Carsten Timm: Theory of superconductivity

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The constant is irrelevant for the following derivation and is omitted from now on. Since H BCS is bilinear in c, c † it describes a non-interacting effective system. But what is unusual is that H BCS contains terms of the form cc and c † c † , which do not conserve the electron number. We thus expect that the eigenstates of H BCS do not have a sharp electron number. We had already seen that the BCS ground state has this property. This is a bit strange since superpositions of states with different electron numbers are never observed. One can formulate the theory of superconductivity in terms of states with fixed electron number, but this formulation is cumbersome and we will not pursue it here. To diagonalize H BCS , we introduce new fermionic operators, which are linear combinations of electron creation and annihilation operators, ( γk↑ γ † −k,↓ ) ( u ∗ = k −v k v ∗ k u k ) ( ck↑ c † −k,↓ ) . (10.8) This mapping is called Bogoliubov (or Bogoliubov-Valatin) transformation. Again, it is not clear yet that u k , v k are related to the previously introduced quantities denoted by the same symbols. For the γ to satisfy fermionic anticommutation relations, we require { } γ k↑ , γ † k↑ = γ k↑ γ † k↑ + γ† k↑ γ k↑ = u ∗ ku k c k↑ c † k↑ − u∗ kv ∗ k c k↑ c −k,↓ − v k u k c † −k,↓ c† k↑ + v kv ∗ k c † −k,↓ c −k,↓ + u k u ∗ k c † k↑ c k↑ − u k v k c † k↑ c† −k,↓ − v∗ ku ∗ k c −k,↓ c k↑ + vkv ∗ k c −k,↓ c † { −k,↓ } } = |u k | 2 c k↑ , c † k↑ −u ∗ kvk ∗ {c k↑ , c −k,↓ } −v k u k {c { } † −k,↓ } {{ } } {{ } , c† k↑ + |v k | 2 c † −k,↓ , c −k,↓ } {{ } } {{ } = 0 = 1 = 0 = 1 = |u k | 2 + |v k | 2 ! = 1. (10.9) Using this constraint, we find the inverse transformation, ( ) ( ) ( ck↑ uk v = k γk↑ c † −k,↓ −v ∗ k u ∗ k γ † −k,↓ ) . (10.10) Insertion into H BCS yields H BCS = ∑ ) ( ) ( ) ) {ξ k (u ∗ kγ † k↑ + v∗ kγ −k,↓ u k γ k↑ + v k γ † −k,↓ + ξ k −vkγ ∗ k↑ + u ∗ kγ † −k,↓ (−v k γ † k↑ + u kγ −k,↓ k ) ( ) ) ( )} −∆ ∗ k (−v k γ † k↑ + u kγ −k,↓ u k γ k↑ + v k γ † −k,↓ − ∆ k (u ∗ kγ † k↑ + v∗ kγ −k,↓ −vkγ ∗ k↑ + u ∗ kγ † −k,↓ = ∑ {( ) ξ k |u k | 2 − ξ k |v k | 2 + ∆ ∗ kv k u k + ∆ k u ∗ kvk ∗ γ † k↑ γ k↑ k ( + ( + + ) −ξ k |v k | 2 + ξ k |u k | 2 + ∆ ∗ ku k v k + ∆ k vku ∗ ∗ k γ † −k,↓ γ −k,↓ ξ k u ∗ kv k + ξ k u ∗ kv k + ∆ ∗ kvk 2 − ∆ k (u ∗ k) 2) γ † k↑ γ† −k,↓ (ξ k vku ∗ k + ξ k vku ∗ k − ∆ ∗ ku 2 k + ∆ k (vk) ∗ 2) } γ −k,↓ γ k↑ + const. (10.11) The coefficients u k , v k should now be chosen such that the γγ and γ † γ † terms vanish. This requires Writing 2ξ k u ∗ kv k + ∆ ∗ kv 2 k − ∆ k (u ∗ k) 2 = 0. (10.12) ∆ k = |∆ k | e iϕ k , (10.13) u k = |u k | e iα k , (10.14) v k = |v k | e iβ k , (10.15) 90
we obtain ( ) 2ξ k |u k | |v k | e i(β k−α k ) + |∆ k | |v k | 2 e i(2β k−ϕ k ) − |u k | 2 e i(ϕ k−2α k ) = 0. (10.16) A special solution of this equation (we do not require the general solution) is given by From the last equation we obtain α k = 0, (10.17) β k = ϕ k , (10.18) ( 2ξ k |u k | |v k | + |∆ k | |v k | 2 − |u k | 2) = 0. (10.19) ( 4ξk 2 |u k | 2 |v k | 2 = |∆ k | 2 |v k | 4 − 2 |v k | 2 |u k | 2 + |u k | 4) (10.20) ⇒ ( 4 ξk 2 + |∆ k | 2) ( |u k | 2 |v k | 2 = |∆ k | 2 |v k | 4 + 2 |v k | 2 |u k | 2 + |u k | 4) ( = |∆ k | 2 |v k | 2 + |u k | 2) 2 = |∆k | 2 (10.21) |∆ k | ⇒ |u k | |v k | = (10.22) 2 √ξk 2 + |∆ k| 2 so that |u k | 2 − |v k | 2 = 2ξ k |u k | |v k | |∆ k | = ξ k √ξ 2 k + |∆ k| 2 . (10.23) Together with |u k | 2 + |v k | 2 = 1 we thus find ⎛ ⎞ |u k | 2 = 1 ξ k ⎝1 + √ ⎠ , (10.24) 2 ξk 2 + |∆ k| 2 ⎛ ⎞ |v k | 2 = 1 ξ k ⎝1 − √ ⎠ . (10.25) 2 ξk 2 + |∆ k| 2 Restoring the phases in Eq. (10.22), we also conclude that ∆ k u k v k = 2 √ξ . (10.26) k 2 + |∆ k| 2 The BCS Hamiltonian now reads, ignoring a constant, ⎛ ⎞ H BCS = ∑ ⎝ ξk 2 √ + |∆ k | 2 √ ⎠ ( ) γ † k ξk 2 + |∆ k| 2 ξk 2 + |∆ k| 2 k↑ γ k↑ + γ † −k,↓ γ −k,↓ = ∑ √ ( ) ξk 2 + |∆ k| 2 γ † k↑ γ k↑ + γ † −k,↓ γ −k,↓ . (10.27) k Using ξ −k = ξ k and the plausible assumption |∆ −k | = |∆ k |, we obtain the simple form H BCS = ∑ kσ E k γ † kσ γ kσ (10.28) with the dispersion E k := √ ξ 2 k + |∆ k| 2 . (10.29) 91
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Theory of Superconductivity Carsten
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Contents 1 Introduction 5 1.1 Scope
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1 Introduction 1.1 Scope Supercondu
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2 Basic experiments In this chapter
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Superconductivity with rather high
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• In 1979, Frank Steglich et al.
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3 Bose-Einstein condensation In thi
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We note the identity g n (y) = ∞
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We find a phase transition at T c ,
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gives a reasonable approximation fo
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We now consider the force F = −eE
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5 Electrodynamics of superconductor
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Thus the supercurrent flows in the
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where φ j is the polar angle of el
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with the penetration depth √ √
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N s is the total number of particle
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6.2 Ginzburg-Landau theory for neut
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scope of this course, though. The i
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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: 10 BCS theory The variational ansat
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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