Carsten Timm: Theory of superconductivity

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Note that the expansion of F up to fourth order is only justified as long as ψ is small. The result is thus limited to temperatures not too far below T c . The scaling |ψ| ∝ (T − T c ) 1/2 is characteristic for mean-field theories. All solutions with this value of |ψ| minimize the free energy. In a given experiment, only one of them is realized. This eqilibrium state has an order parameter ψ = |ψ| e iϕ with some fixed phase ϕ. This state is clearly not invariant under rotations of the phase. We say that the global U(1) symmetry of the system is spontaneously broken since the free energy F has it but the particular equilibrium state does not. It is called U(1) symmetry since the group U(1) of unitary 1 × 1 matrices just contains phase factors e iϕ . Specific heat Since we now know the mean-field free energy as a function of temperature, we can calculate further thermodynamic variables. In particular, the free entropy is S = − ∂F ∂T . (6.10) Since the expression for F used above only includes the contributions of superconductivity or superfluidity, the entropy calculated from it will also only contain these contributions. For T ≥ T c , the mean-field free energy is F (ψ = 0) = 0 and thus we find S = 0. For T < T c we instead obtain S = − ∂ (√− ∂T F α ) = − ∂ (− α2 β ∂T β + 1 2 ∼= ∂ ∂T (α ′ ) 2 2β (T − T c) 2 = (α′ ) 2 α 2 ) = ∂ β ∂T β (T − T c) = − (α′ ) 2 α 2 2β β (T c − T ) < 0. (6.11) We find that the entropy is continuous at T = T c . By definition, this means that the phase transition is continuous, i.e., not of first order. The heat capacity of the superconductor or superfluid is which equals zero for T ≥ T c but is for T < T c . Thus the heat capacity has a jump discontinuity of C = T ∂S ∂T , (6.12) C = (α′ ) 2 β T (6.13) ∆C = − (α′ ) 2 β T c (6.14) at T c . Adding the other contributions, which are analytic at T c , the specific heat c := C/V is sketched here: c normal contribution T c T Recall that Landau theory only works close to T c . 32
6.2 Ginzburg-Landau theory for neutral superfluids To be able to describe also spatially non-uniform situations, Ginzburg and Landau had to go beyond the Landau description for a constant order parameter. To do so, they included gradients. We will first discuss the simpler case of a superfluid of electrically neutral particles (think of He-4). We define a macroscopic condensate wave function ψ(r), which is essentially given by Ψ s (r) averaged over length scales large compared to atomic distances. We expect any spatial changes of ψ(r) to cost energy—this is analogous to the energy of domain walls in magnetic systems. In the spirit of Landau theory, Ginzburg and Landau included the simplest term containing gradients of ψ and allowed by symmetry into the free energy ∫ [ F [ψ] = d 3 r α |ψ| 2 + β ] 2 |ψ|4 + γ (∇ψ) ∗ · ∇ψ , (6.15) where we have changed the definitions of α and β slightly. We require γ > 0 so that the system does not spontaneously become highly non-uniform. The above expression is a functional of ψ, also called the “Landau functional.” Calling it a “free energy” is really an abuse of language since the free energy proper is only the value assumed by F [ψ] at its minimum. If we interpret ψ(r) as the (coarse-grained) condensate wavefunction, it is natural to identify the gradient term as a kinetic energy by writing ∫ [ F [ψ] ∼ = d 3 r α |ψ| 2 + β 2 |ψ|4 + 1 ∣ ∣ ] ∣∣∣ ∣∣∣ 2 2m ∗ i ∇ψ , (6.16) where m ∗ is an effective mass of the particles forming the condensate. From F [ψ] we can derive a differential equation for ψ(r) minimizing F . The derivation is very similar to the derivation of the Lagrange equation (of the second kind) from Hamilton’s principle δS = 0 known from classical mechanics. Here, we start from the extremum principle δF = 0. We write ψ(r) = ψ 0 (r) + η(r), (6.17) where ψ 0 (r) is the as yet unknown solution and η(r) is a small displacement. Then F [ψ 0 + η] = F [ψ 0 ] ∫ + d 3 r + O(η, η ∗ ) 2 by parts = F [ψ 0 ] ∫ + d 3 r [αψ ∗ 0η + αη ∗ ψ 0 + βψ ∗ 0ψ ∗ 0ψ 0 η + βψ ∗ 0η ∗ ψ 0 ψ 0 + 2 2m ∗ (∇ψ 0) ∗ · ∇η + [αψ ∗ 0η + αη ∗ ψ 0 + βψ ∗ 0ψ ∗ 0ψ 0 η + βψ ∗ 0η ∗ ψ 0 ψ 0 − 2 2m ∗ (∇2 ψ 0 ) ∗ η − ] 2 2m ∗ (∇η)∗ · ∇ψ 0 ] 2 2m ∗ η∗ ∇ 2 ψ 0 + O(η, η ∗ ) 2 . (6.18) If ψ 0 (r) minimizes F , the terms linear in η and η ∗ must vanish for any η(r), η ∗ (r). Noting that η and η ∗ are linearly independent, this requires the prefactors of η and of η ∗ to vanish for all r. Thus we conclude that αψ 0 + βψ0ψ ∗ 0 ψ 0 − 2 2m ∗ ∇2 ψ 0 = αψ 0 + β |ψ 0 | 2 ψ 0 − 2 2m ∗ ∇2 ψ 0 = 0. (6.19) Dropping the subscript and rearranging terms we find − 2 2m ∗ ∇2 ψ + αψ + β |ψ| 2 ψ = 0. (6.20) This equation is very similar to the time-independent Schrödinger equation but, interestingly, it contains a nonlinear term. We now apply it to find the variation of ψ(r) close to the surface of a superfluid filling the half space x > 0. We impose the boundary condition ψ(x = 0) = 0 and assume that the solution only depends on x. Then − 2 2m ∗ ψ′′ (x) + αψ(x) + β |ψ(x)| 2 ψ(x) = 0. (6.21) 33
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: N s is the total number of particle
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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