Lecture 15

Question 1

Give the equation for the entropy of a system

Answer 1

S = entropy
∂G = change in Gibbs free energy
∂T = change in temperature

Question 2

Give the equation for the difference in entropy between the superconducting and the normal to the state

Answer 2

∆S = change in entropy
∂G = change in Gibbs free energy
∂T = change in temperature
B = magnetic flux = 0
B_c = critical flux

Question 3

When does the change in entropy for a system equal 0? Why?

Answer 3

At the Curie temperature because the critical field equals 0 at this point.

Question 4

Why isn’t there any latent heat for the transition from the normal to the superconducting state?

Answer 4

Because this transition occurs without a discontinuity in entropy.

Question 5

Give the equation for the difference in the specific heat at constant volume between the superconducting and the normal

Answer 5

∆C_v = change in specific heat
T = temperature
∆S = change in entropy
B_c = magnetic flux

Question 6

Give the equation for the difference in the specific heat at constant volume between the superconducting and the normal to the state at the Curie temperature

Answer 6

∆C_v = change in specific heat
T = temperature
B_c = magnetic flux

Question 7

Describe the graph of specific heat against temperature for a superconducting and a normal state

Answer 7

There is a sharp rise in the specific heat at the Curie temperature of a superconductor due to the onset of ordering of the conduction electrons.

Question 8

Describe the graph of the resistivity and the specific heat against T/T_c for a superconductor

Answer 8

Question 9

Give the equation for electronic specific heat

Answer 9

C_v-el = electronic specific heat
C0 = constant
E_g = bandgap energy ~ (7/2)(k_B)(T_c)
T = temperature

Question 10

What range is the electronic specific heat equation valid for?

Answer 10

Question 11

The electronic specific heat presents the same behaviour as the number of _______ and _____ in a semiconductor.

Answer 11

Electrons
Holes

Question 12

Describe the graph of electronic specific heat against 1/T

Answer 12

Question 13

Compare the energy bands of a semiconductor to the energy states of a superconductor

Answer 13

They are equivalent

Question 14

How can the energy gap of a semiconductor be measured?

Answer 14

By measuring photon absorption

Question 15

How can the bandgap of a superconductor be measured?

Answer 15

By measuring microwave photon absorption (as the bandages are only a few meV).

Question 16

What does the BCS theory of superconductivity explain?

Answer 16

• Attraction between pairs of electrons.
• How bound electron ‘Cooper pairs’ can form.
• How Cooper pairs can form a collective ground state of lower energy than that of the normal electron states.

Question 17

What are Cooper pairs?

Answer 17

Composite bosons made of bound electron pairs. They are formed by attraction between electrons via a lattice.

Question 18

Describe the interaction between two electrons in free space

Answer 18

There is a strong Coulombic repulsion

Question 19

Why do metals have no net charge?

Answer 19

The conduction electrons move through a lattice of positive ions and the charges balance out.

Question 20

Describe how electrons can locally distort a crystal lattice

Answer 20

Question 21

Describe how electrons can attract one another in a lattice

Answer 21

• A single electron distorts a crystal lattice by attracting positive ions towards it.
• The electron then leaves at the Fermi velocity which is faster than the speed of lattice distortion (the speed of sound).
• This leaves a trail of net positive charge behind the electron which attracts other electrons.

Question 22

What is the approximate value of the Fermi velocity

Answer 22

Question 23

What is the approximate value of the speed of sound in a crystal lattice

Answer 23

Question 24

Is attraction between electrons in a crystal direct or indirect?

Answer 24

Indirect

Question 25

What causes the attraction between electrons in a crystal?

Answer 25

Phonon exchange

Question 26

Attraction between electrons via a lattice can produce _____ ________ _____.

Answer 26

Bound electron pairs

Question 27

When is Cooper pair production possible?

Answer 27

- If the electron wavevectors are -k and +k. - If the modulus of the wavevector is equal to the Fermi wavevector. - If the electrons have opposite spin (S = 0 for the pair). - If the electrons have zero angular momentum (L = 0).

Question 28

What is the binding energy per electron of a Cooper pair?

Answer 28

∆(T)

Question 29

What is the binding energy per pair of a Cooper pair?

Answer 29

2∆(T)

Question 30

What is the size of the binding energy of a Cooper pair?

Answer 30

~ a few meV

Question 31

Why are Cooper pairs composite bosons?

Answer 31

Because they have no net spin

Question 32

The superconducting transition is a form of ____-_______ _________.

Answer 32

Bose-Einstein condensation

Question 33

What happens when Cooper pairs condense?

Answer 33

They condense into the superconducting ground state below T_c.

Question 34

Cooper pairs are all described by the same _________ ___________.

Answer 34

Macroscopic wavefunction

Question 35

How does the binding energy per Cooper pair change with an the number of pairs?

Answer 35

The binding energy per Cooper pair increases as the number of pairs increases.

Question 36

Describe a zero supercurrent Cooper pair

Answer 36

The electrons have the wavevectors +k and -k. The centre of mass of the pair is at k = 0.

Question 37

Describe a Cooper pair with a supercurrent

Answer 37

The centre of mass of the pairs has a wavevector, K alongside the electron wavevectors of +k and -k.

Question 38

Give the equation for the average distance between the electrons in a Cooper pair

Answer 38

ζ = BCS coherence length ∆(0) is the energy gap at T = 0

Question 39

What conditions are required for the scattering of a Cooper pair?

Answer 39

The reduction of kinetic energy must be greater than or equal to the pair binding energy (2∆).

Question 40

Describe the wavevectors of a Cooper pair with no supercurrent, with a supercurrent, and after scattering

Answer 40

Question 41

Give the equation for the current density

Answer 41

j = current density n = density of electrons q = charge v = particle velocity

Question 42

Give the equation for the initial total energy of the Cooper pair of electrons before scattering

Answer 42

Ei = initial total energy K = Cooper pair wavevector k = electron wavevector m_e = electron mass

Question 43

Give the equation for the final total energy of the two normal electrons after scattering

Answer 43

Ef = final total energy

Question 44

Give the equation for the critical pair velocity of a Cooper pair

Answer 44

v_c = critical pair velocity ∆ = binding energy

Question 45

Give the equation for the critical current density of a Cooper pair

Answer 45

j_c = critical current density n = density of electrons v_c = critical pair velocity ∆ = binding energy