Carsten Timm: Theory of superconductivity

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Jellium phonons For our discussions we need a specific model for phonons. We use the simplest one, based on the jellium approximation for the nuclei (or ion cores). In this approximation we describe the nuclei by a smooth positive charge density ρ + (r, t). In equilibrium, this charge density is uniform, ρ + (r, t) = ρ 0 +. We consider small deviations Gauss’ law reads ρ + (r, t) = ρ 0 + + δρ + (r, t). (8.65) ∇ · E = 4π δρ + (r, t), (8.66) since ρ 0 + is compensated by the average electronic charge density. The density of force acting on ρ + is f = ρ + E ∼ = ρ 0 +E to leading order in δρ + . Thus ∇ · f ∼ = 4π ρ 0 + δρ + . (8.67) The conservation of charge is expressed by the continuity equation To leading order this reads ∂ ∂t ρ + + ∇ · ρ + v }{{} = j + = 0. (8.68) ⇒ ∂ ∂t δρ + + ρ 0 +∇ · v ∼ = 0 (8.69) ∂2 ∂t 2 δρ ∼ + = −ρ 0 +∇ · ∂ ∂t v Newton = −ρ 0 + ∇ · f ρ m , (8.70) where ρ m is the mass density of the nuclei. With the nuclear charge Ze and mass M we obtain which is solved by with ∂ 2 ∂t 2 δρ + ∼ = − Ze M ∇ · f = −Ze M 4π ρ0 + δρ + , (8.71) Ω = δρ + (r, t) = δρ + (r) e −iΩt (8.72) √ 4π Ze M ρ0 + = √ 4π Z2 e 2 M n0 +, (8.73) where n 0 + is the concentration of nuclei (or ions). We thus obtain optical phonons with completely flat dispersion, i.e., we find the same frequency Ω for all vibrations. One can also obtain the coupling strength g q . It is clear that it will be controlled by the Coulomb interaction between electrons and fluctuations δρ + in the jellium charge density. We refer to the lecture notes on many-particle theory and only give the result: 1 V |g q| 2 = Ω 2 V C(q). (8.74) Consequently, the electron-electron interaction due to phonon exchange becomes V ph (q, iν n ) = 1 V |g q| 2 D 0 q(iν n ) = Ω 2 V C(q) 2Ω (iν n ) 2 − Ω 2 = V Ω 2 C(q) (iν n ) 2 − Ω 2 . (8.75) It is thus proportional to the bare Coulomb interaction, with an additional frequency-dependent factor. retarded form reads V R ph(q, ν) = V ph (q, iν n → ν + i0 + ) = V C (q) Ω 2 (ν + i0 + ) 2 − Ω 2 = V C(q) The Ω 2 ν 2 − Ω 2 + i0 + sgn ν , (8.76) where we have used 2ν i0 + = i0 + sgn ν and have neglected the square of infinitesimal quantities. 78
8.4 Effective interaction between electrons Combining the bare Coulomb interaction and the bare interaction due to phonon exchange, calculated for the jellium model, we obtain the bare effective interaction between electrons, The retarded form is V eff (q, iν n ) := V C (q) + V ph (q, iν n ) = V C (q) + V C (q) = V C (q) Ω 2 (iν n ) 2 − Ω 2 V R eff(q, ν) = V eff (q, iν n → ν + i0 + ) = V C (q) (iν n ) 2 (iν n ) 2 − Ω 2 . (8.77) ν 2 ν 2 − Ω 2 + i0 + sgn ν . (8.78) This expression is real except at ν = Ω and has a pole there. Moreover, V R eff is proportional to V C with a negative prefactor as long as ν < Ω. V R eff V( q) c 0 Ω ν The effective interaction is thus attractive for ν < Ω. The exchange of phonons overcompensates the repulsive Coulomb interaction. On the other hand, for ν → 0, the effective interaction vanishes. This means that in a quasi-static situation the electrons do not see each other at all. What happens physically is that the electrons polarize the (jellium) charge density of the nuclei. The nuclei have a high inertial mass, their reaction to a perturbation has a typical time scale of 1/Ω or a frequency scale of Ω. For processes slow compared to Ω, the nuclei can completely screen the electron charge, forming a polaron, which is charge-neutral. For frequencies ν > 0, we have to think in terms of the response of the system to a test electron oscillating with frequency ν. The jellium acts as an oscillator with eigenfrequency Ω. At the present level of approximation it is an undamped oscillator. The jellium oscillator is excited at the frequency ν. For 0 < ν < Ω, it is driven below its eigenfrequency and thus oscillates in phase with the test electron. The amplitude, i.e., the jellium polarization, is enhanced compared to the ν = 0 limit simply because the system is closer to the resonance at Ω. Therefore, the oscillating electron charge is overscreened. On the other hand, for ν > Ω the jellium oscillator is driven above its eigenfrequency and thus follows the test electron with a phase difference of π. Thus the electron charge is not screened at all but rather enhanced and the interaction is more strongly repulsive than the pure Coulomb interaction. Screening of the effective interaction From our discussion of the Coulomb interaction we know that the real interaction between two electrons in a metal is strongly screened at all except very short distances. This screening is well described within the RPA. We now apply the RPA to the effective interaction derived above. We define −V eff ≡ = + (8.79) 79
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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
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: Note that summing up the more and m
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
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,
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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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