Thermoelectric effect

The thermoelectric effect is the direct conversion of temperature differences to electric voltage and vice versa. A thermoelectric device creates voltage when there is a different temperature on each side. Conversely, when a voltage is applied to it, it creates a temperature difference. At the atomic scale, an applied temperature gradient causes charge carriers in the material to diffuse from the hot side to the cold side.

This effect can be used to generate electricity, measure temperature or change the temperature of objects. Because the direction of heating and cooling is determined by the polarity of the applied voltage, thermoelectric devices can be used as temperature controllers.

The term "thermoelectric effect" encompasses three separately identified effects: the Seebeck effect, Peltier effect, and Thomson effect. Textbooks may refer to it as the Peltier–Seebeck effect. This separation derives from the independent discoveries of French physicist Jean Charles Athanase Peltier and Baltic German physicist Thomas Johann Seebeck. Joule heating, the heat that is generated whenever a voltage is applied across a resistive material, is related though it is not generally termed a thermoelectric effect. The Peltier–Seebeck and Thomson effects are thermodynamically reversible,[1] whereas Joule heating is not.

Seebeck effect

A thermoelectric circuit composed of materials of different Seebeck coefficient (p-doped and n-doped semiconductors), configured as a thermoelectric generator. If the load is removed then the current stops, and the circuit functions as a temperature-sensing thermocouple.

The Seebeck effect is the conversion of temperature differences directly into electricity and is named after the Baltic German physicist Thomas Johann Seebeck, who, in 1821, discovered that a compass needle would be deflected by a closed loop formed by two different metals joined in two places, with a temperature difference between the junctions. This was because the metals responded differently to the temperature difference, creating a current loop and a magnetic field. Seebeck did not recognize there was an electric current involved, so he called the phenomenon the thermomagnetic effect. Danish physicist Hans Christian Ørsted rectified the mistake and coined the term "thermoelectricity".

The Seebeck effect is a classic example of an electromotive force (emf) and leads to measurable currents or voltages in the same way as any other emf. Electromotive forces modify Ohm's law by generating currents even in the absence of voltage differences (or vice versa); the local current density is given by

$\mathbf J = \sigma (-\boldsymbol \nabla V + \mathbf E_{\rm emf})$

where $\scriptstyle V$ is the local voltage[2] and $\scriptstyle \sigma$ is the local conductivity. In general the Seebeck effect is described locally by the creation of an electromotive field

$\mathbf E_{\rm emf} = - S \boldsymbol\nabla T$

where $\scriptstyle S$ is the Seebeck coefficient (also known as thermopower), a property of the local material, and $\scriptstyle \boldsymbol \nabla T$ is the gradient in temperature $\scriptstyle T$.

The Seebeck coefficients generally vary as function of temperature, and depend strongly on the composition of the conductor. For ordinary materials at room temperature, the Seebeck coefficient may range in value from −100 μV/K to +1,000 μV/K (see Thermoelectric materials)

If the system reaches a steady state where $\scriptstyle \mathbf J \;=\; 0$, then the voltage gradient is given simply by the emf: $\scriptstyle -\boldsymbol \nabla V \;=\; S \boldsymbol\nabla T$. This simple relationship, which does not depend on conductivity, is used in the thermocouple to measure a temperature difference; an absolute temperature may be found by performing the voltage measurement at a known reference temperature. A metal of unknown composition can be classified by its thermoelectric effect if a metallic probe of known composition is kept at a constant temperature and held in contact with the unknown sample that is locally heated to the probe temperature. It is used commercially to identify metal alloys. Thermocouples in series form a thermopile. Thermoelectric generators are used for creating power from heat differentials.

Peltier effect

The Seebeck circuit configured as a thermoelectric cooler

The Peltier effect is the presence of heating or cooling at an electrified junction of two different conductors and is named for French physicist Jean Charles Athanase Peltier, who discovered it in 1834. When a current is made to flow through a junction between two conductors A and B, heat may be generated (or removed) at the junction. The Peltier heat generated at the junction per unit time, $\scriptstyle \dot{Q}$, is equal to

$\dot{Q} = \left( \Pi_\mathrm{A} - \Pi_\mathrm{B} \right) I$

where $\scriptstyle \Pi_A$ ($\scriptstyle \Pi_B$) is the Peltier coefficient of conductor A (B), and $\scriptstyle I$ is the electric current (from A to B). Note that the total heat generated at the junction is not determined by the Peltier effect alone, as it may also be influenced by Joule heating and thermal gradient effects (see below).

The Peltier coefficients represent how much heat is carried per unit charge. Since charge current must be continuous across a junction, the associated heat flow will develop a discontinuity if $\scriptstyle \Pi_A$ and $\scriptstyle \Pi_B$ are different. The Peltier effect can be considered as the back-action counterpart to the Seebeck effect (analogous to the back-emf in magnetic induction): if a simple thermoelectric circuit is closed then the Seebeck effect will drive a current, which in turn (via the Peltier effect) will always transfer heat from the hot to the cold junction. The close relationship between Peltier and Seebeck effects can be seen in the direct connection between their coefficients: $\scriptstyle \Pi \;=\; T S$ (see below).

A typical Peltier heat pump device involves multiple junctions in series, through which a current is driven. Some of the junctions lose heat due to the Peltier effect, while others gain heat. Thermoelectric heat pumps exploit this phenomenon, as do thermoelectric cooling devices found in refrigerators.

Thomson effect

In many materials, the Seebeck coefficient is not constant in temperature, and so a spatial gradient in temperature can result in a gradient in the Seebeck coefficient. If a current is driven through this gradient then a continuous version of the Peltier effect will occur. This Thomson effect was predicted and subsequently observed by Lord Kelvin in 1851. It describes the heating or cooling of a current-carrying conductor with a temperature gradient.

If a current density $\scriptstyle \mathbf J$ is passed through a homogeneous conductor, the Thomson effect predicts a heat production rate $\scriptstyle \dot q$ per unit volume of:

$\dot q = -\mathcal K \mathbf J \cdot \boldsymbol \nabla T$

where $\scriptstyle \boldsymbol \nabla T$ is the temperature gradient and $\scriptstyle \mathcal K$ is the Thomson coefficient. The Thomson coefficient is related to the Seebeck coefficient as $\scriptstyle \mathcal K \;=\; T\, \frac{dS}{dT}$ (see below). This equation however neglects Joule heating, and ordinary thermal conductivity (see full equations below).

Full thermoelectric equations

Often, more than one of the above effects is involved in the operation of a real thermoelectric device. The Seebeck effect, Peltier effect, and Thomson effect can be gathered together in a consistent and rigorous way, described here; the effects of Joule heating and ordinary heat conduction are included as well. As stated above, the Seebeck effect generates an electromotive force, leading to the current equation[3]

$\mathbf J = \sigma (-\boldsymbol \nabla V - S \boldsymbol\nabla T)$

To describe the Peltier and Thomson effects we must consider the flow of energy. To start we can consider the dynamic case where both temperature and charge may be varying with time. The full thermoelectric equation for the energy accumulation, $\scriptstyle \dot e$ is[3]

$\dot e = \boldsymbol \nabla \cdot (\kappa \boldsymbol \nabla T) - \boldsymbol \nabla \cdot (V + \Pi)\mathbf J + \dot q_{\rm ext}$

where $\scriptstyle \kappa$ is the thermal conductivity. The first term is the Fourier's heat conduction law, and the second term shows the energy carried by currents. The third term $\scriptstyle \dot q_{\rm ext}$ is the heat added from an external source (if applicable).

In the case where the material has reached a steady state, the charge and temperature distributions are stable so one must have both $\scriptstyle \dot e \;=\; 0$ and $\scriptstyle \boldsymbol \nabla \,\cdot\, \mathbf J \;=\; 0$. Using these facts and the second Thomson relation (see below), the heat equation then can be simplified to

$-\dot q_{\rm ext} = \boldsymbol \nabla \cdot (\kappa \boldsymbol \nabla T) + \mathbf J \cdot \left(\sigma^{-1} \mathbf J\right) - T \mathbf J \cdot\boldsymbol \nabla S$

The middle term is the Joule heating, and the last term includes both Peltier ($\scriptstyle \boldsymbol \nabla S$ at junction) and Thomson ($\scriptstyle \boldsymbol \nabla S$ in thermal gradient) effects. Combined with the Seebeck equation for $\scriptstyle \mathbf J$, this can be used to solve for the steady state voltage and temperature profiles in a complicated system.

If the material is not in a steady state, a complete description will also need to include dynamic effects such as relating to electrical capacitance, inductance, and heat capacity.

Physical origin of the thermoelectric coefficients

A material's temperature, crystal structure, and impurities influence the value of the thermoelectric coefficients. The Seebeck effect can be attributed to two things[citation needed]: charge-carrier diffusion and phonon drag. Typically metals have small Seebeck coefficients because of partially filled bands, with a conductivity that is relatively insensitive to small changes in energy. In contrast, semiconductors can be doped with impurities that donate excess electrons or electron holes, allowing the value of S to be varied over a large range (both negative and positive). The sign of the Seebeck coefficients can be used to determine whether the electrons or the holes dominate electric transport in a semiconductor or semimetal.

Thomson relations

In 1854, Lord Kelvin found relationships between the three coefficients, implying that the Thomson, Peltier, and Seebeck effects are different manifestations of one effect (uniquely characterized by the Seebeck coefficient).

The first Thomson relation is[3]

$\mathcal K \equiv {d\Pi \over dT} - S$

where $\scriptstyle T$ is the absolute temperature, $\scriptstyle \mathcal K$ is the Thomson coefficient, $\scriptstyle \Pi$ is the Peltier coefficient, and $\scriptstyle S$ is the Seebeck coefficient. This relationship is easily shown given that the Thomson effect is a continuous version of the Peltier effect. Using the second relation (described next), the first Thomson relation becomes $\scriptstyle \mathcal K \;=\; T \frac{dS}{dT}$.

The second Thomson relation is

$\Pi = TS$

This relation expresses a subtle and fundamental connection between the Peltier and Seebeck effects. It was not satisfactorily proven until the advent of the Onsager relations, and it is worth noting that this second Thomson relation is only guaranteed for a time-reversal symmetric material; if the material is placed in a magnetic field, or is itself magnetically ordered (ferromagnetic, antiferromagnetic, etc.), then the second Thomson relation does not take the simple form shown here.[4]

The Thomson coefficient is unique among the three main thermoelectric coefficients because it is the only one directly measurable for individual materials. The Peltier and Seebeck coefficients can only be easily determined for pairs of materials; hence, it is difficult to find values of absolute Seebeck or Peltier coefficients for an individual material.

If the Thomson coefficient of a material is measured over a wide temperature range, it can be integrated using the Thomson relations to determine the absolute values for the Peltier and Seebeck coefficients. This needs to be done only for one material, since the other values can be determined by measuring pairwise Seebeck coefficients in thermocouples containing the reference material and then adding back the absolute thermopower of the reference material.

Charge carrier diffusion

On a fundamental level, an applied voltage difference refers to a difference in the thermodynamic chemical potential of charge carriers, and the direction of the current under a voltage difference is determined by the universal thermodynamic process in which (given equal temperatures) particles flow from high chemical potential to low chemical potential. In other words, the direction of the current in Ohm's law is determined via the thermodynamic arrow of time (the difference in chemical potential could be exploited to produce work, but is instead dissipated as heat which increases entropy). On the other hand, for the Seebeck effect not even the sign of the current cannot be predicted from thermodynamics, and so to understand the origin of the Seebeck coefficient it is necessary to understand the microscopic physics.

Charge carriers (such as thermally excited electrons) constantly diffuse around inside a conductive material. Due to thermal fluctuations, some of these charge carriers travel with a higher energy than average, and some with a lower energy. When no voltage differences or temperature differences are applied, the carrier diffusion perfectly balances out and so on average one sees no current: $\scriptstyle\mathbf J = 0$. A net current can be generated by applying a voltage difference (Ohm's law), or by applying a temperature difference (Seebeck effect). To understand the microscopic origin of the thermoelectric effect, it is useful to first describe the microscopic mechanism of the normal Ohm's law electrical conductance—to describe what determines the $\scriptstyle\sigma$ in $\scriptstyle\mathbf J = -\sigma\boldsymbol\nabla V$. Microscopically, what is happening in Ohm's law is that higher energy levels have a higher concentration of carriers per state, on the side with higher chemical potential. For each interval of energy, the carriers tend to diffuse and spread into the area of device where there are less carriers per state of that energy. As they move, however, they occasionally scatter dissipatively, which re-randomizes their energy according to the local temperature and chemical potential. This dissipation empties out the carriers from these higher energy states, allowing more to diffuse in. The combination of diffusion and dissipation favours an overall drift of the charge carriers towards the side of the material where they have a lower chemical potential.[5]

For the thermoelectric effect, now, consider the case of uniform voltage (uniform chemical potential) with a temperature gradient. In this case, at the hotter side of the material there is more variation in the energies of the charge carriers, compared to the colder side. This means that high energy levels have a higher carrier occupation per state on the hotter side, but also the hotter side has a lower occupation per state at lower energy levels. As before, the high-energy carriers diffuse away from the hot end, and produce entropy by drifting towards the cold end of the device. However, there is a competing process: at the same time low-energy carriers are drawn back towards the the hot end of the device. Though these processes both generate entropy, they work against each other in terms of charge current, and so a net current only occurs if one of these drifts is stronger than the other. The net current is given by $\scriptstyle\mathbf J = \sigma S\boldsymbol\nabla T$, where (as shown below) the thermoelectric coefficient $\scriptstyle\sigma S$ depends literally on how conductive high-energy carriers are, compared to low-energy carriers. The distinction may be due to a difference in rate of scattering, a difference in speeds, a difference in density of states, or a combination of these effects.

Mott formula

The processes described above apply in materials where each charge carrier sees an essentially static environment so that its motion can be described independently from other carriers, and independent of other dynamics (such as phonons). In particular, in electronic materials with weak electron-electron interactions, weak electron-phonon interactions, etc. it can be shown in general that the linear response conductance is

$\sigma = \int c(E) \Bigg( -\frac{df(E)}{dE} \Bigg) \, dE,$

and the linear response thermoelectric coefficient is

$\sigma S = \frac{k_{\rm B}}{-e} \int \frac{E - \mu}{k_{\rm B}T} c(E) \Bigg( -\frac{df(E)}{dE} \Bigg) \, dE$

where $\scriptstyle c(E)$ is the energy-dependent conductivity, and $\scriptstyle f(E)$ is the Fermi-Dirac distribution function. These equations are known as the Mott relations, of Sir Nevill Francis Mott.[6] The derivative $\scriptstyle -\frac{df(E)}{dE} = \frac{1}{4kT} \operatorname{sech}^2 \tfrac{E-\mu}{2kT}$ is a function peaked around the chemical potential (Fermi level) $\scriptstyle \mu$ with a width of approximately $\scriptstyle 3.5 kT$. The energy-dependent conductivity (a quantity that cannot actually be directly measured — one only measures $\scriptstyle\sigma$) is calculated as $\scriptstyle c(E) = e^2 D(E) \nu(E)$ where $\scriptstyle D(E)$ is the electron diffusion constant and $\scriptstyle \nu(E)$ is the electronic density of states (in general, both are functions of energy).

In materials with strong interactions, none of the above equations can be used since it is not possible to consider each charge carrier as a separate entity. It is worth noting that the Wiedemann-Franz law can also be exactly derived using the non-interacting electron picture, and so in materials where the Wiedemann-Franz law fails (such as superconductors), the Mott relations also generally tend to fail.[7]

The formulae above can be simplified in a couple of important limiting cases:

• In semimetals and metals, where transport only occurs near the Fermi level and $\scriptstyle c(E)$ changes slowly in the range $E \approx \mu \pm kT$, one can perform a Sommerfeld expansion $\scriptstyle c(E) = c(\mu) + c'(\mu) (E-\mu) + O[(E-\mu)^2]$, which leads to
$S_{\rm metal} = \frac{\pi^2 k^2 T}{-3 e} \frac{c'(\mu)}{c(\mu)} + O[(kT)^3], \quad \sigma_{\rm metal} = c(\mu) + O[(kT)^2].$
This expression is sometimes called "the Mott formula", however it is much less general than Mott's original formula expressed above. In the Drude-Sommerfeld degenerate free electron gas with scattering, the value of $\scriptstyle c'(\mu) / c(\mu)$ is of order $\scriptstyle 1/(kT_{\rm F})$, where $T_{\rm F}$ is the Fermi temperature, and so a typical value of the Seebeck coefficient in the Fermi gas is $\scriptstyle S_{\rm Fermi~gas} \approx \tfrac{\pi^2 k}{-3e} T/T_{\rm F}$ (the prefactor varies somewhat depending on details such as dimensionality and scattering). In highly conductive metals the Fermi temperatures are typically around 104 – 105 K, and so it is understandable why their absolute Seebeck coefficients are only of order 1 – 10 μV/K at room temperature. Note that whereas the free electron gas is expected to have a negative Seebeck coefficient, real metals actually have complicated band structures and may exhibit positive Seebeck coefficients (examples: Cu, Ag, Au). The fraction $\scriptstyle c'(\mu) / c(\mu)$ in semimetals is sometimes calculated from the measured derivative of $\scriptstyle \sigma_{\rm metal}$ with respect to some energy shift induced by field effect. This is not necessarily correct and the estimate of $\scriptstyle c'(\mu) / c(\mu)$ can be incorrect (by a factor of two or more), since the disorder potential depends on screening which also changes with field effect.[8]
• In semiconductors at low levels of doping, transport only occurs far away from the Fermi level. At low doping in the conduction band (where $\scriptstyle E_{\rm C} - \mu \gg kT$, where $\scriptstyle E_{\rm C}$ is the minimum energy of the conduction band edge), one has $\scriptstyle -\frac{df(E)}{dE} \approx \tfrac{1}{kT} e^{-(E-\mu)/(kT)}$. Approximating the conduction band levels' conductivity function as $\scriptstyle c(E) = A_{\rm C} (E - E_{\rm C})^{a_{\rm C}}$ for some constants $\scriptstyle A_{\rm C}$ and $\scriptstyle a_{\rm C}$,
$S_{\rm C} = \frac{k}{-e} \Big[ \frac{E_{\rm C} - \mu}{kT} + a_{\rm C} + 1\Big], \quad \sigma_{\rm C} = A_{\rm C} (kT)^{a_{\rm C}} e^{-\frac{E_{\rm C} - \mu}{kT}} \Gamma(a_{\rm C}+1).$
whereas in the valence band when $\scriptstyle \mu - E_{\rm V}\gg kT$ and $\scriptstyle c(E) = A_{\rm V} (E_{\rm V} - E)^{a_{\rm V}}$,
$S_{\rm V} = \frac{k}{e} \Big[ \frac{\mu - E_{\rm V}}{kT} + a_{\rm V} + 1\Big], \quad \sigma_{\rm V} = A_{\rm V} (kT)^{a_{\rm V}} e^{-\frac{\mu - E_{\rm V}}{kT}} \Gamma(a_{\rm V}+1).$
The values of $\scriptstyle a_{\rm C}$ and $\scriptstyle a_{\rm V}$ depend on material details; in bulk semiconductor these constants range between 1 and 3, the extremes corresponding to acoustic-mode lattice scattering and ionized-impurity scattering.[9] In extrinsic (doped) semiconductors either the conduction or valence band will dominate transport, and so one of the numbers above will give the measured values. In general however the semiconductor can be intrinsic in which case the bands conduct in parallel, and so the measured values will be
$S_{\rm semi} = \frac{\sigma_{\rm C} S_{\rm C} + \sigma_{\rm V} S_{\rm V}}{\sigma_{\rm C} + \sigma_{\rm V}}, \quad \sigma_{\rm semi} = \sigma_{\rm C} + \sigma_{\rm V}$
The highest Seebeck coefficient is obtained when the semiconductor is lightly doped, however a high Seebeck coefficient is not necessarily useful. For thermoelectric power devices (coolers, generators) it is more important to maximize the thermoelectric power factor $\scriptstyle \sigma^2 S$, or the thermoelectric figure of merit, and the optimum generally occurs at high doping levels.[10]

Spin

Recently, researcher from Keio and Tohoku University have reported the observation of the thermal generation of driving power,[11] or voltage, for electron spin: the spin Seebeck effect. Using a recently developed spin-detection technique that involves the spin Hall effect, they measured the spin voltage generated from a temperature gradient in a metallic magnet. One significant property of this thermally induced spin voltage is that it persists even at distances far from the sample ends, and spins can be extracted from every position on the magnet simply by attaching a metal. The spin Seebeck effect observed is directly applicable to the production of spin-voltage generators, which are crucial for driving spintronic devices.This discovery allows people to pass a pure spin current, a flow of electron spins without electric currents, over a long distance. These innovative capabilities will invigorate spintronics research.

Phonon drag

Phonons are not always in local thermal equilibrium; they move against the thermal gradient. They lose momentum by interacting with electrons (or other carriers) and imperfections in the crystal. If the phonon-electron interaction is predominant, the phonons will tend to push the electrons to one end of the material, hence losing momentum and contributing to the thermoelectric field. This contribution is most important in the temperature region where phonon-electron scattering is predominant. This happens for

$T \approx {1 \over 5} \theta_\mathrm{D}$

where $\scriptstyle \theta_D$ is the Debye temperature. At lower temperatures there are fewer phonons available for drag, and at higher temperatures they tend to lose momentum in phonon-phonon scattering instead of phonon-electron scattering. This region of the thermopower-versus-temperature function is highly variable under a magnetic field.[citation needed]

Relationship with entropy

The thermopower or Seebeck coefficient, represented by S, of a material measures the magnitude of an induced thermoelectric voltage in response to a temperature difference across that material, and the entropy per charge carrier in the material.[12] S has units of V/K, though μV/K is more common.

Superconductors have S = 0 since the charged carriers produce no entropy. This allows a direct measurement of the absolute thermopower of the material of interest, since it is the thermopower of the entire thermocouple.

Applications

Thermoelectric generators

The Seebeck effect is used in thermoelectric generators, which function like heat engines, but are less bulky, have no moving parts, and are typically more expensive and less efficient. They have a use in power plants for converting waste heat into additional electrical power (a form of energy recycling), and in automobiles as automotive thermoelectric generators (ATGs) for increasing fuel efficiency. Space probes often use radioisotope thermoelectric generators with the same mechanism but using radioisotopes to generate the required heat difference.

Peltier effect

The Peltier effect can be used to create a refrigerator which is compact and has no circulating fluid or moving parts; such refrigerators are useful in applications where their advantages outweigh the disadvantage of their very low efficiency.

Temperature measurement

Thermocouples and thermopiles are devices that use the Seebeck effect to measure the temperature difference between two objects, one connected to a voltmeter and the other to the probe. The temperature of the voltmeter, and hence that of the material being measured by the probe, can be measured separately using cold junction compensation techniques.

• Nernst and Ettingshausen effects – special thermoelectric effects in a magnetic field.
• Pyroelectricity – the creation of an electric polarization in a crystal after heating/cooling, an effect distinct from thermoelectricity.

References

1. ^ As the "figure of merit" approaches infinity, the Peltier–Seebeck effect can drive a heat engine or refrigerator at closer and closer to the Carnot efficiency. Disalvo, F. J. (1999). "Thermoelectric Cooling and Power Generation". Science 285 (5428): 703–6. doi:10.1126/science.285.5428.703. PMID 10426986. Any device that works at the Carnot efficiency is thermodynamically reversible, a consequence of classical thermodynamics.
2. ^ The voltage in this case does not refer to electric potential but rather the 'voltmeter' voltage $\scriptstyle V \;=\; -\mu/e$, where $\scriptstyle \mu$ is the Fermi level.
3. ^ a b c "A.11 Thermoelectric effects". Eng.fsu.edu. 2002-02-01. Retrieved 2013-04-22.
4. ^ There is a generalized second Thomson relation relating anisotropic Peltier and Seebeck coefficients with reversed magnetic field and magnetic order. See, for example, Rowe, D.M., ed. (2010). Thermoelectrics Handbook: Macro to Nano. CRC Press. ISBN 9781420038903. edit
5. ^ Datta, Supriyo (2005). Quantum Transport: Atom to Transistor. Cambridge University Presss. ISBN 9780521631457. edit Chapter 11.
6. ^ Cutler, M.; Mott, N. (1969). "Observation of Anderson Localization in an Electron Gas". Physical Review 181 (3): 1336. Bibcode:1969PhRv..181.1336C. doi:10.1103/PhysRev.181.1336. edit
7. ^ Jonson, M.; Mahan, G. (1980). "Mott's formula for the thermopower and the Wiedemann-Franz law". Physical Review B 21 (10): 4223. doi:10.1103/PhysRevB.21.4223. edit
8. ^ Hwang, E. H.; Rossi, E.; Das Sarma, S. (2009). "Theory of thermopower in two-dimensional graphene". Physical Review B 80 (23). doi:10.1103/PhysRevB.80.235415. edit
9. ^ Semiconductor Physics: An Introduction， Karlheinz Seeger
10. ^ G. Jeffrey Snyder, "Thermoelectrics". http://www.its.caltech.edu/~jsnyder/thermoelectrics/
11. ^ K. Uchida, S. Takahashi; K. Harii, J.Ieda (2008). "Observation of the spin Seebeck effect". Nature 455 (3): 778. doi:10.1038/nature07321. edit
12. ^ Rockwood, Alan L. (1984). "Relationship of thermoelectricity to electronic entropy". Phys. Rev. A 30 (5): 2843–4. Bibcode:1984PhRvA..30.2843R. doi:10.1103/PhysRevA.30.2843.