Carsten Timm: Theory of superconductivity

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Note that the external legs do not represent electronic Green functions but only indicate the states of incoming and outgoing electrons. The series is very similar to the one for the RPA susceptibility. Indeed, the effective interaction is V eff (q, iν n ) = U + Uχ +− 0 (q, iν n ) U + Uχ +− 0 (q, iν n ) Uχ +− 0 (q, iν n ) + · · · = U + U 2 [χ +− 0 (q, iν n ) + χ +− 0 (q, iν n ) Uχ +− 0 (q, iν n ) + · · · ] = U + U 2 χ +− RPA (q, iν n). (12.36) Typically one goes beyond the RPA at this point by including additional diagrams. In particular, also charge fluctuations are included through the charge susceptibility and the bare Green function G 0 is replaced by a selfconsistent one incorporating the effect of spin and charge fluctuations on the electronic self-energy. This leads to the fluctuation-exchange approximation (FLEX ). One could now obtain the Cooper instability due to the FLEX effective interaction in analogy to Sec. 9.1 and use a BCS mean-field theory to describe the superconducting state. However, since the system is not in the weak-coupling limit—the typical interaction times the electronic density of states is not small—one usually employs a strong-coupling generalization of BCS theory known as Eliashberg theory. Since the effective interaction is, like the spin susceptibility, strongly peaked close to (π/a, π/a), it favors d x2 −y2-wave pairing, as we have seen. The numerical result of the FLEX for T c and for the superfluid density n s are sketched here: T c n s 0 doping x The curve for T c vs. doping does not yet look like the experimentally observed dome. There are several aspects that make the region of weak doping (underdoping) difficult to treat theoretically. One is indicated in the sketch: n s is strongly reduced, which indicates that superconductivity may be in some sense fragile in this regime. Furthermore, the cuprates are nearly two-dimensional solids. In fact we have used a two-dimensional model so far. If we take this seriously, we know from chapter 7 that any mean-field theory, which Eliashberg theory with FLEX effective interaction still is, fails miserably. Instead, we expect a BKT transition at a strongly reduced critical temperature. We reinterpret the FLEX critical termperature as the mean-field termperature T MF and the FLEX superfluid density as the unrenormalized superfluid density n 0 s. We have seen in chapter 7 that the bare stiffness K 0 is proportional to n 0 s. The BKT transition temperature T c is defined by K(l → ∞) = 2/π. It is thus reduced by small n 0 s, corresponding to a small initial value K(0) ≡ K 0 . This is physically clear: Small stiffness makes it easy to create vortex-antivortex pairs. T c is of course also reduced by T MF and can never be larger than T MF . A BKT theory on top of the FLEX gives the following phase diagram, which is in qualitative agreement with experiments: T ns T MF vortex fluctuations superconductor T c 0 x This scenario is consistent with the Nernst effect (an electric field measured normal to both an applied magnetic field and a temperature gradient) in underdoped cuprates, which is interpreted in terms of free vortices in a broad temperature range, which we would understand as the range from T c to T MF . 128
Also note that spin fluctuations strongly affect the electronic properties in underdoped cuprates up to a temperature T ∗ significantly higher than T MF . For example, below T ∗ the electronic density of states close to the Fermi energy is suppressed compared to the result of band-structure calculations. The FLEX describes this effect qualitatively correctly. This suppression is the well-known pseudogap. Quantum critical point The previously discussed approach relies on a resummation of a perturbative series in U/t. This is questionable for cuprates, where U/t is on the order of 3. Many different approaches have been put forward that supposedly work in this strong-coupling regime. They emphasize different aspects of the cuprates, showing that it is not even clear which ingredients are the most important for understanding the phase diagram. Here we will review a lign of thought represented by Chandra Varma and Subir Sachdev, among others. Its starting point is an analysis of the normal region of the phase diagram. T T * ’’strange metal’’ ρ ~ T 0 AFM pseudo− gap superconductor Fermi liquid Roughly speaking, there are three regimes in the normal-conducting state: • a pseudo-gap regime below T ∗ at underdoping, in which the electronic density of states at low energies is suppressed, • a “strange-metal” regime above the superconducting dome, without a clear suppression of the density of states but with unusual temperature and energy dependencies of various observables, for example a resistivity linear in temperature, ρ ∝ T , • an apparently ordinary normal-metal (Fermi-liquid) regime at overdoping, with standard ρ ∝ const + T 2 dependence. The two crossover lines look very much like what one expects to find for a quantum critical point (QCP), i.e., a phase transition at zero temperature. T affected by phase I quantum critical affected by phase II x phase I QCP δ c phase II tuning parameter δ The regions to the left and right have a characteristic energy scale ϵ(δ) inherited from the ground state (T = 0), which dominates the thermal fluctuations. The energy scales go to zero at the QCP, i.e., for δ → δ c from both sides. Right at the QCP there then is no energy scale and energy or temperature-dependent quantities have to be power laws. The emerging idea is that the superconducting dome hides a QCP between antiferromagnetic (spindensity-wave) and paramagnetic order at T = 0. The spin fluctuations associated with this QCP become stronger 129
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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
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It will be useful to write the elec
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(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 and 124: The fact that the gap closes at som
Page 125 and 126: where τ is the imaginary time, T
Page 127: At lower temperatures, details beco
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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