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And unquiet flows the entropy

The International Year of Light aims to raise global awareness about how light-based technologies can promote sustainable development and provide solutions to global challenges in energy, education, agriculture, and health. To celebrate this special year over the course of March, three of our authors will write about light-based technologies and their importance to us. This week, Kamran Behnia discusses thermoelectricity.

The business of condensed-matter physics is to explain why the world appears as it does to our naked eyes. This is a field lacking the glamour of high-energy physics or the poetry of astrophysics. The general public is quick to forget that smartphones owe much to the manipulation of electron herds in the Silicon Forest and the quantum theory of solids.

Contrary to popular belief, quantum mechanics is not restricted to the microscopic world. As far as we know, it prevails no matter how many trillions of atoms gather together. One needs it to explain why some solid bodies are shiny while others are transparent or why electricity flows easily in some bodies while not in others.

Thermoelectricity, discovered almost a century before the quantum revolution of physics, is the entanglement between an electric current and a thermal flow. It appears whenever a solid body hosts traveling electrons. We can measure the Seebeck coefficient of a solid body by applying a thermal gradient and quantifying the electric field it generates. It is a transport property, a macroscopic response depending on what happens microscopically to a huge number of itinerant electrons during their trip across the solid. In a crystal, such properties are very sensitive to imperfections, in contrast to what are known as thermodynamic properties.

A firm conceptual frame for thermoelectric phenomena was formulated in 1948 by Herbert Callen, based on Lars Onsager’s reciprocity relations. Callen distinguished two components of the heat flow in a solid body. The first is driven by a thermal gradient and the other is driven by particle flow. The magnitude of the first is set by the thermal conductance of the solid and the second is set by its Seebeck coefficient. In absence of a thermal gradient, the first component vanishes and only the second survives. In this picture, the Seebeck coefficient becomes the ratio of entropy flow to particle flow and thus a measure of entropy carried by traveling electrons.

This fundamental conclusion has important consequences. For example, there are “heavy” electrons, which acquire their mass by living in an environment rich in entropy. When they travel, the entropy they carry is not seen in a thermal conductivity measurement (which measures the flow of entropy produced by the thermal gradient), but in their Seebeck coefficient (which measures the flow of entropy due to traveling particles). It is widely known that thermodynamic entropy is measured in a specific-heat experiment; on the other hand, it is not as obvious that this entropy, in its mobile version, shows up in the thermoelectric response.

Following Rolf Landauer’s work, electric conduction is now seen as a transmission. In this language, electric conductivity of a solid body is expressed in terms of the quantum of electric conductance and two material-dependent length scales that are the Fermi wavelength and the mean free path of electrons. Electrons transmit charge depending on their momentum and the average distance between two scattering events. The thermoelectric response can be formulated in this language. It depends on the quantum of thermoelectric conductance, and the appropriate length scale is the de Broglie thermal wavelength. It quantifies the amount of information that an electron subject to a Fermi-Distribution can carry.

Simple metals are considered to have been domesticated long ago by the quantum theory of solids. Yet many like copper present an enigmatic thermoelectricity. We know almost everything about the way electrons organize themselves around copper atoms and certainly enough to explain its electric conductivity, its thermal conductivity, its mechanical properties, and its color. When you attach two electrodes to a piece of copper and apply a current, heat flows. Not a surprise! This is copper’s thermoelectric response. However, heat flows opposite to the direction one expects given the sign of the electrons’ electric charge. The wrong sign of the Seebeck coefficient in copper is a puzzle that mysteriously faded away from the collective memory of condensed-matter physicists during the last decades of the twentieth century. This erasure is by itself worthy of anthropological investigation.

Image Credit: “Volts of electricity” by JimmyMac210. CC by NC-2.0 via Flickr.

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