4.5 Thermoelectric effects

Thermoelectric effects describe the direct conversion between thermal and electrical energy in materials. These phenomena arise from the coupling of heat and charge transport, and they underpin devices that generate electricity from heat (thermoelectric generators) or provide cooling through electrical input (Peltier coolers).

The three foundational effects are the Seebeck, Peltier, and Thomson effects. Together with the thermoelectric figure of merit $ZT$, they define how we evaluate and optimize thermoelectric materials for energy harvesting, cooling, and thermal management.

Thermoelectric phenomena

Thermoelectric phenomena occur because charge carriers (electrons or holes) carry both charge and thermal energy. When a temperature gradient or electric current is present, these two forms of transport become coupled. That coupling is what makes it possible to convert heat into voltage or current into a temperature difference.

Three distinct effects capture this coupling:

• The Seebeck effect converts a temperature difference into a voltage
• The Peltier effect converts an electric current into heating or cooling at a junction
• The Thomson effect describes heat exchange when current flows through a single material that has a temperature gradient

All three are thermodynamically related through the Kelvin relations, which connect their respective coefficients.

Seebeck effect

Discovered by Thomas Johann Seebeck in 1821, this effect occurs when a temperature gradient across a conductor or semiconductor causes charge carriers to diffuse from the hot side to the cold side. That diffusion builds up a net charge at the cold end, producing a measurable voltage.

Seebeck coefficient

The Seebeck coefficient (also called thermopower) quantifies how much voltage a material generates per unit temperature difference:

$S = \frac{\Delta V}{\Delta T}$

Typical units are $\mu V/K$. Metals have small Seebeck coefficients (a few $\mu V/K$), while good thermoelectric semiconductors reach hundreds of $\mu V/K$. The sign of $S$ tells you the dominant carrier type: negative for electrons (n-type), positive for holes (p-type).

Temperature gradient and voltage generation

Here's the physical picture of how the Seebeck effect works:

1. A temperature difference $\Delta T$ is maintained across the material.
2. Charge carriers on the hot side have more kinetic energy and diffuse toward the cold side.
3. This diffusion creates a charge imbalance, which sets up an internal electric field opposing further diffusion.
4. At steady state, the electric field exactly balances the diffusion tendency, and a measurable open-circuit voltage $\Delta V = S \cdot \Delta T$ appears across the material.

This voltage can drive current through an external load, which is the operating principle of thermoelectric generators.

Peltier effect

Discovered by Jean Charles Athanase Peltier in 1834, this is essentially the reverse of the Seebeck effect. When an electric current passes through a junction between two dissimilar materials, heat is absorbed at one junction and released at the other.

Peltier coefficient

The Peltier coefficient $\Pi$ gives the rate of heat exchange per unit current at a junction:

$\Pi = \frac{Q}{I}$

where $Q$ is the rate of heat absorbed or emitted and $I$ is the current. The Kelvin relation connects it directly to the Seebeck coefficient:

$\Pi = S \cdot T$

where $T$ is the absolute temperature of the junction.

Current flow and heat absorption/emission

• At the junction where current flows from material A to material B, heat is absorbed (the junction cools down).
• At the opposite junction, where current flows from B to A, heat is released (the junction heats up).
• Reversing the current direction swaps which junction heats and which cools.

This reversibility is what makes Peltier devices useful as solid-state heat pumps: you can switch between cooling and heating mode just by flipping the current direction.

Thomson effect

Discovered by William Thomson (Lord Kelvin) in 1851, this effect is subtler than the other two. It describes the heat absorbed or released when current flows through a single material that has a temperature gradient along it.

Thomson coefficient

The Thomson coefficient $\mu$ is defined as:

$\mu = \frac{Q}{J \cdot \nabla T}$

where $J$ is the current density and $\nabla T$ is the temperature gradient. The Kelvin relation ties it to the Seebeck coefficient:

$\mu = T \frac{dS}{dT}$

This means the Thomson coefficient is nonzero only if the Seebeck coefficient varies with temperature.

Heat generation/absorption

• Current flowing from a hot region to a cold region absorbs heat (cooling).
• Current flowing from a cold region to a hot region releases heat (heating).

The Thomson effect is generally smaller than the Seebeck and Peltier effects, but it becomes significant in materials where $S$ changes rapidly with temperature. It also plays an important role in the thermodynamic consistency of thermoelectric theory.

Figure of merit

The thermoelectric figure of merit $ZT$ is the single most important parameter for evaluating thermoelectric material performance. It captures the fundamental trade-offs between the three transport properties that matter.

Definition

$ZT = \frac{S^2 \sigma}{\kappa} T$

where:

• $S$ is the Seebeck coefficient
• $\sigma$ is the electrical conductivity
• $\kappa$ is the total thermal conductivity (electronic + lattice contributions)
• $T$ is the absolute temperature

The numerator $S^2 \sigma$ is often called the power factor. A good thermoelectric material needs a large power factor (high voltage generation with low electrical resistance) and low thermal conductivity (so the temperature gradient isn't short-circuited by heat flow).

Thermoelectric efficiency

The maximum efficiency of a thermoelectric generator operating between hot-side temperature $T_H$ and cold-side temperature $T_C$ is:

$\eta_{max} = \frac{T_H - T_C}{T_H} \cdot \frac{\sqrt{1 + ZT_{avg}} - 1}{\sqrt{1 + ZT_{avg}} + T_C / T_H}$

The first factor is the Carnot efficiency. The second factor, which depends on $ZT_{avg}$ (averaged over the operating temperature range), determines how close you get to Carnot. As $ZT \to \infty$, the device approaches Carnot efficiency. Current state-of-the-art materials reach $ZT \approx 1$ to $2.5$ at their optimal operating temperatures.

Optimization strategies

Maximizing $ZT$ is difficult because $S$, $\sigma$, and $\kappa$ are all interrelated through carrier concentration:

• Increasing carrier concentration raises $\sigma$ but typically decreases $S$
• Higher $\sigma$ also increases the electronic contribution to $\kappa$

Strategies to navigate these trade-offs include:

• Doping optimization: Tuning carrier concentration to maximize the power factor (typically $10^{19}$ to $10^{21}$ carriers/cm³ for good thermoelectrics)
• Nanostructuring: Introducing grain boundaries or nanoparticles that scatter phonons more than electrons, reducing lattice thermal conductivity
• Band structure engineering: Manipulating band convergence or resonant states to enhance $S$ without sacrificing $\sigma$

Thermoelectric materials

The best thermoelectric materials are typically narrow-gap semiconductors or heavily doped semiconductors. Metals have too-low Seebeck coefficients, and insulators have too-low electrical conductivity. Semiconductors sit in the sweet spot.

Key material families

Different materials excel in different temperature ranges:

• Bismuth telluride ($Bi_2Te_3$): The workhorse for near-room-temperature applications (200–400 K). $ZT \approx 1$ at 300 K.
• Lead telluride ($PbTe$): Optimal for mid-temperature range (500–900 K). Used in some power generation applications.
• Silicon germanium ($SiGe$): Best at high temperatures (>900 K). Used in NASA's radioisotope thermoelectric generators.
• Skutterudites and half-Heusler alloys: Promising mid-to-high temperature materials with $ZT$ values approaching 1.5 in optimized compositions.

Balancing transport properties

• Low thermal conductivity is desirable because it maintains the temperature gradient that drives the Seebeck voltage. Lattice thermal conductivity can be reduced through phonon scattering at grain boundaries, point defects, or nanostructured inclusions.
• High electrical conductivity minimizes resistive (Joule) losses. It's controlled primarily through carrier concentration via doping.
• High Seebeck coefficient maximizes voltage output per degree of temperature difference. It depends on band structure details and carrier concentration, and can be enhanced through band convergence or energy filtering of carriers.

Applications of thermoelectrics

Thermoelectric generators

These devices convert waste heat into electrical power with no moving parts. Applications include:

• Automotive: Recovering energy from exhaust heat to improve fuel efficiency
• Industrial: Harvesting heat from furnaces, pipelines, or process streams
• Space missions: Radioisotope thermoelectric generators (RTGs) have powered NASA's Voyager probes since 1977 and the Curiosity Mars rover. RTGs use the heat from radioactive decay of $^{238}Pu$ to generate electricity reliably for decades.

Thermoelectric coolers

Peltier-based coolers provide precise, compact, solid-state cooling:

• Stabilizing the temperature of laser diodes and infrared detectors in optoelectronics
• Small-scale refrigeration (portable coolers, CPU cooling)
• Temperature cycling in PCR machines for biological applications

Their main advantage is the absence of moving parts or refrigerant fluids, which makes them highly reliable and compact.

Waste heat recovery and space exploration

Thermoelectric generators are well-suited for waste heat recovery because they scale easily, require no maintenance, and work with any heat source. In space, RTGs are preferred over solar panels for missions beyond Jupiter, where sunlight is too weak. The Voyager 1 and 2 probes, launched in 1977, still operate on RTG power more than 45 years later.

Measurement techniques

Accurate characterization of $S$, $\sigma$, and $\kappa$ is essential for calculating $ZT$ and comparing materials.

Seebeck coefficient measurement

1. Apply a small, controlled temperature difference $\Delta T$ across the sample.
2. Measure the open-circuit voltage $\Delta V$ that develops.
3. Repeat for several values of $\Delta T$.
4. Plot $\Delta V$ vs. $\Delta T$; the slope gives $S$.

Care must be taken to ensure the voltage probes are at the same temperature as the points where $T$ is measured, or systematic errors result.

Electrical conductivity measurement

The four-point probe technique is standard:

1. Place four equally spaced probes in a line on the sample surface.
2. Pass a known current $I$ through the two outer probes.
3. Measure the voltage drop $V$ across the two inner probes.
4. Calculate the resistivity from $V$, $I$, and the sample geometry, then invert to get $\sigma$.

Using separate current and voltage probes eliminates contact resistance from the measurement.

Thermal conductivity measurement

Two common methods:

• Laser flash method: A short laser pulse heats one face of a thin disc sample. An infrared detector on the opposite face records the temperature rise over time. From the time profile, you extract the thermal diffusivity $\alpha$, then calculate $\kappa = \alpha \rho c_p$ (where $\rho$ is density and $c_p$ is specific heat).
• 3ω method: A narrow metal strip deposited on the sample serves as both heater and thermometer. An AC current at frequency $\omega$ causes heating at $2\omega$, producing a voltage signal at $3\omega$ that depends on the sample's thermal conductivity. This method is especially useful for thin films.

Challenges in thermoelectrics

Material optimization

The central challenge is that $S$, $\sigma$, and $\kappa$ are coupled through the electronic structure. Increasing carrier concentration to boost $\sigma$ tends to decrease $S$ and increase the electronic part of $\kappa$. Breaking these trade-offs requires creative materials design, and achieving $ZT > 3$ (which would make thermoelectrics competitive with mechanical heat engines for many applications) remains an open goal.

Thermal management and device engineering

Even with high-$ZT$ materials, device performance depends on:

• Minimizing thermal contact resistance at interfaces between the thermoelectric legs and heat exchangers
• Designing heat exchangers that efficiently deliver heat to (and remove heat from) the thermoelectric module
• Managing thermal stress from repeated heating/cooling cycles, which can cause cracking or delamination

Cost-effectiveness

Many high-performance thermoelectric materials contain tellurium, germanium, or other elements that are scarce or expensive. For thermoelectrics to see widespread commercial adoption (e.g., in every car's exhaust system), either these materials need to be replaced with earth-abundant alternatives, or the required material volume must be dramatically reduced.

Future prospects

Nanostructured materials

Nanostructuring remains one of the most effective strategies for improving $ZT$. Quantum dots, nanowires, and superlattices introduce interfaces that scatter heat-carrying phonons much more effectively than charge-carrying electrons. This partially decouples $\kappa$ from $\sigma$, which is exactly what you need.

High-temperature thermoelectrics

Materials that perform well above 1000 K would unlock large-scale waste heat recovery from industrial processes and concentrated solar power. Skutterudites, clathrates, and half-Heusler alloys are active research targets in this space, with some compositions reaching $ZT > 1.5$ at elevated temperatures.

Flexible and organic thermoelectrics

Flexible thermoelectric devices that conform to curved surfaces could harvest body heat for wearable electronics or capture waste heat from irregularly shaped industrial equipment. Organic semiconductors and conducting polymers (such as PEDOT:PSS) offer low cost and solution processability, though their $ZT$ values are still well below those of inorganic materials. Improving electrical conductivity and long-term stability in these organic systems is a key research frontier.