14.4. the Ginzburg–Landau Theory the BCS Theory Answered the Question Why Electrons Pair Up

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This is indeed what one observes experimentally for convectional superconductors.

14.3.7. Experimental evidence of the BCS theory III: isotope effect Because the attraction is mediated by phonons in the BCS theory, the transition temperature should depend on the mass of nucleons. As shown above, for the BCS theory, Tc ∝ ϵD, where ϵD is the Debye energy. For an isotropic elastic medium, it is 6 π2 13 ϵD = ℏωD = ℏ v (14.9) VC

-1/2 Here, VC is the volume of a unit cell and v is the sound velocity. The sound velocity is typically proportional to M , where M is the mass of the nucleons. For example, in Chapter 3, we calculated before the sound velocity for a 1D crystal, which has

K v = a (14.10) M Here, a is the lattice spacing, K is the spring constant of the bond and M is the mass of the atom.

-1/2 Therefore, we found that Tc ∝ ϵD ∝ v ∝ M , so the BCS theory predicts that α = 1/2 in the isotope effect, which is indeed what observed in experiments.

14.3.8. Experimental evidence of the BCS theory IV (the direct evidence): charge are carried by particles with charge -2 e, instead of -e Q: How to measure the charge of the carriers? A: the Aharonov–Bohm effect. Aharonove and Bohm told us that if we move a charged particle around a closed loop, the quantum wave function will pick up a phase Δϕ

Δϕ = q ΦB /ℏ (14.11) where q is the charge of the particle and ΦB is the magnetic flux

→ → → → ΦB =   B·ⅆ S =  A·ⅆ r (14.12) S We know that in quantum physics, particles are waves and thus they have interference phenomenon. For interference, the phase of a wave plays the crucial role, and thus one can detect this phase Δϕ using interferometers. By measuring Δϕ as a function of B, one can determine the charge of the particle. For superconductors, it is found to be q = -2 e (or +2 e for paring of holes). We will come back to this point later with more details.

14.4. the Ginzburg–Landau theory The BCS theory answered the question why electrons pair up. However, to understand why these pairs lead to the interesting phenomena of superconductivity and the Meissner effect, one needs to do something more. This task will be address in this section via investigating the Ginzburg–Landau theory of superconductivity. This theory is actually developed before the BCS theory, but in this lecture, we will use the BCS theory to guide us to write down this Ginzburg-Landau theory. Please keep in mind that when Ginzburg and Landau wrote down this theory, they didn’t know the microscopic mechanism of superconductivity and they didn’t know the existence of the Cooper pair. However, they still successfully developed the correct phenomenological theory, based on their physics intuitions.

14.4.1. Wave function and super fluid density → Let’s first write down the wave function for the ground state of a Cooper pair ψGr. Because this ground state have many Cooper pairs (at low T, a large portion of Cooper pairs are on this ground state), we can combine all these Cooper pairs together and describe them with one wave function

→ → ψr = NS ψGr (14.13)

→ Here, NS is the number of Cooper pairs in the ground state. With this normalization factor, the mode square of ψr has a simple physical meaning → → * → → * → ρr = ψr ψr = NS ψGr ψGr (14.14)

→ * → → Here, ψGr ψGr is the probability density of finding a Cooper pair at position r if we just have one Cooper pair in the ground state. There- → → * → fore, ρr = NS ψGr ψGr is the probability density to find one Cooper pair at this position when we have a condensate (here NS Cooper pairs → is in the ground state). In other words, ρr is the density of Cooper pairs in the ground states. Because these Cooper pairs result in superconduc- → tivity, ρr is called the super fluid density. 120 Phys520.nb

The integer of ρ give us the total number of particles in the ground state, which is just NS

→ →  ⅆ r ρr = NS (14.15)

→ Notice that ψr is a complex function, we can write it in terms of absolute value and a complex phase

→ → → ψr = ρr ⅇⅈ ϕr (14.16)

The amplitude here is just the square root of the super fluid density, because ψ 2 = ρ.

→ Deep inside the superconductor, we assume that ρr is a constant, so that we can write

→ → ψr = ρ ⅇⅈ ϕr (14.17)

14.4.2. Ginzburg–Landau free energy For such a wavefunciton, one can write down the energy of this quantum state and this energy is known as the Ginzburg–Landau free energy. It is NOT the free energy we used in thermal dynamics, but it is similar. Very typically, the system want to go to the state which minimize the GL free energy. The energy of this state relies on the detailed properties of the materials, which we don’t know here. For Ginzburg and Landau, they didn’t know the BCS theory when they developed this theory, so they didn’t even know the clear physical meaning of ψ, and therefore, there is no way that they can obtain the accurate form of the enregy. However, they are smart enough to identify the “structure” of the energy, which is enough → → to explain many important phenomena. We know that the energy must relies on the wavefunction ψ r and its derivatives ∇ ψ r, so we can write it down as → → F = Fψr, ∇ψr (14.18)

Then, we expand the GL free energy as a power series of ψ and ∇ψ

ℏ2 → * → → * → → → * → → 2 F = F0 +  ⅆ r  ∇ψ r ∇ψr + a ψ r ψr +  ⅆ r b ψ r ψr (14.19) 2 M Here the first term is the zeroth order term. The second and third terms are of the order ψ2 and ψ4 respectively. Because F must be real, we cannot have other terms up to Oψ4.

For the normal phase (NS = 0 and thus ρS = 0), we have ψ(r) = 0. Therefore, F = F0. This tells us that the zeroth order term F0 is just the GL free energy for the normal state. Therefore, we will now rename F0 as Fn, where n stands for the normal state 1 → * * * 2 F = Fn +  ⅆ r  (-ⅈ ℏ ∇ψ) (-ⅈ ℏ ∇ψ) + a ψ ψ + b (ψ ψ)  (14.20) 2 M The first term in the integral is just the kinetic energy p2 2 M, and therefore the value of M should be the mass of the particles (cooper pairs). In other words, M should be twice of the effective mass of electrons (or holes) M = 2 m* (14.21) This is something Ginzburg and Landau didn’t know when they develop this theory. Because Cooper pairs are charged, the wave function should couple to E and M field. We have discussed before how to couple a wavefunction to E and M field before in quantum mechanics (the minimal coupling). Basically, we just need to substitute momentum p into

→ → q → p → p + A (14.22) c Here we don’t need to consider electric potential , because E=0 inside a perfect metal. Phys520.nb 121

1 q → * q → → * * 2 F = Fn +  ⅆ r   -ⅈ ℏ ∇+ A ψ  -ⅈ ℏ ∇+ A ψ + a ψ ψ + b (ψ ψ)  (14.23) 2 M c c Now, we use the fact

→ → ψr = ρ ⅇⅈ ϕr (14.24)

So the free energy can be rewrite as

ρ q → 2 → 2 F = Fn +  ⅆ r  ℏ ∇ϕ + A + a ρ + b ρ  (14.25) 2 M c

14.4.3. The superconductor transition We need to minimize the GL free energy to find the proper quantum state.

2 ℏ ρ q → 2 → 2 F = Fn +  ⅆ r  ∇ϕ + A + a ρ + b ρ  (14.26) 2 M ℏ c The first term inside the integer is no negative. So its minimum value is zero. This minimum is reached when

q → ∇ϕ + A = 0. (14.27) ℏ c This result has very important impact and we will revisit it in the next section.

→ 2 F = Fn +  ⅆ r a ρ + b ρ  (14.28)

The problem here is to minimize the function a ρ + b ρ2 for ρ ≥ 0. For stability reasons, we assume b > 0 (if b < 0, the minimum of the free energy is -∞ at ρ = ±∞, which is unphysical). If a > 0, the function a ρ + b ρ2 reaches its minimum at ρ = 0. In other words, the system has zero super fluid density, so this is the normal state. If a < 0, the function a ρ + b ρ2 reaches its minimum at ρ = -a/2 b, and the minimum value is -a2 4 b. In other words, this is the superconductor state with super fluid density ρ = -a/2 b. With this intuition, we know that the coefficient a must depend on temperature a(T).

At T > TC, a(T) is positive, so the normal state has the lowest GL free energy

At T < TC, a(T) is negative, so the superconductor state has the lowest GL free energy.

The transition temperature TC is determined by the equation α(T) = 0. [or say, a(TC) = 0] If we expand a(T) near the transition point,

a(T) = a(TC) + a' (T = TC) (T - TC) + … (14.29) If we ignore higher order term, we find that

a(T) = a0(T - TC) (14.30)

Here we used the fact that a(TC) = 0 and we defined a0 = a' (T = TC).

Because ρ = -a/2 b, near the transition temperature (T close but slightly smaller than TC)

ρ = -a/2 b ≈ -a0(T - TC)/2 b = a0 /2 b(TC - T) (14.31)

Therefore, the super fluid density creases as TC - T for T ~TC.

14.4.4. The Meissner effect Now, let’s consider the superconducting phase where ρ = -a/2 b > 0.

2 2 2 → ℏ ρ q → a F = Fn +  ⅆ r  ∇ϕ + A -  (14.32) 2 M ℏ c 4 b To minimize the free energy, the first term must be zero 122 Phys520.nb

q → ∇ϕ + A = 0 (14.33) ℏ c If we take a curl on both sides, we find that

q → q → 0 = ∇ ×∇ϕ + ∇ × A = - B (14.34) ℏ c ℏ c Therefore, B=0 inside a superconductor. This is the Meissner effect.

14.4.5. Quantization of magnetic flux

Fig. 5. Magnetic flux through a superconducting ring

Consider a superconducting ring and we draw a loop around the ring. We can compute the loop integral around this loop

→ q → →  ⅆ r · ∇ϕ + A =  ⅆ r 0 = 0 (14.35) ℏ c Therefore,

→ → q →  ⅆ r ·∇ϕ = - ⅆ r · A (14.36) ℏ c

→  ⅆ r ·∇ϕ = ϕθ=2 π - ϕθ=0 (14.37)

→ Here, ∮ ⅆ r ·∇ϕ measures the change of the phase ϕ when we go around the loop. Because the final point and the initial point are the same point in real space, the phase must differ by 2 π n.

→ → q → q → → q → → q → → q  ⅆ r ·∇ϕ = 2 π n = - ⅆ r · A = -  ⅆ r · A = -   ⅆ r ∇ × A = -   ⅆ r B = - ΦB (14.38) ℏ c ℏ c ℏ c ℏ c ℏ c

where ΦB is the magnetic flux inside the ring.

ℏ c h c ΦB = 2 n π = n . (14.39) q q Notice that this quantization has the charge of the particle q invovled. Here, the charge q is ±2e, instead of e because electrons pairs up. This is the direct evidence of the existence of Cooper pairs.

14.4.6. London equation The electric current is

→ q → q → → q → j = ψ* -ⅈ ℏ ∇+ A ψ + ψ ⅈ ℏ ∇+ A ψ* (14.40) 2 M c c

→ → For ψr = ρ ⅇⅈ ϕr, we have

→ q → q → → q → q → q → j = ρ ℏ ∇ϕ + A + ℏ ∇ϕ + A  = ρ ℏ ∇ϕ + A (14.41) 2 M c c M c This relation is known as the London equation. If we take a curl for both sides, Phys520.nb 123

2 2 → q q → q → q → ∇ × j = ρ ℏ ∇ × ∇ϕ + ∇ × A = ρ ∇ × A = - ρ B (14.42) M c M c M c Take another curl on both sides

2 → q → ∇ ×∇ × j = - ρ ∇ × B (14.43) M c → → For the l.h.s., we know that ∇ ×∇ × j = -∇2 j → → For the r.h.s., the Maxwell’s equation tells us that ∇ × B = 4 π/c j (CGS unit) 2 → 4 π q ρ → -∇2 j = - j (14.44) M c2

2 → 4 π q ρ → ∇2 j = j (14.45) M c2 We can define London penetration depth

M c2 M c λL = = (14.46) 4 π q2 ρ 4 π ρ q

So,

→ → j ∇2 j = (14.47) 2 λL

→ → M c Because B = - ∇ × j, we can take another curl on both sides and get q2 ρ

→ → B ∇2 B = (14.48) 2 λL Consider a superconductor as shown in the figure below. the gray area is a superconductor, the white region is out side.

ⅆ2 B B = (14.49) 2 2 ⅆ x λL The solution for this equaiton is 124 Phys520.nb

B = B0 exp(-x/λL) (14.50) The magnetic field decays exponentially. Because the current follows the same equation, we can also show that the current in a superconductor mostly flows along the surface. And the current inside the bulk is very small (essential 0). Why is that? If there is some current in the bulk, the current with generate magnetic field in the bulk (Amperes circuital law), which will violate B = 0! At the end of the day, all the currents in a superconductor shall flow on the surface. By doing so, the generated B fields only exists near the surface (within the London penetration depth), which is allowed. Note: If you want a superconducting wire to carry a huge current, you want to maximize the surface area, instead of the cross section! So a wire with circular cross-section is not the best option, because for the same cross-section area, the surface is minimized.