Carsten Timm: Theory of superconductivity

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T 0 AFM SC hole doping x A glance at typical phase diagrams shows that the undoped cuprates tend to be antiferromagnetic. Weak hole doping or slightly stronger electron doping destroy the antiferromagnetic order, and at larger doping, superconductivity emerges. At even larger doping (the “overdoped” regime), superconductivity is again suppressed. Also in many other unconventional superconductors superconductivity is found in the vicinity of but rarely coexisting with magnetic order. This is true for most pnictide and heavy-fermion superconductors. The vicinity of a magnetically ordered phase makes itself felt by strong magnetic fluctuations and strong, but short-range, spin correlations. These are seen as an enhanced spin susceptibility. At a magnetic second-order phase transition, the static spin susceptibility χ q diverges at q = Q, where Q is the ordering vector. It is Q = 0 for ferromagnetic order and Q = (π/a, π/a) for checkerboard (Néel) order on a square lattice. Even some distance from the transition or at non-zero frequencies ν, the susceptibility χ q (ν) tends to have a maximum close to Q. Far away from the magnetic phase or at high frequencies this remnant of magnetic order becomes small. This discussion suggests that the exchange of spin fluctuations, which are strong close to Q, could provide the attractive interaction needed for Cooper pairing. The Hubbard model The two-dimensional, single-band, repulsive Hubbard model is thought (by many experts, not by everyone) to be the simplest model that captures the main physics of the cuprates. The Hamiltonian reads, in real space, H = − ∑ ijσ t ij c † iσ c jσ + U ∑ i c † i↑ c i↑c † i↓ c i↓ (12.18) with U > 0 and, in momentum space, H = ∑ kσ ϵ k c † kσ c kσ + U ∑ c † k+q,↑ N c† k ′ −q,↓ c k ′ ↓c k↑ . (12.19) kk ′ q Also, the undoped cuprate parent compounds have, from simple counting, an odd number of electrons per unit cell. Thus for them the single band must be half-filled, while for doped cuprates it is still close to half filling. The underlying lattice in real space is a two-dimensional square lattice with each site i corresponding to a Cu + ion. Cu + O 2− The transverse spin susceptibility is defined by χ +− (q, τ) = − ⟨ T τ S + (q, τ) S − (−q, 0) ⟩ , (12.20) 124
where τ is the imaginary time, T τ is the time-ordering directive, and with S ± (q, τ) := S x (q, τ) ± iS y (q, τ) (12.21) S α (q, τ) := √ 1 ∑ c † k+q,σ (τ) σα σσ ′ N 2 c kσ ′(τ) (12.22) kσσ ′ are electron-spin operators. σ = (σ x , σ y , σ z ) is the vector of Pauli matrices. The susceptibility can be rewritten as χ +− (q, τ) = − 1 ∑ ⟨ ) ( ) ⟩ T τ (c † k+q,↑ N c k↓ (τ) c † k ′ −q,↓ c k ′ ↑ (0) . (12.23) kk ′ In the non-interacting limit of U → 0, the average of four fermionic operators can be written in terms of products of averages of two operators (Wick’s theorem). The resulting bare susceptibility reads χ +− 0 (q, τ) = − 1 ∑ ⟨ ⟩ ⟩ T τ c † k+q,↑ N (τ) c k+q,↑(0) ⟨T τ c k↓ (τ) c † k↓ (0) 0 0 k = 1 ∑ ⟩ ⟩ ⟨T τ c k+q,↑ (0) c † k+q,↑ ⟨T N (τ) τ c k↓ (τ) c † k↓ (0) 0 0 k = 1 ∑ G 0 N k+q,↑(−τ) Gk↓(τ). 0 (12.24) The Fourier transform as a function of the bosonic Matsubara frequency iν n is χ +− 0 (q, iν n ) = ∫ β 0 dτ e iν nτ χ +− 0 (q, τ) ∫ β k = 1 dτ e ∑ iν 1 ∑ nτ e −iω′ n (−τ) G N β k+q,↑(iω 0 n) ′ 1 ∑ e −iωnτ G β k↓(iω 0 n ) 0 k iω n ′ iω n = 1 ∑ 1 ∑ N β 2 β δ νn +ω n ′ ,ω n Gk+q,↑(iω 0 n) ′ Gk↓(iω 0 n ) = 1 ∑ 1 ∑ G N β k+q,↑(iω 0 n − iν n ) Gk↓(iω 0 n ) k iω n ,iω n ′ k iω n = 1 ∑ 1 ∑ 1 1 . (12.25) N β iω iω n − iν n − ξ k+q iω n − ξ k n k This expression can be written in a more symmetric form by making use of the identity ξ k = ξ −k and replacing the summation variables k by −k − q and iω n by iω n + iν n . The result is χ +− 0 (q, iν n ) = 1 ∑ 1 ∑ 1 1 . (12.26) N β iω k iω n − ξ k iω n + iν n − ξ k+q n The Matsubara frequency sum can be evaluated using methods from complex analysis. We here give the result without proof, χ +− 0 (q, iν n ) = 1 ∑ n F (ξ k ) − n F (ξ k+q ) . (12.27) N iν n + ξ k − ξ k+q k Note that the same result is found for the bare charge susceptibility except for a spin factor of 2. Diagramatically, the result can be represented by the bubble diagram χ +− 0 (q, iν n ) = − 1 2 Π 0(q, iν n ) = − k , , iω n . (12.28) k+ q , , iω n + iν n 125
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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
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6.2 Ginzburg-Landau theory for neut
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scope of this course, though. The i
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As above, we keep only terms up to
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Within the mean-field theories we h
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Including the superconducting contr
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The Gibbs free energy of the domain
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The magnetic flux Φ through this l
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Another useful quantity is the free
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We obtain e −ik yy (− 2 2m ∗
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has a simple interpretation: It is
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This is a Gaussian average since F
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Note that in two dimensions the cur
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Thus the energy is ∫ E 2 = 2E cor
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The energy of an isolated vortex-an
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Next, we have to express Z ′ in t
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quasi−long−range order free vor
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which is valid within the film and
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attraction of the magnetic monopole
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Finally, the voltage measured for a
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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
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: The fact that the gap closes at som
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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