Carsten Timm: Theory of superconductivity

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6 Ginzburg-Landau theory Within London and Pippard theory, the superfluid density n s is treated as given. There is no way to understand the dependence of n s on, for example, temperature or applied magnetic field within these theories. Moreover, n s has been assumed to be constant in time and uniform and in space—an assumption that is expected to fail close to the surface of a superconductor. These deficiencies are cured by the Ginzburg-Landau theory put forward in 1950. Like Ginzburg and Landau we ignore complications due to the nonlocal electromagnetic response. Ginzburg-Landau theory is developed as a generalization of London theory, not of Pippard theory. The starting point is the much more general and very powerful Landau theory of phase transitions, which we will review first. 6.1 Landau theory of phase transitions Landau introduced the concept of the order parameter to describe phase transitions. In this context, an order parameter is a thermodynamic variable that is zero on one side of the transition and non-zero on the other. In ferromagnets, the magnetization M is the order parameter. The theory neglects fluctuations, which means that the order parameter is assumed to be constant in time and space. Landau theory is thus a mean-field theory. Now the appropriate thermodynamic potential can be written as a function of the order parameter, which we call ∆, and certain other thermodynamic quantities such as pressure or volume, magnetic field, etc. We will always call the potential the free energy F , but whether it really is a free energy, a free enthalpy, or something else depends on which quantities are given (pressure vs. volume etc.). Hence, we write F = F (∆, T ), (6.1) where T is the temperature, and further variables have been suppressed. The equilibrium state at temperature T is the one that minimizes the free energy. Generally, we do not know F (∆, T ) explicitly. Landau’s idea was to expand F in terms of ∆, including only those terms that are allowed by the symmetry of the system and keeping the minimum number of the simplest terms required to get non-trivial results. For example, in an isotropic ferromagnet, the order parameter is the three-component vector M. The free energy must be invariant under rotations of M because of isotropy. Furthermore, since we want to minimize F as a function of M, F should be differentiable in M. Then the leading terms, apart from a trivial constant, are F ∼ = α M · M + β 2 (M · M)2 + O ( (M · M) 3) . (6.2) Denoting the coefficients by α and β/2 is just convention. α and β are functions of temperature (and pressure etc.). What is the corresponding expansion for a superconductor or superfluid? Lacking a microscopic theory, Ginzburg and Landau assumed based on the analogy with Bose-Einstein condensation that the superfluid part is described by a single one-particle wave function Ψ s (r). They imposed the plausible normalization ∫ d 3 r |Ψ s (r)| 2 = N s = n s V ; (6.3) 30
N s is the total number of particles in the condensate. They then chose the complex amplitude ψ of Ψ s (r) as the order parameter, with the normalization |ψ| 2 ∝ n s . (6.4) Thus the order parameter in this case is a complex number. They thereby neglect the spatial variation of Ψ s (r) on an atomic scale. The free energy must not depend on the global phase of Ψ s (r) because the global phase of quantum states is not observable. Thus we obtain the expansion F = α |ψ| 2 + β 2 |ψ|4 + O(|ψ| 6 ). (6.5) Only the absolute value |ψ| appears since F is real. Odd powers are excluded since they are not differentiable at ψ = 0. If β > 0, which is not guaranteed by symmetry but is the case for superconductors and superfluids, we can neglect higher order terms since β > 0 then makes sure that F (ψ) is bounded from below. Now there are two cases: • If α ≥ 0, F (ψ) has a single minimum at ψ = 0. Thus the equilibrium state has n s = 0. This is clearly a normal metal (n n = n) or a normal fluid. • If α < 0, F (ψ) has a ring of minima with equal amplitude (modulus) |ψ| but arbitrary phase. We easily see √ ∂F ∂ |ψ| = 2α |ψ| + 2β |ψ|3 = 0 ⇒ |ψ| = 0 (this is a maximum) or |ψ| = − α (6.6) β Note that the radicand is positive. F α > 0 α = 0 α < 0 0 Re ψ Imagine this figure rotated around the vertical axis to find F as a function of the complex ψ. F (ψ) for α < 0 is often called the “Mexican-hat potential.” In Landau theory, the phase transition clearly occurs when α = 0. Since then T = T c by definition, it is useful to expand α and β to leading order in T around T c . Hence, Then the order parameter below T c satisfies √ |ψ| = ψ α/ β α ∼ = α ′ (T − T c ), α ′ > 0, (6.7) β ∼ = const. (6.8) − α′ (T − T c ) β = √ α ′ √ T − Tc . (6.9) β T c T 31
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: with the penetration depth √ √
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
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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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