Superconductivity Notes

Superconductivity and superfluidity

Notes from the University La Sapienza course held by Prof. Luciano Pietronero

Tomarchio Luca

July 23, 2019

Contents

1. London equations
   1. Example: Long cylinder in solenoid
   2. Coherence length
   3. Broken homogeneity symmetry
2. Electrodynamic quantization
   1. Flux quantization and experimental results
3. The Josephson effect
   1. DC Josephson
   2. Macroscopic quantum interference (SQUID)
4. Development of a microscopic theory
   1. Instabilities of low dimensional systems
   2. Second quantization for electrons
   3. Understanding the Cooper pairs
5. Electron-phonon interaction
   1. Resistivity theory
   2. Diffraction model for resistivity
6. Cooper pairs theory
   1. Second quantization development
7. The BCS theory
   1. Number of particles
   2. Critical temperature
8. Ginzburg-Landau theory
9. High-T superconductivity
10. Superfluidity
    1. He superfluidity and cold atoms
    2. SF wave-function and phenomena
11. Interaction BEC

Superconductivity has been discovered in mercury in 1911 by K. Onnes in his Leiden laboratory, thanks to the early developments in cryogenic technology, while studying the resistivity behavior at low temperatures. The complexity of the phenomenon brought a mystery veil upon the microscopic interpretation for more than forty years, and before the discovery of high critical temperature cuprates, Matthias rules were followed: avoid magnetism, oxygen, insulators, and theorists.

The properties of superconductive materials are, in principle, four:

• Zero resistance, which has been the first to be discovered, but it can be confused with multiple other phenomena, like breakdown or experimental errors.
• Meissner effect, or perfect diamagnetism, this is the most astounding proof for superconductivity, there is no other analogous known property. It states that a superconductor expels, up to a penetration length, all the external magnetic fields outside the material. The SC state is also broken for too high magnetic fields.
• Flux quantization, of charges q = −2e, which are identified with the Cooper pairs of BCS theory. It means that two electrons form a bounded boson state, such that SC looks like BEC, but with the main difference of having Fermi particles above the critical temperature. This feature is universal for every type of superconductors known.
• Energy gap between the Fermi level and the Landau excitations at low energies for independent quasi-particles.

Before superconductivity, all theories of electrons in metals were developed in an independent electron picture, from Bloch theory to Landau Fermi liquid theory. But, this approach can’t be used to understand the transition to the SC state, in particular, a SC theory has to avoid a perturbative theory starting from an independent picture at zero order. A similar effort had been made by Heisenberg for the Coulomb potential, but with little results. A first insight, instead, had come from the isotopic effect, for which the critical temperature scales like M^−1/2, where M is the ion's mass. This property explains how important the electron-phonon coupling is for the creation of the flux quantization.

Other approaches have been developed with a pure electrodynamic phenomenology to describe the Meissner effect, such as the London equations or the two-fields theory. But only with BCS a microscopic theory for the property came out, containing all the previous results, including the Landau-Ginzburg phenomenological theory.

The first features to understand are the differences between superconducting states and ideal materials, since both show off a null resistivity. For the latter systems, Maxwell equations impose the condition ∂B = 0 inside the medium, while SC requires a more stringent one: B = 0.

Considering two systems, an ideal and a superconductive one, which reach their state below a critical temperature. Supposing the presence of an external magnetic field, we can describe the different responses to it at different temperatures. For an ideal metal (τ→∞), we see that the magnetic field satisfies the condition, inside the medium, of ∂B/∂t = 0. The differences of the final properties are zero, in terms of external field and temperatures, but we find different states, meaning a hysteresis behavior (which is a process possible only for first-order transitions). For a SC state, the first sequence doesn’t change, but the second does in response to the more strict relation of B = 0 inside the medium.

London equations

The first electrodynamic approach to superconductivity can be found in the two-fluids model, where we consider two types of electrons: normal and superconductive. The latter are considered to behave like free particles, following the Newton law, while the former can be represented through the Drude model:

2dv_neτ_nSC −eE → −ni, −nim = J = ehv J = ehv = ESC SC SC n n ndt m

The two electron states identify two different free energies, derived experimentally from the specific heat data. We define the condensation energy as the difference in free energy at T = 0 between F_s, the energy for a complete superconducting system, and F_n, for a normal one. This difference can be shown to be smaller than kT_B, in particular: ∆F = (βE_F)⁻¹10⁻³eV⁻¹, since the Fermi energy is just 1 eV. This means that, considering a naive interpretation of single electron behavior, only such a fraction of electrons will be in the new condensed state. The free energy of these free particles will take the form under a magnetic field:

2Z Z Z 1 B2F = F_s dr + E_kin + E_mag E = dr mv_sn_SCE_mag= drSC2 2µ₀

where the magnetization contribution can be omitted due to the non-magnetic nature of the medium, i.e., its volume is considered as free space with the B field taken to be non-null only outside the conductor. The supercurrents give an additional energy because they aren’t associated to the orientation of pre-existing dipoles.

Figure 1: The external magnetic field is compensated exactly by the demagnetizing −4πM, one H = where the magnetization is given by the supercurrents loops. Type two superconductors avoid the singularity by achieving a nonhomogeneous state.

By taking a stationary condition ∂J_SC/∂t = 0, it follows a null electric field, hence J_SC = 0, meaning that the entire current, in a stationary state, comes from superconductive electrons. Starting from the Maxwell-Faraday local equation for a superconductive conductor, in which Ohm’s law is replaced by an accelerative supercurrent, that is dJ_s/dt = (n_se²/m)E, we obtain:

2∂J_sn_se²∂B s→ ∇× −∇× − =E = ∂t ∂t m ∂t

Taking the spatial and time derivative of the fourth Maxwell equation for non-magnetic materials:

∂B 1 ∂Bm²2 2∇ × ∇ × −∇ ∇ × → ∇B = B = µ₀J_s = λ² =0 s s2 2∂t λ² ∂t µ₀ n_se²

where the contribution ∂D/∂t to the current has been approximated to zero. A solution to the equation is of course ∂B/∂t = 0, which only represents a stationary field inside the medium. The parameter λ is known as the penetration length. Given a simple geometry of a medium with a surface coincident with the xy plane (extending along z > 0), if we suppose a magnetic field dependant only on z, the solution for the internal magnetic field would be:

∂B (z) ∂Bx 0 −z/λ= e_s∂t ∂t

where z is defined from the surface of the material (where the currents are non-zero). The only time derivative component that survives is the one along x, since the equations wouldn’t permit the field to have a component normal to the surface. Indeed, the Maxwell equations seem not to solve the property for the superconductive state, since we need the more stringent feature B = 0. An empirical intuition can be found in the modification of the precedent relation, where the derivative is omitted:

22 ∂J n en e s ss∇ × − B = EJ =s m ∂t m

These two relations are known as the London equations (1935), and they predict a correct superconductive feature B_x(z) = B₀exp(−z/λ_s). The same result can be obtained through a variational principle. Using the fourth Maxwell equation, we can write the total free energy for the superconductive electrons as:

Z1 2 2 2 |∇ ×F dr B + λ_s² B|= F₀+2µ₀

where an average value of v_s has been used. It is minimized by a field inside the medium that satisfies the first London equation, hence, the Meissner effect (taken for granted while writing the London equation) is a consequence of the equilibrium condition in the presence of persistent currents with slow variations in space.

If we insert the effective density of conduction electrons (as determined from surface impedance measurements of the skin depth) for n_s, the predicted value of λ_s is found to be 200 Å in typical classic metallic conductors, against the experimental values of typically 500 Å for T ≤ T_c. This quantitative discrepancy is found to be the result of the non-local behavior of electrodynamics in pure superconductors, which is kept in mind in the BCS theory. A frequently used empirical relation, called the two-fluids temperature dependence, is:

λ(T ) ≈ λ(0)/(1 − t⁴)^1/2 t = T /T_c

1.1 Example: Long cylinder in solenoid

Suppose a cylinder of radius r inside a solenoid of radius r₀ > r₁, N number of turns and length L. Through the flow of a current in the latter specimen, London equations highlight how the magnetic field generated is localized only on the surface of the cylinder, since r₀ λ_s. In particular, inside the specimen, we will have:

N I−r/λB(r) = Be_z B = µ₀ L

applying the fourth Maxwell equation B −r/λ∇ × → −B = µ₀ J_SC J_SC(r) = e_θ

meaning that the superconducting state will create a screening current on the surface of the cylinder, rotating in such a way to compensate the internal magnetic field (total diamagnetism). Experimentally, by reaching an external critical field H_c = N I_c/L, it will become uniform across the entire section and the specimen is no longer superconducting (loss of total diamagnetism). Such a limit is also related to a critical current for type I SCs, I_c ∼≤ 2πLH_c, i.e., the energy required to generate the reverse field is too high to maintain the state. In general, working conditions are I ≤ I_c and T ≤ T_c. The transition is reversible, and we can calculate the difference between the free energies of the two states for non-magnetic materials:

πr₀²LF + π(r₁² − r₀²)Lµ₀n − πr₀²LF_SC −ΔF = 2H_c−2 2 2 2 − −− μ2

where we have neglected the slightly penetration of the field and the kinetic terms of the surface currents, which are small compared to bulk properties. The field lines will be more or less packed in some spots near the medium, however, they will maintain their average density.

A thermodynamic description of the SC state means considering macroscopic averaged parameters, and it makes sense only considering normal Gaussian distributions; if we are near a critical transition described by power laws, this picture will fail. From the free energy variation, we see the production of a work to expel the magnetic field from the medium, hence reducing the total flux φ across the coil. The energy source maintaining the constant field has to pay an activation energy price for the deformation of the field lines. The differential work done by the coil, under a change of current I + dI, will be:

N 0dφ ·Idt = N Idφ = N AIdB = V H dB = µ V (H dM + H dH)dL =·

where the H · dH term is the energy required to vary the flux without sample (not measured in an experiment), while the H · dM contribution is the energy associated to the presence of medium. Therefore, the differential internal energy of the medium will be:

·dU = T dS + µ₀ BH dM

where the latter term is the magnetic work analogous to the P · dV one. Since S and M are not the physical parameters computable from the experiments, we want to switch the dependencies to T and H. This can be done through the computation of the Gibbs free energy:

−− ·−G(T, H) = U T S μ₀ V H B

from which we see that the ambiguity of the work contribution of the generator disappears exactly. Differentiating, we would obtain:

∂G 1 ∂G−SdT − · → − −dG = µ₀ V M dH S = , M =∂T µ₀ V ∂H

To find the condensation energy (at a given temperature) of the system we want to compute the energy required to change the field from 0 to H_c(T ). In particular:

HZ Z µ₀ Vc ·− −µ₀ HM dH =ΔG_s(T, H_c) − G_s(T, 0) = dG = 20 −H

where the result is obtained given the Meissner effect M ≈0. Since the following equality holds G_s(T, H_c) = G_n(T, H_c), and since for normal metals M ≈0, thus obtaining the condensation energy:

2H_c− −µ₀ΔG = G_s(T, 0) − G_n(T, 0) = V2

which is associated to the energy of the Cooper pairs production. The energy per atom will be given by the volume of the atom itself, and this will be similar to the energy gap, highlighting how these macroscopic considerations can be related to the microscopic ones. The associated entropy difference will be:

∂H_c−ΔS = S_s(T, 0) − S_n(T, 0) =µ₀ V ∂T

Multiplying this latter result by T, we obtain a positive latent heat per unit volume required for the transition (first order) from SC to normal. The subtle consequence is the double nature of the process, which becomes a second order transition at zero external fields (H = 0 at T = T_c).

1.2 Coherence length

In the derivation of the London equations, we have assumed that v(r), or the supercurrents, are slow varying functions in space. But, what do we mean with slow?

In a condensate state, the velocities of two electrons are correlated if their distance is small compared to a certain range. For pure metals, this correlation length is called ξ₀. Our derivation, therefore, which defines variations of v, h, and j over a scale of order λ, requires λ ξ₀. To estimate ξ₀, we notice that the important energy domain is given by the creation of a gap between the Fermi level and the higher states:

2p − < E + ∆ ∆ EE ∆ < F FF 2m

The thickness of the shell in momentum space is δp ∼ 2∆/v_F, hence, a wave packet of plane waves with maximum uncertainty momentum δp has a minimum spatial extent ∼δx ∼ h/δp. This leads us to take

}v_F∼δx ξ₀ = π∆

to maintain the SC state. The π factor has been introduced from the reduced Planck constant. The quantum in SCs is composed of two electrons, hence we need a correct modulation to maintain the SC state. In fact, the combination of two plane waves to obtain such a state requires energy, in particular:

22 21 k }}i(k+q)x ikx 2√ ·→ψ = e + e + k q + O(q )E = m 2m

where the additional energy is given by the second term. This has to be smaller than the energy gap to maintain the SC state, obtaining again the same relation for ξ₀ as the inverse of the maximum possible value of q.

Figure 2: The increase in energy is given by the alternating occupation due to the density modulation, i.e., by the Pauli principle.

For simple (nontransition) metals, λ is too small to satisfy the relation, and London equations become only qualitative. These specimens are called Pippard or I type superconductors. Transition metals and intermetallic compounds which satisfy the condition, instead, are known as London or II type superconductors.

1.3 Broken homogeneity symmetry

In general, when a magnetic flux is expelled, its density lines increase and decrease in density around the medium (maintaining the average lines density), creating spots on the surface where the magnetic field acquires greater intensities with respect to others. This means that, in these spots, the critical field condition will be reached earlier. This effect, which can be described through the calculus of the demagnetization of the external volume, is traduced into a homogeneity break, where the SC prefers to create alternations...