Carsten Timm: Theory of superconductivity

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with the density of states per spin direction and per unit cell, D(ϵ). approximately constant close to the Fermi energy we get (with = 1) Assuming the density of states to be · · · ≈ V 0 k B T ∑ ∫ ∞ D(E F ) iω 1 n −∞ |iωn| 1 < ω D βω D /2π ∑ = V 0 k B T D(E F ) π 2 [ ∼= V 0 D(E F ) γ + ln n=0 ( 4 βω D 2π dξ (ωn) 1 2 + ξ 2 } {{ } = π/|ωn 1 | 1 (2n+1)π β βω = V 0 ✟k ✟ D /2π ∑ B T D(E F ) ✁β n + 1 n=0 2 )] . (9.9) In the last step we have used an approximation for the sum over n that is valid for βω D ≫ 1, i.e., if the sum has many terms. Since T c for superconductors is typically small compared to the Debye temperature ω D /k B (a few hundred Kelvin), this is justified. γ ≈ 0.577216 is the Euler constant. Altogether, we find −V 0 Λ ≈ ( ) (9.10) 1 − V 0 D(E F ) γ + ln 2βω D π for |iω n | < ω D . Coming from high temperatures, but still satisfying k B T ≪ ω D , multiple scattering enhances Λ. Λ diverges at T = T c , where ( V 0 D(E F ) γ + ln 2ω ) D = 1 (9.11) πk B T c ⇒ ln 2eγ ω D 1 = (9.12) πk B T c V 0 D(E F ) 2e γ ( ) 1 ⇒ k B T c = ω D exp − . (9.13) }{{} π V 0 D(E F ) ≈ 1.13387 ∼ 1 This is the Cooper instability. Its characteristic temperature scale appears to be the Debye temperature of a few hundred Kelvin. This is disturbing since we do not observe an instability at such high temperatures. However, the exponential factor tends to be on the order of 1/100 so that we obtain T c of a few Kelvin. It is important that k B T c is not analytic in V 0 at V 0 = 0 (the function has an essential singularity there). Thus k B T c cannot be expanded into a Taylor series around the non-interacting limit. This means that we cannot obtain k B T c in perturbation theory in V 0 to any finite order. BCS theory is indeed non-perturbative. 9.2 The BCS ground state We have seen that the Fermi sea becomes unstable due to the scattering of electrons in states |k, ↑⟩ and |−k, ↓⟩. Bardeen, Cooper, and Schrieffer (BCS) have proposed an ansatz for the new ground state. It is based on the idea that electrons from the states |k, ↑⟩ and |−k, ↓⟩ form (so-called Cooper) pairs and that the ground state is a superposition of states built up of such pairs. The ansatz reads ( ) u k + v k c † k↑ c† −k,↓ |0⟩ , (9.14) |ψ BCS ⟩ = ∏ k 1 84
where |0⟩ is the vacuum state without any electrons and u k , v k are as yet unknown complex coefficients. Normalization requires 1 = ⟨ψ BCS |ψ BCS ⟩ = ⟨0| ∏ k = ⟨0| ∏ k = ∏ k (u ∗ k + v ∗ k c −k,↓ c k↑ ) ∏ k ′ (u k ′ + v k ′ c † k ′ ↑ c† −k ′ ,↓ ) |0⟩ ( |u k | 2 + u ∗ kv k c † k↑ c† −k,↓ + u kv ∗ k c −k,↓ c k↑ + |v k | 2) |0⟩ ( |u k | 2 + |v k | 2) . (9.15) This is certainly satisfied if we demand |u k | 2 + |v k | 2 = 1 for all k, which we will do from now on. Note that the occupations of |k, ↑⟩ and |−k, ↓⟩ are maximally correlated; either both are occupied or both are empty. Also, |ψ BCS ⟩ is peculiar in that it is a superposition of states with different total electron numbers. (We could imagine the superconductor to be entangled with a much larger electron reservoir so that the total electron number in superconductor and reservoir is fixed.) As a consequence, expressions containing unequal numbers of electronic creation and annihilation operators can have non-vanishing expectation values. For example, ⟨ ⟨ ∣ c † ∣∣c k↑ c† −k,↓ ⟩BCS ≡ † ψ BCS k↑ c† −k,↓ ⟩ ∣ ψ BCS = ⟨0| ∏ k ′ (u ∗ k ′ + v∗ k ′ c −k ′ ,↓c k ′ ↑) c † k↑ c† −k,↓ ∏ ( ) u k ′′ + v k ′′ c † k ′′ ↑ c† −k ′′ ,↓ |0⟩ k ′′ ∏ = ⟨0| vku ∗ k (|u k ′| 2 + |v k ′| 2) |0⟩ = vku ∗ k . (9.16) k ′ ≠k } {{ } =1 The coefficients u k , v k are chosen so as to minimize the expectation value of the energy, ⟨ψ BCS |H| ψ BCS ⟩, under the constraint |u k | 2 + |v k | 2 = 1 for all k. |ψ BCS ⟩ is thus a variational ansatz. We write the Hamiltonian as H = ∑ ξ k c † kσ c kσ + V int (9.17) kσ and, in the spirit of the previous section, choose the simplest non-trivial approximation for V int that takes into account that 1. only electrons with energies |ξ k | ω D relative to the Fermi energy are important and 2. the instability is due to the scattering between electrons in single-particle states |k ↑⟩ and |−k, ↓⟩. This leads to with V kk ′ = V int = 1 N ∑ kk ′ V kk ′ c † k↑ c† −k,↓ c −k ′ ,↓c k′ ↑ (9.18) { −V 0 for |ξ k | < ω D and |ξ k ′| < ω D , 0 otherwise. (9.19) We assume that scattering without momentum transfer, k ′ = k, contributes negligibly compared to k ′ ≠ k since there are many more scattering channels for k ′ ≠ k. We also assume that u −k = u k , v −k = v k , which is, at worst, 85
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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
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 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: diagrams describing this situation.
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 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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