Chapter 9 - Thermoelectrics

Question 1

What is the motivation of using thermoelectric generators?

Answer 1

There is often a lot of waste heat which theoretically could be extracted using thermoelectric generators. For example in combustion engines, where only 25% of the energy is used to power the vehicle, and the rest goes out the exhaust gas or to cooling the coolant.

Question 2

What are Joule losses?

Answer 2

Joule losses are energy lost to heat due to electron scattering on phonons.

Question 3

What is the Seebeck effect?

Answer 3

The development of a voltage in a conductor as a result of a temperature gradient.

Question 4

What is the Peltier effect?

Answer 4

The absorption or release of heat at a junction of dissimilar conductors owing to a change in heat capacity of carriers when they leave one medium and enter a different one.

Or: reverse Seebeck effect, apply a voltage and get a temperature gradient.

Question 5

What is the definition of the Seebeck coefficient?

Answer 5

alpha = -∆phi/∆T, that is the ratio of the change of potential to the change of temperature.

Question 6

How is the thermal voltage defined?

Answer 6

Uth = -alpha * ∆T, where alpha is the Seebeck coefficient.

Question 7

Explain the Seebeck effect in terms of the temperature dependence of the Fermi-Dirac distribution.

Answer 7

See slide 5. As the temperature increases, there are more electron-like states and hole-like states. In the case of symmetric carrier transport (holes and electrons have same mobility) there will be no net transport of holes and electrons in a system with two ends held at different temperatures.

Question 8

What is the energy transported proportional to?

Answer 8

(E-E_F) * l(E), where l(E) is the mean free path.

Question 9

How do we achieve a net current flowing between two terminals held at different temperatures?

Answer 9

In metals: We need different effective masses for holes andelectrons.

In semiconductors: we dope it.

Question 10

For a metal, draw the different situation of electron and hole-like states for metals with Seebeck-coefficient 0, 0.

Answer 10

See notes.

Question 11

What is the explanation of the Seebeck-effect in semiconductors?

Answer 11

The side with higher T will have more excited electrons. There will therefore be a concentration gradient in the conduction band, and we have a diffusion of electrons. Since there is a diffusion of electrons from the hot side, there is a net positive charge, which is counteracted by a drift of electrons from cold to hot side. In steady state, these currents cancel each other, but we still have a voltage.

Question 12

What can the sign of the thermal voltage tell us?

Answer 12

The type of charge carrier.

Question 13

What happens when we want to measure the thermal voltage?

Answer 13

We connect metals which also produce a thermal voltage.

Question 14

Draw a schematic of a thermocouple.

Answer 14

See notes.

Question 15

What can be said of the Seebeck-coefficients dependence on T?

Answer 15

It is generally a strong function of T.

Question 16

How would you measure the Seebeck-coefficient?

Answer 16

You would need a reference material with a known Seebeck-coefficient for contacts. This is usually Pt, which is almost symmetric and has a Seebeck-coefficient of almost 0.

Question 17

For a Pelter-element, what is the heat flow given as?

Answer 17

dq/dt = πj, where π is the Peltier coefficient, π = alphaT.

Question 18

When solving the Boltzmann transport equation for the thermoelectrics, what results do we get for i) metals, ii) n-type semiconductors and iii) p-type semiconductors?

Answer 18

i) alpha = - π^2 / 2 * k/e * kT/E_F (for e-like)
ii) alpha = -k/e * [(E_C-E_F) / kT + 5/2 + r]
iii) alpha = +k/e * [(E_F - E_V)/kT + 5/2 +r]

where r is the scattering coefficient.

Question 19

What are the orders of magnitudes of the Seebeck-coefficient for metals, n-type SC and p-type SC?

Answer 19

Metals: µV/K
Semiconductors: mV/K

Question 20

In thermoelectrics, what is the scattering coefficient?

Answer 20

It denotes the contributions of scattering to the Seebeck-coefficient. It is e.g. -1/2 for phonon scattering, 3/2 for scattering at charged defects. The total contributions usually order of 1 when all is combined.

Question 21

Draw a typical layout of a thermoelectric power generator.

Answer 21

See notes.

Question 22

Why must we have a series of many thermoelectric power generators to get a high voltage?

Answer 22

Because they are low-impedance devices, which means they have a low voltage and high current. Many are needed in series to create a high voltage.

Question 23

What are the requirements of the “leg”-materials in a thermoelectric power generator?

Answer 23

• large Seebeck-coefficient, alpha = V/K
• large electrical conductivity, sigma = en/pµ = A/Vm
• small thermal conductivity, kappa = W/Km

Question 24

What is the figure of merit of a thermoelectric device?

Answer 24

Z*T = (alpha^2 * sigma)/kappa * T

Question 25

What is the maximum theoretical efficiency of a thermoelectric generator?

Answer 25

eta = (T_H - T_C)/T_H * (sqrt(1+ZT_M) - 1) / (sqrt(1+ZT_M) + T_C/T_H),

where the first term is the Carnot efficiency of the system. T_M is the average temperature, that is T_M = 1/2 * (T_H - T_C).

Question 26

When is the expression of the maximum efficiency of a thermoelectric generator valid?

Answer 26

When the difference in temperature is way lower than the cold reservoir.

Question 27

For useful power conversions, what does the figure of merit has to be?

Answer 27

Above 1. World record is about 3.

Question 28

What is a common factor among good ZT-materials?

Answer 28

That they all have heavy atoms and are complex alloys (to reduce thermal conductivity).

Question 29

How does the figure of merit depend on the carrier concentration? Draw a plot.

Answer 29

For metals, the Seebeck-coefficient is proportional to kT/E_F, with E_F being proportional to n^2/3.

For semiconductors it is proportional to E_C-E_F/kT = ln(N_C / n).

This means that it is inversely proportional to the conductivity.

But the figure of merit has conductivity shown up in the numerator.

Thus the Seebeck-coefficient decreases with carrier concentration, and the conductivity increases. This can be shown in a plot (see notes).

When it comes to kappa, there is no real dependence on carrier concentration before it reaches 10^21.

The max is a carrier concentration of around 10^20

Question 30

How is the figure of merit dependent on temperature?

Answer 30

Seebeck-coefficient: for metals directly proportional, for semiconductors inversely proportional.

Conductivity: for metals inversely proportional, for semiconductors directly proportional.

Kappa: Phonons goes as T^3 until the Debye-temperature. After that 1/T (since all phonon modes are excited, and we only loose phonons to scattering when moving to higher T). Electrons goes as T.

Question 31

Name three strategies that are investigated to increase ZT.

Answer 31

i) Reduction of kappa in materials with complex unit cells (phonon glass - electron crystal, “ratteling” atoms)
ii) Superlattices of two different materials, e.g. Bi2Te3 / Sb2Te3. Here we get acoustic impedance change at each interface, which reflects phonons (like light reflection), while electrons can pass through.

iii) Nanostructures (nanowires, quantum dots)
- decrease kappa due to interface/surface scattering of phonons.
- increase of alpha and sigma due to a change in electronic density of states: quantum confinement => higher electron energy.