Carsten Timm: Theory of superconductivity

More documents

Recommendations

Info

Then the harmonic oscillator is centered at x 0 = − 2π m ∗ q = − Φ 0 q. (6.126) ω c a y Ha y Since the lowest-energy solution is obtained for n = 0 (the ground state of the harmonic oscillator), Abrikosov only considered the n = 0 solutions ( ψ 0 (x, y) = exp iq 2πy ) ( f 0 (x) = C a y }{{} exp iq 2πy ) ( exp − 1 m ∗ [ ω c x + Φ 0 q a y 2 Ha y normalization } {{ } −x 0 ] ) 2 . (6.127) In the Gauss function we find the quantity as long as H ≈ H c2 . Thus m ∗ = c ω c 2eH ≈ c = − 2 2e H c2 2m ∗ α = ξ2 (T ), (6.128) ( ψ 0 (x, y) = C exp iq 2πy ) ( exp − 1 [ a y 2ξ 2 x + Φ ] ) 2 0 q . (6.129) Ha y This is a set of functions enumerated by q ∈ Z. Abrikosov considered linear combinations ∞∑ ( ψ(x, y) = C q exp iq 2πy ) ( exp − 1 [ a y 2ξ 2 x + Φ ] ) 2 0 q . (6.130) Ha y q=−∞ To be periodic in x with period a x , up to a plane wave (the corresponding discussion in Tinkham’s book is not fully correct), this ansatz has to satisfy ∞∑ ( ψ(x + a x , y) = C q exp iq 2πy ) ( exp − 1 [ a y 2ξ 2 x + a x + Φ ] ) 2 0 q Ha y = q=−∞ ∞∑ q=−∞ ( C q exp iq 2πy ) exp a y ( − 1 2ξ 2 [ x + Φ 0 Ha y ( q + Ha y Φ 0 a x )] 2 ) ! ∝ ψ(x, y) ∀ x, y. (6.131) This requires Ha x a y =: ν ∈ N. (6.132) Φ 0 (Note that this quantity is positive.) Then ∞∑ ( ψ(x + a x , y) = C q exp iq 2πy ) ( exp − 1 [ a q=−∞ y 2ξ 2 x + Φ ] ) 2 0 (q + ν) Ha y ( = exp iν 2πy ) ∑ ∞ ( C q−ν exp iq 2πy ) ( exp − 1 [ a y a y 2ξ 2 x + Φ ] ) 2 0 q . (6.133) Ha y q=−∞ This equals ψ(x, y) up to a plane-wave factor if C q−ν = C q ∀ q, (6.134) i.e., if C q is periodic. Abrikosov considered the case ν = 1, which leads to a square lattice. The lowest-free-energy solution is obtained for ν = 2 and C 1 = iC 0 , which gives a triangular lattice, sketched above. Note that ν = Ha xa y Φ 0 (6.135) 50
has a simple interpretation: It is the number of flux quanta passing through a rectangular unit cell of the vortex lattice. The vortex lattice is a rather complex system: It is a lattice of interacting lines with a line tension ϵ v . At non-zero temperatures, the vortices fluctuate, which can lead to the melting of the vortex lattice. The resulting vortex liquid can be pictured as a pot of boiling spaghetti, with the constraint that the vortex lines must either terminate at the surface or form closed loops (!). Moving vortices lead to ohmic resistance, even though most of the sample is still superconducting. To complicate matters, interaction of vortices with defects (“pinning”) plays an important role. There is an extensive research literature on vortex matter, which we cannot review here. H 51
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 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: We obtain e −ik yy (− 2 2m ∗
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?