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A collaborative effort breaks Fourier's law

MIT researchers find that Fourier's law of heat conduction breaks down at lengths much longer than previously thought.
Alexei A. Maznev (left) and Jeffrey K. Eliason with time-resolved optical spectroscopy used to demonstrate ballistic acoustic phonon transport in silicon.
Alexei A. Maznev (left) and Jeffrey K. Eliason with time-resolved optical spectroscopy used to demonstrate ballistic acoustic phonon transport in silicon.
photo: Denis Paiste/Materials Processing Center

Time-resolved optical spectroscopy experiments showing that heat-carrying acoustic phonons in silicon travel much longer distances at room temperature than previously thought grew partly out of theory and partly out of serendipity, according to MIT researchers who participated in the work.

"I think the idea of this experiment resulted partially from accidental discovery, but also partially from (MIT Professor) Gang Chen and others developing an advanced understanding of the thermal transport," explains Alexei A. Maznev, a staff scientist in the lab of Professor Keith A. Nelson who developed the model and planned the experiment. “A number of things came together that resulted in this work.”

Previously, scientists had thought that heat transport in a solid was governed by thermal diffusion, which is expressed mathematically by Fourier's law of heat conduction, at distances larger than a mean free path — the distance a phonon travels before dissipating — which was estimated to be about 40 nanometers in silicon at room temperature.

But the MIT researchers' experimental work found that Fourier's law breaks down at lengths below approximately 5 microns, much longer than previously thought. Jeremy A. Johnson PhD ’11, Maznev, Nelson, Chen, chemistry graduate student Jeffrey K. Eliason, and colleagues published their findings in Physical Review Letters in January 2013 in the article, "Direct Measurement of Room-Temperature Nondiffusive Thermal Transport Over Micron Distances in a Silicon Membrane." (See related article.)

Two sides of the puzzle

Phonons can be identified by wavelength or mean free path, explains Eliason, a fourth-year doctoral student in chemistry who was a first-year graduate student when he began the measurements for the paper. "We have these two sides of the puzzle, and through theory, we try to connect the two," he says. "The general idea is that if we know which heat carriers are important, then we can begin to structure the materials in a way that can prohibit those specific phonons from traveling well in the material."

"The mean free path is a more direct measure of how effectively those phonons contribute to the thermal conductivity, because the mean free path describes how far that phonon will travel before it scatters,” Eliason says. “So you want to know both the wavelength of the phonon, to determine how you want to scatter it, and the mean free path, so you know how well that phonon contributes to the thermal conductivity."

“The question is not what’s the average mean free path, but what is the mean free path of the phonons that contribute significantly to the thermal transport,” Maznev says. “I have done this experiment that has demonstrated quite nicely that this 40-nanometer estimate is wrong.”

Maznev says the mechanisms of thermal conductivity in solid state have been understood since the 1930s, and phonons that contribute to heat transfer at room temperature are all high frequency, above 1 terahertz. However, quantitative calculations of the mechanism at the atomic level lagged. With Fourier's law, scientists and engineers could analyze thermal transport without going into the details of what happens at the microscopic level. "What has changed in the past decade is that there are several developments which really are a departure from this kind of established picture,” Maznev says. “People are looking at small things. The scale in microelectronics is already on the level of tens of nanometers. At these distances, the diffusion model doesn't work."

Engineering desired properties

The desire to engineer materials with specific thermal properties, such as low heat transfer for thermoelectric devices, and low heat buildup for electronic devices, is a strong motivation for the work. The work has also been stimulated by first-principles calculations of phonon relaxation times, such as work by physicist David Broido of Boston College, who collaborates through the MIT-based Solid State Solar Thermal Energy Conversion Center (S3TEC), a DOE-funded Energy Frontier Research Center. "They know how to calculate from microscopic dynamics for a number of materials," Maznev says.

Eliason and Maznev's research uses laser-induced transient grating spectroscopy, a specialty of Nelson's lab. The group also used the method to measure thermal conductivity of thin film PbTe and GaAs/AlAs superlattices.

Using a pair of crossed laser pulses and a separate probe laser beam, the researchers measured the thermal grating decay as the material cooled back to its initial state. The thermal grating, i.e., the sinusoidal temperature profile in the target material, can be read from changes in its refractive index, which are proportional to the temperature changes. The probe laser beam is diffracted by the pattern, and the diffracted signal decays as heat moves from the heated to unheated regions. In the silicon experiment, the grating period was changed from about 2 microns to 20 microns.

Breaking Fourier's law

Normally, transporting heat over longer distances takes more time. "According to the Fourier law, the time is proportional to square of the distance; that is, if you increase the distance by factor of 10, then the time needed for it to come across this distance is increased by a factor of 100," Maznev explains.

It was previously thought that if the distance was longer than 40 nanometers, heat transport in silicon at room temperature should obey the Fourier law.

"What we observed in this experiment is that when the grating spacing becomes smaller than about 5 microns, then this law is no longer followed,” Maznev says. “What we have found is that at small distances, transport is slower than this Fourier law would predict — that is, we see a deviation from this quadratic dependence. This is really a direct indication of the deviation from the Fourier law. It turns out that it breaks down at distances of a few microns, which is much longer than previously thought."

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