r B ∮ dr · B = 4π c ∫ da · j (7.104) and with the vector character restored Note that the sheet current is thus ⇒ r+∆r ∫ 2 dr ′ Φ 0 2π(r ′ ) 2 = 4π ∆r d j(r) c (7.105) ⇒ r c ∆r d j(r) = Φ ( ) 0 1 π r − 1 ∼= Φ 0 ∆r r + ∆r π r 2 (7.106) ⇒ j(r) = Φ 0 1 4π 2 r 2 d (7.107) j(r) = Φ 0 4π 2 K(r) = Φ 0 4π 2 ˆφ r 2 d . (7.108) ˆφ r 2 . (7.109) For large r we have |ψ| = ψ 0 = √ −α/β (note that typically λ ⊥ ≫ ξ). Then the second Ginzburg-Landau equation gives j = i q 2m ∗ (−i∇ϕ − i∇ϕ)ψ2 0 − q2 m ∗ c ψ2 0A = ψ0 2 q ( m ∗ ∇ϕ − q ) c A . (7.110) Thus we can rewrite the gradient term in the free energy as ∫ d d 2 r The contribution from large r is 1 2m ∗ ψ2 0 (∇ϕ − q ) 2 ∫ c A = d d 2 r ψ2 0(m ∗ ) 2 2m ∗ ψ 4 0 q2 j · j = d ∫ m ∗ d 2 r 2q 2 ψ0 2 j · j. (7.111) dm ∗ ∫ 2q 2 ψ0 2 d 2 r Φ2 0 1 16π 4 r 4 d 2 = m∗ Φ 2 ∫∞ 0 32π 4 q 2 ψ0 2d 2π λ ⊥ dr r 3 = m∗ Φ 2 0 1 8π 3 q 2 ψ0 2d , (7.112) λ 2 ⊥ which is also finite for an infinite film. We conclude that the free energy of an isolated vortex is finite in a superconducting film. It is then plausible and indeed true that the interaction energy of a vortex-antivortex pair does not diverge for large separations r but saturates for r ≫ λ ⊥ . Since for the far field of a single vortex the magnetic-field energy, Eq. (7.103), dominates over the energy due to the gradient term, we expect the large-r interaction to be dominated by the Coulomb-type 66
attraction of the magnetic monopoles in the upper and lower half spaces. This supposition is borne out by a proper analysis. Consequently, for large r the interaction behaves like V int = const − ( ) 2 Φ0 1 2π r (7.113) (Φ 0 /2π is the monopole strength, according to Eq. (7.102)). N S All this showes that, strictly speaking, there will be a non-zero concentration of free vortices at any temperature T > 0. Thus there is no quasi-long-range order. However, the relevant length scale is λ ⊥ = λ 2 /d, which can be very large for thin films, even compared to the lateral size L of the sample. In this case the large-r limit is experimentally irrelevant. But for vortex separations r ≪ λ ⊥ , the magnetic-field expulsion on the scale r is very weak since λ ⊥ is the effective penetration depth. Then the fact that the condensate is charged is irrelevant and we obtain the same logarithmic interaction as for a neutral superfluid. Thus for thin films of typical size we can use the previously discussed BKT theory. For superconducting films we even have the advantage of an additional observable, namely the voltage for given current. We give a hand-waving derivation of V (I). The idea is that a current exerts a Magnus force on a vortex, in the direction perpendicular to the current. The force is opposite for vortices and antivortices and is thus able to break vortexantivortex pairs. As noted above, free vortices lead to dissipation. A vortex moving through the sample in the orthogonal direction between source and drain contacts leads to a change of the phase difference ∆ϕ by ±2π. We will see in the chapter on Josephson effects why this corresponds to a non-zero voltage. Since free vortices act independently, it is plausible to assume that the resistance is R ∝ n v , (7.114) where n v now denotes the concentration of free vortices. To find it, note that the total potential energy due to vortex-antivortex attraction and Magnus force can be written as with V = V int − 2F Magnus r (7.115) V int = 2π k B T K ln r r 0 . (7.116) 67
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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) = ∞
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: which is valid within the film and
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
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
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 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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