Smooth-walled wire traps high energy phonons, low energy phonons carry heat.
Tiny wires may boost heat flow.
Getting rid of heat is one of the central challenges with modern technology. It doesn’t matter whether the technology is a high-end server CPU or some pathetically anemic processor in a no-brand set-top box—someone has had to think about thermal management. One of the central issues in thermal management is thermal resistance, a material’s tendency to limit the flow of heat. The thicker a material, the larger the temperature gradient required to achieve the same amount of cooling because the thermal resistance increases with thickness.
Except when it doesn’t. If the heat is carried by ballistic phonons, thermal resistance stays constant.
Energy in motion
Heat is basically energy. In a solid material, energy is stored in two places: the motion of electrons, and the motion of the nuclei. The motion of electrons can pull nuclei into motion, while likewise, nuclei kick electrons around, so energy travels back and forth between the two.
The very nature of how energy is stored also implies that energy moves. If a nucleus moves, it will naturally set its neighbors into motion, so the energy travels outward from wherever it was initially injected. Provided they are in an electrically conducting material, electrons are always in motion, so they also carry energy from place to place. For today’s story, electrons are not important.
Energy transport via the motion of nuclei takes the form of vibrations with fixed packets of energy, called phonons—analogous to light waves and photons. Phonons are very sensitive: they are easily scattered by any imperfection in the structure of the material. An atom not quite at the right location, or an impurity like the wrong atom in the material will cause the phonon to scatter.
Over a distance of a few micrometers, a phonon will scatter many many times. The result is that the energy flow from hot to cold is slow. Heat diffuses like the stain growing up the wall of a teenager’s bedroom.
When phonons travel long distances with minimal scattering—perhaps only reflections off a material’s surfaces—we call that ballistic transport. This is exactly what the researchers observed in gallium phosphide wires. In fact, what the researchers show is that the material enables ballistic transport for distances up to 15µm, which is an astonishingly long way for phonons. I assume longer wires are possible, but the data ends at 15µm.
However, the length over which ballistic transport is possible depends on the diameter of the wire. There is a sharp transition between 40nm and 50nm. Below 40nm, ballistic transport up to 15µm seems possible; above 50nm, no ballistic transport is observed.
Why is this the case? The wires act as waveguides for the phonons, just like fiber-optic cables act as waveguides for light. The phonons travel along the wire by reflecting off its walls. If the wall is perfectly smooth, then the reflections will be just like light reflecting off a mirror, and the phonon will travel onward as if it were going straight along the wire with a slightly reduced speed.
If the wall is rough, the phonon reflection could be at any angle—they may even end up going back the direction they came from. Each phonon will take a different amount of time to traverse the wire. This is typical heat diffusion.
But smooth and rough are a matter of perspective. For low-energy phonons, a mirror-like reflection can be obtained from a rougher surface than for a high-energy phonon. Think of it this way: your bathroom mirror is much smoother than a satellite dish, but the satellite dish works like a mirror for radio waves. Radio waves are low energy (and therefore long wavelength), so the dish surface appears smooth, while for visible light, the surface of the dish looks like the Himalayas.
Because things are smoother for long-wavelength, low-energy phonons, these can travel ballistically in the wires, while high-energy, short-wavelength phonons diffuse. In the confines of a narrow wire, the roughness of the walls traps the high-energy phonons; they are scattered with equal frequency in the direction of the cold end as the hot end of the wire with the result that, on average, they never go anywhere.
As the wire diameter grows, the number of ways a phonon can be trapped falls (or more precisely, the number of ways a phonon can travel along the wire increases), so the high-energy phonons start to flow. The diffusing phonons carry more energy than the ballistic phonons, so the heat transport is dominated by diffusion instead of faster ballistic phonons.
Which is bad news for those looking for a super-excellent heat-sucking new material. Under ballistic transport conditions, both the thermal conductivity is high and the thermal resistance is high. The difference (compared to diffuse transport) being that the thermal resistance doesn’t increase as the wire length increases. That means that, for now, you are probably better off with short fat wires with slow transport than long thin wires with fast transport.