Carsten Timm: Theory of superconductivity

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Since not all electrons are reflected and in addition some holes are generated, where does the missing charge go? The quasiparticle states in the superconductor are evanescent and thus cannot accommodate the missing charge. The only possible explanation is that the charge is added to the superconducting condensate, i.e., that additional Cooper pairs are formed. (The whole process can also run backwards, in which case Cooper pairs are removed.) Recall that the condensate does not have a sharp electron number and can therefore absorb or emit electrons without changing the state. But it can only absorb or emit electrons in pairs. The emerging picture is that if an incoming electron is not specularly reflected, a Cooper pair is created, which requires a second electron. This second electron is taken from the normal region, creating a hole, which, as we have seen, travels in the direction the original electron was coming from. N S e h Cooper pair Andreev bound states An interesting situation arises if a normal region is delimited by superconductors on two sides. We here only qualitatively consider a superconductor-normal-superconductor (SNS) hetero structure. Similar effects can also occur for example in the normal core of a vortex. If no voltage is applied between the two superconductors, an electron in the normal region, with energy within the gap, is Andreev reflected as a hole at one interface. It is then Andreev reflected as an electron at the other interface. It is plausible that multiple reflections can lead to the formation of bound states. The real physics is somewhat more complicated since the electron is also partially specularly reflected as an electron. It is conceptually clear, though, how to describe Andreev bound states within the Bogoliubov-de Gennes formalism: We just have to satisfy continuity conditions for both interfaces. S N S 2∆ 0 electron hole 2∆ 0 If Andreev reflection dominates, as assumed for the sketch above, a Cooper pair is emitted into the right superconductor for every reflection at the right interface. Conversely, a Cooper pair is absorbed from the left superconductor for every reflection at the left interface. This corresponds to a supercurrent through the device. Andreev bound states thus offer a microscopic description of the Josephson effect in superconductor-normalsuperconductor junctions. If we apply a voltage V , the situation changes dramatically: If an electron moving, say, to the right, increases its kinetic energy by eV due to the bias voltage, an Andreev reflected hole traveling to the left also increases its kinetic energy by eV since it carries the opposite charge. An electron/hole Andreev-reflected multiple times can thus gain arbitrarily high energies for any non-vanishing bias voltage. 118
S N S 2∆ 0 2eV eV In particular, an electron-like quasiparticle from an occupied state below the gap in, say, the left superconductor can after multiple reflections emerge in a previously unoccupied state above the gap in the right superconductor. A new transport channel becomes available whenever the full gap 2∆ 0 is an odd integer multiple of eV : 2∆ 0 = (2n + 1) eV, n = 0, 1, . . . (11.104) ⇒ eV = ∆ 0 n + 1 , n = 0, 1, . . . (11.105) 2 The case n = 0 corresponds to direct quasiparticle transfer from one superconductor to the other, similar to quasiparticle tunneling in a superconductor-insulator-superconductor junction. The opening of new transport channels for n = 0, 1, . . . , i.e., at eV = 2 3 ∆ 0, 2 5 ∆ 0, 2 7 ∆ 0, . . . (11.106) leads to structures in the current-voltage characteristics below the gap, specifically to peaks in the differential conductance dI/dV . dI dV ∆ 0 2 3 ∆ 0 2 5 ∆ 0 − 2 0 2 eV ∆ 0 119
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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
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
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: with ˜k 1 := (−k 1x , k 1y , k 1
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,
Page 135: It is plausible that in a crystal t
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Carsten Timm: Theory of superconductivity

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