Carsten Timm: Theory of superconductivity

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y B j As noted above, the magnetic flux starts to penetrate the superconductor at the lower critical field H c1 . Furthermore, since flux expulsion in type-II superconductors need not be perfect, they can withstand stronger magnetic fields than type-I superconductors, up to an upper critical field H c2 > H c , H c1 . x H H c2 ( T) Shubnikov (vortex) phase H c1 ( T) 2nd order normal metal Meißner phase 2nd order T c T We will now review the basic ideas of Abrikosov’s approach. Abrikosov’s results are quantitatively valid only close to H c2 since he assumed the magnetic flux density B to be uniform, which is valid for where λ ≫ l, (6.111) l = √ Φ0 B (6.112) is the typical distance between vortices (B/Φ 0 is the two-dimensional concentration of vortex lines). For B = Bẑ = const = Hẑ we can choose the gauge A = ŷ Hx. Then the first Ginzburg-Landau equation becomes (note m ∗ = 2m) ( 1 2m ∗ i ∇ + 2eH 2 ŷx) ψ + α ψ + β |ψ| 2 ψ = 0. (6.113) c Slightly below H c2 , |ψ| is expected to be small (this should be checked!) so that we can neglect the non-linear term. Introducing the cyclotron frequency of a superconductor, ω c := 2eH m ∗ c , (6.114) we obtain ( − 2 2m ∗ ∇2 − iω c x ∂ ∂y + 1 ) 2 m∗ ωc 2 x 2 ψ(r) = −α ψ(r). (6.115) }{{} > 0 This looks very much like a Schrödinger equation and from the derivation it has to be the Schrödinger equation for a particle of mass m ∗ and charge q = −2e in a uniform magnetic field H. This well-known problem is solved by the ansatz ψ(x, y) = e ik yy f(x). (6.116) 48
We obtain e −ik yy (− 2 2m ∗ ∇2 − iω c x ∂ ∂y + 1 ) 2 m∗ ωc 2 x 2 e ikyy f(x) = − 2 d 2 f 2m ∗ dx 2 + 2 ky 2 2m ∗ f − iω c x ik y f + 1 2 m∗ ωc 2 x 2 f = − 2 2m ∗ d 2 f dx 2 + 1 2 m∗ ω 2 c ( x 2 + 2 k y m ∗ ω c + 2 k 2 y (m ∗ ) 2 ω 2 c ) f = −αf. (6.117) Defining x 0 := −k y /m ∗ ω c , we get − 2 2m ∗ d 2 f dx 2 + 1 2 m∗ ω 2 c (x − x 0 ) 2 f = −αf. (6.118) This is the Schrödinger equation for a one-dimensional harmonic oscillator with shifted minimum. Thus we obtain solutions f n (x) as shifted harmonic-oscillator eigenfunctions for ( −α = ω c n + 1 ) , n = 0, 1, 2, . . . (6.119) 2 Solution can only exist if −α = −α ′ (T − T c ) = α ′ (T c − T ) ≥ ω c 2 = eH m ∗ c . (6.120) Keeping T ≤ T c fixed, a solution can thus only exist for H ≤ H c2 with the upper critical field Using ξ 2 = − 2 /2m ∗ α, we obtain H c2 (T ) = H c2 = − m∗ c e α. (6.121) c 2e ξ 2 (T ) = Φ 0 2π ξ 2 (T ) . (6.122) Note that since ξ ∼ 1/ √ T c − T close to T c , H c2 (T ) sets in linearly, as shown in the above sketch. H c2 should be compared to the thermodynamic critical field H c , Hc 2 = − mc2 1 2e 2 λ 2 α = 2 c 2 1 8e 2 λ 2 ξ 2 = 1 ( ) 2 hc 1 8π 2 2e λ 2 ξ 2 Φ 0 ⇒ H c = 2π √ 2 λξ = H c2 ξ √ 2 λ = H c2 √ 2 κ ⇒ H c2 = √ 2 κ H c . (6.123) For κ = 1/ √ 2, H c2 equals H c . This is the transition between a type-II and a type-I superconductor. So far, our considerations have not told us what the state for H H c2 actually looks like. To find out, one in principle has to solve the full, not the linearized, Ginzburg-Landau equation. (We have seen that the linearized equation is equivalent to the Schrödinger equation for a particle in a uniform magnetic field. The solutions are known: The eigenfunctions have uniform amplitude and the eigenenergies form discrete Landau levels, very different from what is observed in the Shubnikov phase.) Abrikosov did this approximately using a variational approach. He used linear combinations of solutions of the linearized equation as the variational ansatz and assumed that the solution is periodic in two dimensions, up to a plane wave. We call the periods in the x- and y-directions a x and a y , respectively. The function ψ n (x, y) = e ikyy f n (x) (6.124) has the period a y in y if k y = 2π a y q, q ∈ Z. (6.125) 49
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: Another useful quantity is the free
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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