Carsten Timm: Theory of superconductivity

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The fugacity has to be chosen to give the correct particle number N ! = ∑ k ⟨n k ⟩ = ∑ k 1 e β(ϵ k−µ) − 1 = ∑ k 1 y −1 e βϵ k − 1 . (3.6) Since ⟨n k ⟩ must be non-negative, µ must satisfy µ ≤ ϵ k ∀ k. (3.7) For a free particle, the lowest possible eigenenergy is ϵ k = 0 for k = 0 so that we obtain µ ≤ 0. For a large volume V , the allowed vectors k become dense and we can replace the sums over k by integrals according to ∑ ∫ · · · → V k d 3 k (2π) 3 · · · = 2πV ∫∞ h 3 (2m)3/2 dϵ √ ϵ · · · (3.8) In the last equation we have used the density of states (DOS) of free particles in three dimensions. This replacement contains a fatal mistake, though. The DOS for ϵ = 0 vanishes so that any particles in the state with k = 0, ϵ k = 0 do not contribute to the results. But that is the ground state! For T = 0 all bosons should be in this state. We thus expect incorrect results at low temperatures. Our mistake was that Eq. (3.8) does not hold if the fraction of bosons in the k = 0 ground state is macroscopic, i.e., if N 0 /N := ⟨n 0 ⟩ /N remains finite for large V . To correct this, we treat the k = 0 state explicitly (the same would be necessary for any state with macroscopic occupation). We write Then ∑ · · · → 2πV ∫∞ h 3 (2m)3/2 dϵ √ ϵ · · · + (k = 0 term). (3.9) k ln Z = − 2πV ∫∞ h 3 (2m)3/2 dϵ √ ϵ ln ( 1 − ye −βϵ) − ln(1 − y) with Defining 0 0 0 by parts = 2πV h 3 N = 2πV ∫∞ h 3 (2m)3/2 dϵ √ 1 ϵ y −1 e −βϵ − 1 + 1 y −1 − 1 = 2πV ∫∞ h 3 (2m)3/2 g n (y) := 1 Γ(n) ∫ ∞ 0 dx x n−1 y −1 e x − 1 0 2 ∫∞ (2m)3/2 3 β dϵ 0 dϵ 0 ϵ 3/2 y −1 e βϵ − ln(1 − y) − 1 (3.10) √ ϵ y −1 e −βϵ − 1 + y 1 − y . (3.11) (3.12) for 0 ≤ y ≤ 1 and n ∈ R, and the thermal wavelength √ λ := h 2 2πmk B T , (3.13) we obtain ln Z = V λ 3 g 5/2(y) − ln(1 − y), (3.14) N = V λ 3 g 3/2(y) + y . (3.15) } {{ } 1 − y } {{ } =: N ϵ =: N 0 14
We note the identity g n (y) = ∞∑ k=1 y k k n , (3.16) which implies g n (0) = 0, (3.17) ∞∑ 1 g n (1) = = ζ(n) kn for n > 1 (3.18) k=1 with the Riemann zeta function ζ(x). Furthermore, g n (y) increases monotonically in y for y ∈ [0, 1[. g n ( y) g 3/2 2 1 g g 5/2 oo 0 0 0.5 1 y We now have to eliminate the fugacity y from Eqs. (3.14) and (3.15) to obtain Z as a function of the particle number N. In Eq. (3.14), the first term is the number of particles in excited states (ϵ k > 0), whereas the second term is the number of particles in the ground state. We consider two cases: If y is not very close to unity (specifically, if 1 − y ≫ λ 3 /V ), N 0 = y/(1 − y) is on the order of unity, whereas N ϵ is an extensive quantity. Thus N 0 can be neglected and we get N ∼ = N ϵ = V λ 3 g 3/2(y). (3.19) Since g 3/2 ≤ ζ(3/2) ≈ 2.612, this equation can only be solved for the fugacity y if the concentration satisfies To have 1 − y ≫ λ 3 /V in the thermodynamic limit we require, more strictly, N V ≤ ζ(3/2) λ 3 . (3.20) N V < ζ(3/2) λ 3 . (3.21) Note that λ 3 ∝ T −3/2 increases with decreasing temperature. Hence, at a critical temperature T c , the inequality is no longer fulfilled. From N ! ζ(3/2) = V ( ) 3/2 (3.22) h 2 2πmk B T c we obtain k B T c = 1 h 2 ( ) 2/3 N . (3.23) [ζ(3/2)] 2/3 2πm V If, on the other hand, y is very close to unity, N 0 cannot be neglected. Also, in this case we find N ϵ = V λ 3 g 3/2(y) = V λ 3 g 3/2(1 − O(λ 3 /V )), (3.24) 15
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: 3 Bose-Einstein condensation In thi
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
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Carsten Timm: Theory of superconductivity

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