Superconductivity Notes

Z

1 2 2 2

|∇ ×

F dr B + λ B|

= F +

0 s

2µ 0

where an average value of v has been used. It is minimized by a field inside the

s

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 sur-

face impedance measurements of the skin depth) for n , the predicted value of λ is

s s

∼

found to be 200 Ȧ in typical classic metallic conductors, against the experimental

values of typically 500 Ȧ for T T . This quantitative discrepancy is found to be the

c

result of the non local behaviour 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 4 1/2

≈ −

λ(T ) λ(0)/(1 t ) t = T /T

c

1.1 Example: Long Cylinder in Solenoid

Suppose a cylinder of radius r inside a solenoid of radius r > r , N number of turns

0 1 0

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 cylin-

der, since r λ. In particular, inside the specimen, we will have

0 s

N I

−r/λ

B(r) = Be ẑ B = µ 0 L

applying the the fourth Maxwell equation B −r/λ

∇ × → −

B = µ J J (r) = e θ̂

0 SC SC µ λ

0

meaning that the superconducting state will create a screening current on the sur-

face of the cylinder, rotating in such a way to compensate the internal magnetic field

6

(total diamagnetism). s /L, it will become

Experimentally, by reaching an external critical field H = N I

c c

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 I type

'

SCs, I 2πLH , i.e., the energy required to generate the reverse field is too high to

c c 12

34 ∼

≤ I and T T . The

maintain the state. In general, working conditions are I c c

transition is reversible, and we can calculate the difference between the free energies

of the two states for non-magnetic materials 2

2 H

H 2 2

2 2 2 − −

− πr LF πr Lµ

∆F = πr LF + π(r r )Lµ n 0

SC 0 0 1

0 1 0 2 2

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 distribu-

tions; 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

dφ · · ·

Idt = N Idφ = N AIdB = V H dB = µ V (H dM + H dH)

dL = N 0

dt

·

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 + µ BH dM

0

where the latter term is the magnetic work analogous at 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

7

free energy − − ·

G(T, H) = U T S µ V H B

0

from which we see that the ambiguity of the work contribution of the generator dis-

appears exactly. Differentiating, we would obtain ∂G 1 ∂G

−SdT − · → − −

dG = µ V M dH S = , M =

0 ∂T µ V ∂H

0

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 (T ). In particular

c

H

Z Z µ V

c 0 2

·

− −µ H

M dH =

G (T, H ) G (T, 0) = dG = V

s c s 0 c

2

0 −H.

where the result is obtained given the Meissner effect M = Since the following

≈

equality holds G (T, H ) = G (T, H ), and since for normal metals M 0

s c n c − ≈

G (T, H ) G (T, 0) 0

n c n

thus obtaining the condensation energy 2

H

c

− −µ

∆G = G (T, 0) G (T, 0) = V

s n 0 2

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

−

∆S = S (T, 0) S (T, 0) = µ V H

s n 0 c ∂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 c 8

1.2 Coherence Length

In the derivation of the London equations we have assumed that v(r), or the super-

currents, 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

ξ . Our derivation, therefore, which defines variations of v, h and j over a scale of

0 s

order λ, requires λ ξ . To estimate ξ , we notice that the important energy domain

0 0

is given by the creation of a gap between the Fermi level and the higher states

2

p

− < E + ∆ ∆ E

E ∆ < F F

F 2m '

The thickness of the shell in momentum space is δp 2∆/v , hence, a wave packet of

F

plane waves with maximum uncertainty momentum δp has a minimum spatial extent

∼

δx h/δp. This leads us to take }v F

∼

δx ξ =

0 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 2

2 2

1 k }

}

i(k+q)x ikx 2

√ ·

→

ψ = e + e + k q + O(q )

E = m 2m

2

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 ξ as

0

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.

9

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

alternate layers of SC and normal state, known as intermediate state.

This result is true for both I and II types. However, the latter specimens show an

additional similar behaviour after reaching the exact critical magnetic field, hence, it

will break the homogeneity to create a periodic pattern of fluxons which stabilizes

the superconductive state outside the lattice until a second critical magnetic field is

reached. This kind of media can be studied using the Landau-Ginsburg theory, which

has been firstly applied by Abrikosov in 1957.

Figure 3: Fluxons are elongated structures which form a Wigner lattice (hexagonal),

due to their behaviour as interacting (repulsion) charged particles, and where

the magnetic field can flow as in a normal non magnetic metal. These are

bounded by the formation of a circular current, with a diameter proportional

to two times the coherence length, which screens the internal magnetic field.

∼ ∼

The structures can form only when λ 80 nm > ξ 4 nm.

For impure metals, both ξ and λ depend upon the free mean path, meaning that they

0

are proportional to the density of impurities in the medium.

Having used the concept of stable and permanent supercurrents in our theory, we

would like to prove how long these motions last. Our first basic, and incorrect, as-

sumption might be to analyse the behaviour of a single particle, and state that thermal

fluctuations causes a transition probability through the gap for the single electrons.

Such a process would happen with a probability exp(−2β∆), obtaining an erroneous

result by orders of magnitude. A better result is instead obtained if we introduce the

concept of broken coherence length, which is needed to destroy the SC state. Since

there can’t be fluctuations smaller than this length, we can calculate the minimum

free energy difference as the energy required to destroy an entire volume given by the

10

coherence length 4

2 2 ∼ ×

≈ /2) 6 10 eV

∆F = (min .volume)(energy gap) (Rξ )(µ H

0 c

7 −2

−10 ∼

(k T 10 eV ), meaning the

which gives a probability exp(−β∆F ) of order e B

18

existence of supercurrents for approximately 10 s! In the volume many Cooper pairs

will be present.

2 E