Carsten Timm: Theory of superconductivity

More documents

Recommendations

Info

Version: February 6, 2012 L A TEX & Figures: S. Lange and C. Timm
Contents 1 Introduction 5 1.1 Scope . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.2 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.3 Books . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 2 Basic experiments 7 2.1 Conventional superconductors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 2.2 Superfluid helium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 2.3 Unconventional superconductors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 2.4 Bose-Einstein condensation in dilute gases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 3 Bose-Einstein condensation 13 4 Normal metals 18 4.1 Electrons in metals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 4.2 Semiclassical theory of transport . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 5 Electrodynamics of superconductors 23 5.1 London theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 5.2 Rigidity of the superfluid state . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 5.3 Flux quantization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 5.4 Nonlocal response: Pippard theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 6 Ginzburg-Landau theory 30 6.1 Landau theory of phase transitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 6.2 Ginzburg-Landau theory for neutral superfluids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 6.3 Ginzburg-Landau theory for superconductors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 6.4 Type-I superconductors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 6.5 Type-II superconductors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 7 Superfluid and superconducting films 52 7.1 Superfluid films . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 7.2 Superconducting films . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64 8 Origin of attractive interaction 70 8.1 Reminder on Green functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70 8.2 Coulomb interaction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 8.3 Electron-phonon interaction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77 8.4 Effective interaction between electrons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79 3
Page 1: Theory of Superconductivity Carsten
Page 5 and 6: 1 Introduction 1.1 Scope Supercondu
Page 7 and 8: 2 Basic experiments In this chapter
Page 9 and 10: Superconductivity with rather high
Page 11 and 12: • In 1979, Frank Steglich et al.
Page 13 and 14: 3 Bose-Einstein condensation In thi
Page 15 and 16: We note the identity g n (y) = ∞
Page 17 and 18: We find a phase transition at T c ,
Page 19 and 20: gives a reasonable approximation fo
Page 21 and 22: We now consider the force F = −eE
Page 23 and 24: 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 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 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
show all

Carsten Timm: Theory of superconductivity

You also want an ePaper? Increase the reach of your titles

Delete template?

Save as template?