Superconductivity Notes

Superconductivity and Superfluidity

Notes from the University La Sapienza course held by prof.

Luciano Pietronero

Tomarchio Luca

July 23, 2019

1

Contents

1 London equations 4

1.1 Example: Long Cylinder in Solenoid . . . . . . . . . . . . . . . . . . . 6

1.2 Coherence Length . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

1.3 Broken Homogeneity Symmetry . . . . . . . . . . . . . . . . . . . . . . 10

2 Electrodynamic Quantization 11

2.1 Flux Quantization and Experimental results . . . . . . . . . . . . . . . 12

3 The Josephson Effect 12

3.1 DC Josephson . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

3.2 Macroscopic Quantum Interference (SQUID) . . . . . . . . . . . . . . . 14

4 Development of a Microscopic Theory 15

4.1 Instabilities of Low Dimensional Systems . . . . . . . . . . . . . . . . . 17

4.2 Second Quantization for Electrons . . . . . . . . . . . . . . . . . . . . . 18

4.3 Understanding the Cooper Pairs . . . . . . . . . . . . . . . . . . . . . . 21

5 Electron-Phonon Interaction 23

5.1 Resistivity Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

5.2 Diffraction Model for Resistivity . . . . . . . . . . . . . . . . . . . . . . 27

6 Cooper Pairs Theory 29

6.1 Second Quantization Development . . . . . . . . . . . . . . . . . . . . . 31

7 The BCS Theory 33

7.1 Number of Particles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

7.2 Critical Temperature . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

8 Ginzburg-Landau Theory 40

9 High-T Superconductivity 42

c

10 Superfluidity 44

4

10.1 He Superfluidity and Cold Atoms . . . . . . . . . . . . . . . . . . . . . 48

10.2 SF Wave-Function and Phenomena . . . . . . . . . . . . . . . . . . . . 51

11 Interaction BEC 53

2

Superconductivity has been discovered in mercury in 1911 by K. Onnes in his Leida

laboratory, thanks to the early developments in cryogenic technology, while studying

the resistivity behaviour 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 super

conductive materials are, in principle, four:

ˆ Zero resistance, which has been the first to be discovered, but it can be confused

with multiple other phenomena, like breakdown or experimental errors.

ˆ 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.

ˆ −2e,

Flux quantization, of charges q = 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.

ˆ 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 inde-

pendent 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

−1/2

which the critical temperature scales like M , where M is the ions’ 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 re-

t

quires 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 3

where 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 an hysteresis behaviour (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

1 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 2

dv n e τ

n

SC −eE → −n i, −n i

m = J = ehv J = ehv = E

SC SC SC n n n

dt 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 , the energy for a complete superconducting system,

s

and F , for a normal one. This difference can be shown to be smaller than k T , in

n B

−1 −3

'

particular: ∆F = (βE ) 10 eV , since the Fermi energy is just 1 eV . This

F

means that, considering a naive interpretation of single electron behaviour, 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 2

Z Z Z

1 B

2

F = F dr + E + E E = dr mv n E = dr

s kin mag kin SC mag

SC

2 2µ 0

4

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 ori-

entation 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

d

loops. Type two superconductors avoid the singularity by achieving a non

homogeneous state.

By taking a stationary condition ∂ J = 0, it follows a null electric field, hence

t SC

J = 0, meaning that the entire current, in a stationary state, comes from supercon-

n

ductive electrons. Starting from the Maxwell-Faraday local equation for a supercon-

ductive conductor, in which Ohm’s law is replaced by an accelerative supercurrent,

2

that is dJ /dt = (n e /m)E, we obtain

s s 2

∂J n e ∂B

∂B s s

→ ∇× −

∇× − =

E = ∂t ∂t m ∂t

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

materials ∂B 1 ∂B m

2

2 2

∇ × ∇ × −∇ ∇ × → ∇

B = B = µ J = λ =

0 s s

2 2

∂t λ ∂t µ n e

0 s

s

where the contribution ∂D/∂t to the current has been approximated to zero. A solu-

tion 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

s

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) ∂B

x 0 −z/λ

= e s

∂t ∂t

where z is defined from the surface of the material (where the currents are non-zero).

5

The only time derivative component that survives is the one along x, since the equa-

tions 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

2

2 ∂J n e

n e s s

s

∇ × − B = E

J =

s m ∂t m

These two relations are known as the London equations (1935), and predict a correct

superconductive feature B (z) = B exp(−z/λ ). The same result can be obtained

x 0 s

through a variational principle. Using the fourth Maxwell equation, we can write

the total free energy for the superconductive electrons as

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

alternat