Carsten Timm: Theory of superconductivity

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where O(λ 3 /V ) is a correction of order λ 3 /V ≪ 1. Thus, by Taylor expansion, N ϵ = V λ 3 g 3/2(1) − O(1) = V ζ(3/2) − O(1). (3.25) λ3 The intensive term O(1) can be neglected compared to the extensive one so that N ϵ ∼ V = ζ(3/2). (3.26) λ3 This is the maximum possible value at temperature T . Furthermore, N 0 = y/(1 − y) is solved by For y to be very close to unity, N 0 must be N 0 ≫ 1. Since y = N 0 N 0 − 1 = 1 1 + 1/N 0 . (3.27) must be positive, we require N 0 = N − N ϵ ∼ V = N − ζ(3/2) (3.28) λ3 We conclude that the fraction of particles in excited states is The fraction of particles in the ground state is then N V > ζ(3/2) λ 3 (3.29) ⇒ T < T c . (3.30) N ϵ N ∼ = V ( ) 3/2 Nλ 3 ζ(3/2) = λ3 (T c ) T λ 3 (T ) = . (3.31) T c In summery, we find in the thermodynamic limit ( ) 3/2 N 0 N ∼ T = 1 − . (3.32) T c (a) for T > T c : (b) for T < T c : N ϵ N ∼ = 1, ( ) 3/2 N ϵ N ∼ T = , T c N 0 N ≪ 1, (3.33) ( ) 3/2 N 0 N ∼ T = 1 − . (3.34) T c 1 N / N 0 N / N ε 0 0 0.5 1 T / T c 16
We find a phase transition at T c , below which a macroscopic fraction of the particles occupy the same singleparticle quantum state. This fraction of particles is said to form a condensate. While it is remarkable that Bose-Einstein condensation happens in a non-interacting gas, the BEC is analogous to the condensate in strongly interacting superfluid He-4 and, with some added twists, in superfluid He-3 and in superconductors. We can now use the partition function to derive equations of state. As an example, we consider the pressure p = − ∂ϕ ∂V = + ∂ ∂V k BT ln Z = k BT λ 3 g 5/2(y) (3.35) (ϕ is the grand-canonical potential). We notice that only the excited states contribute to the pressure. The term − ln(1 − y) from the ground state drops out since it is volume-independant. This is plausible since particles in the condensate have vanishing kinetic energy. For T > T c , we can find y and thus p numerically. For T < T c we may set y = 1 and obtain Remember that for the classical ideal gas at constant volume we find p = k BT λ 3 ζ(5/2) ∝ T 5/2 . (3.36) p ∝ T. (3.37) For the BEC, the pressure drops more rapidly since more and more particles condense and thus no longer contribute to the pressure. 17
Page 1 and 2: Theory of Superconductivity Carsten
Page 3 and 4: Contents 1 Introduction 5 1.1 Scope
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: We note the identity g n (y) = ∞
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
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Carsten Timm: Theory of superconductivity

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