Thermoelectricity

## Introduction

Thermoelectricity effect, by definition (wikipedia), thermoelectricity refers to a class of phenomena in which a temperature difference creates an electric potential or an electric potential creates a temperature difference.

Seebeck Effect
If a small temperature difference, ∇T is applied to the couple, then the derivative, ~ ∇T/∇T, defines the thermoelectric power of the thermocouple concerned. If the generated potential difference, ∇V, has the same direction shown in the figure above, the absolute thermoelectric power (S_1) of the first conductor is positive with respect to the other conductor,

(1)
\begin{align} S_{1} - S_{2} \equiv \frac{dV}{dT} > 0 \end{align}

If a temperature gradient is applied to a conductor, without an electric field, an energy flow (heat flow) and an electric current called the thermoelectric current will be produced, if somehow we have a closed circuit. Another condition required to observe the thermoelectric current is to have two different materials in the circuit. This is known as the Seebeck effect, in honor of the man to whom the discovery is attributed.

Peltier Effect

If an electric current passes from one material to another, it was also observed that the heat could be either absorbed or released in the junction region, depending on the direction of the current. It is important to distinguish between the Peltier cooling or heat and the joules. The joules heat is a direct consequence of the electric resistivity of the material and is an irreversible effect which depends only on the square of the current density. On the other hand, the Peltier effect depends linearly on the magnitude of the current flow. The temperature of the junction (heat absorbed or emitted depends on the direction of the current flow and the temperature gradient. This implies that this effect is reversible as joule heat is not.

Peltier heat Π at the junction is defined as “heat developed per unit time per unit electric current flowing from left to right."

Thomson Relation

William Thomson, in 1854, related the Peltier heat Πwith the Seebeck Potential V in his postulate called Thomson heat in a conductor. It is express as follows: If an electric current with a current density Jx is passing through a conductor in presence of a temperature gradient dT/dx, then the net heat produced in the conductor per unit volume per second Q,is given by

(2)
\begin{align} Q = \frac{J_{x}^{2}}{\sigma } - \mu J_{x}\frac{dT}{dx} \end{align}

Where σ is the conductivity, $J_{x}$ is the current density, dT/dx is the temperature gradient and μ is the Thomson heat coefficient. Thomson derived a relation between Thompson heat (μ), absolute thermoelectric power (S) and the Peltier heat (Π) of a conductor,

$\mu = \frac{T ds}{dT}$ , $\Pi = TS$

## Estimates of magnitude of thermoelectric properties.

First approach is assume the Thomson heat be equal to the equilibrium specific heat. For this analysis, is going to be used to represent a conductor as a constant number of free electrons in a rigid box.

$\mu = C_{q}$
$\mu$ = Thomson effect; $C_{q}$ = equilibrium specific heat of conduction per unit positive charge

For our model $\mu = \frac{C_{q}}{e}$
$C_el$= specific heat per electron
e = charge of an electron

With Maxwellian Statistics $\mu \approx \frac{3}{2}\frac{K}{e}$
$k = 1.4x10^{-23} joules/k^{0}$
$e = -1.6x10^{-19} coulomb$
$\mu \approx -1.3x10^{-4} volt/deg$

In the case of the metal, the value of the Thomson effect doesn’t match with the experimental data at approximate 0⁰C. This difference indicate that Thomson heat don’t obey the classical theory. The density of conduction electrons in a metal is very high, and they obey the Pauli Exclusion Principle.

$K.E. = p^2/2m$, for a given momentum the kinetic is large by consequence of the small mass of the electron, because of this, the highest states occupied by the conduction electrons correspond to a very high energy. At normal temperature (T « ζ₀/k), a small portion of the electrons, the once near the top of energy distribution can exchange thermal energy with the surroundings. On the lower state doesn’t occur energy transition because all the neighboring state are already occupied. The smaller fraction of electrons who contribute to thermal exchange is given by KT/ ζ₀ where ζ₀ is the Fermi-Energy.

The electronic specific heat in a metal as a dense assembly of free electrons

(3)
\begin{align} C_{el} = \frac{\pi^{2}}{2}k(\frac{kT}{\zeta _{0}}) \equiv \frac{ \pi^2 T}{2 T_0}k \end{align}

$kT_0 = {\zeta _{0}}$
$T_0 = Fermi degeneracy temperature of the elctron$

If metals are compared with semiconductors, will notice that a band gap limited the electrons in the conduction band. At normal temperature just few electrons are available; they must be excited to provide electronic conduction. At the time that electrons are thermal excited from either valance band or from localized impurity levels to the conduction band, their will left back holes (kind of positive electrons).

The mobility of holes is much lower than electron mobility, that is why if it is neglected should not affect the approximation. “ The situation may also arise where there is a large energy gap between the valance and conduction bands, and it is much more profitable to excite electrons from the full valance band into localized atomic orbital around impurity atoms (so called acceptor sites) calling for much less energy than excitation on the conduction band”.

In the semiconductor, how was mentioned before, are excited by thermal energy, consequently, it may be well expect to have a variation of the electrons concentration if temperature change.

$N \propto exp (-U/kT)$
N = conduction electron in an intrinsic material
U = excitation energy per electron

then Thomson heat

$\mu \approx \frac{k}{e}(\frac{3}{3}-\frac{T}{N}\frac{dN}{dT})$

$\mu \approx \frac{k}{e}(\frac{3}{2}-\frac{U}{kT})$

This second term can then be expected to contribute quite markedly to the Thomson heat.

## Non-Equilibrium or Phonon-Drag Effects

The previews discussions were based in a free electron model, no collision inside the conductor. For this model is possible say that the only mechanism giving rise to a thermoelectric current are electrons in the hot end of the conductor to have more thermal energy that the ones in the cold end of the conductor and the electrons diffusion will generated the thermoelectric current.

“However, a temperature gradient causes energy flow through the atomic lattice in the form of thermally generated lattice waves (Debye waves or phonons) with a net energy flow the hotter to the colder end”. If the phonons interact significantly with the conduction electrons, their will behave like the electrical resistance of a conductor which depend on the temperature. The streaming phonons will tend to “sweep” the electrons from the hotter side to the colder side of the conductor (phonon-drag effects).

When electrons interact with phonons strongly enough the specific heat have to be included in the lattice as well.

$S \approx \frac{Cg}{Ne}\alpha$

$C_g$ = specific heat of the lattice per unit volume
$N$ = density of conduction electrons
$\alpha$ = lies between 0 and about 1, id s measure of the relative chance of a phonon "colliding" with a conduction electron

The preview's equation may be rewritten

(4)
\begin{align} S = \frac{C_g}{\beta e}\alpha \end{align}

$C_g$ = lattice specific heat per atom
$\beta$ = number of conduction electrons per atom

A more precise analysis

(5)
\begin{align} S = \frac{1}{3}\frac{C_g}{Ne}\alpha \end{align}

$\alpha$ measure the probability of a phonon to collide with a conduction electron. But phonon not only interact with electrons, their also interact with other phonons due to anharmonic coupling. As the temperature increase the anharmonic coupling effect become more significant, the phonon – phonon collision will predominate rather than phonon – drag effect. As the temperature increase, phonon – phonon interaction increase and α should diminish as 1/T.

For pure metals at low temperature, collision between phonons and conduction electrons will predominate, in the other hand, the lattice specific heat will decrease as the temperature go down as $(T/\theta)^3$, where ѳ is a temperature characteristic of the mass and binding energy of the lattice atoms.

The specific heat per atom is given by Debye model

(6)
\begin{align} C_g = \frac{12\pi^4}{5}k (\frac{T}{\theta })^3 \approx 200k(\frac{T}{\theta })^3 \end{align}

For details about Debye Model of Specific Heat
http://electrons.wikidot.com/debye-model-for-specific-heat

Semiconductors at low temperature can give rise to phonon drag effect, providing thermoelectric power as high as several milivolt per degree. For metals, “it appears unlikely that the phonon – drag effect would ever exceed about 10µV/deg., although when and if this occurs at low temperature (say around 10⁰K), this value is then extremely large compare with ordinary (“diffusion”) thermoelectric power.”
The shape of the Fermi-surface of a metal in relation to the First Brillouin zone structure of a metal will provide details about the angular characteristic of the scattering.

Details of Fermi Surface
http://electrons.wikidot.com/fermi-energy-and-fermi-surface

Detail of First Brillouin Zone
http://electrons.wikidot.com/brillouin-zones

This link show a brief animation about the main idea of thermoelectricity

http://student.britannica.com/lm/animations/othermv001d4/product.html

This link show a video of the new BMW technology using thermo-electricity to reduce fuel consumption

### Reference

Thermoelectricity: An introduction to the Principles, D.K.C. MacDonald

Elementary Solid State Physics, M. Ali Omar

Thermoelectrics Handbook: Macro to Nano, edited by D.M. Rowe

http://en.wikipedia.org/wiki/Thermoelectricity

Theory of Transport Effects in Semiconductors: Thermoelectricity, P. J. Price

Seebeck Effect, Pertier Effect and Thomson Effect images are from :
http://www.chem.cornell.edu/fjd3/thermo/intro.html