Partial view of a Mandelbrot set, one example of a fractal pattern. (Image courtesy of Wolfgang Beyer.)
Most people think of bright and intricate images when they think of fractals. Now, fractal mathematics are being used to investigate heat transfer in semiconductor materials.
Thermoelectric devices generate electricity from heat and their performance hinges on the presence of a pronounced temperature gradient from one side to the other.
Lower thermal conductivity preserves a greater temperature gradient, which improves performance of the thermoelectric device.
Typically, heat transport in solids is described in terms of the random chaotic motion of “energy carriers” gradually transferring heat from hot to cold regions.
However, over tiny distances of a few nanometers the motion of thermal energy behaves in a way that resembles the structure of fractals. Fractals are mathematical sets that generate repeating patterns that appear identical at any scale.
Researchers have used a modern theory of heat transport in experiments with semiconductors used in computers, lasers and thermoelectrics. The left image shows a rendering of heat spreading in a semiconductor using the modern transport theory. The image on the right shows a rendering using the conventional heat-transport theory. (Image courtesy of Purdue University/Bjorn Vermeersch and Ali Shakouri.)
A research team from Purdue University and the University of California Santa Barbara has developed a theory based on the work of mathematician Paul Lévy.
The team applied this theory in experiments with the semiconductor iridium gallium aluminum arsenide, which is used in high-speed transistors and lasers.
The research shows that inserting erbium arsenide alloy nanoparticles reduces thermal conductivity and doubles the semiconductor’s thermoelectric efficiency.
The nanoparticles cause thermal conductivity of the material to decrease up to three-fold without changing the fractal dimension.
The energy carriers are quasiparticles called phonons. These phonons experience "quasiballistic" motion, meaning they are transported without colliding with many other particles. This motion causes the heat to conduct with "superdiffusion."
The approach is similar to effects seen in superdiffusive glasses, sometimes called “Lévy glasses,” which are materials containing spheres of glass that change the diffusion of light passing through.
The potential applications of this research include systems to harvest waste heat in vehicles or power generation plants, to control overheating in miniaturized and high-speed electronics components and to improve the performance of high-power lasers.
“When we look at the problem of heat transport, what is surprising is that the theory we use dates back to Fourier, which was 200 years ago, and he developed it to explain how the temperature of the Earth changes," said Ali Shakouri, professor of electrical and computer engineering at Purdue University.
"However, we still use the same theory at the smallest size scale, say tens of nanometers, and the fastest time scale of hundreds of picoseconds. The work we have done is applying Lévy theory for the first time to heat transport in actual materials experimental work," Shakouri said.
Understanding and modeling how nanomaterials improve heat transfer will enable engineers to optimize their designs.
Imagine how powerful it will be to marry the nanoscale to the macroscale in a thermal simulation package like FloTherm. For example, optimizing a heat sink for shape and nanoparticle placements could allow processors to run longer and faster.
The team’s paper was published in the July 2015 issue of the journal NanoLetters and is available to read here.