Physicists at the California Institute of Technology have announced the first observation of the quantum of thermal conductance. This discovery reveals a fundamental limit to the heat that can be conducted by objects of atomic dimensions.
The findings, reported in the April 27 issue of the journal Nature, could have profound implications for the future design of microscopic electronic devices and for the transmission of information, according to the research team leader, Caltech physics professor Michael Roukes.
The quantum of thermal conductance is best understood by beginning with a simple explanation of heat flow. In the everyday world, the amount of heat carried by an object can vary in a smooth and continuous way. Heat actually flows by means of collective, wavelike vibrations of the atoms that make up a solid material. Usually immense numbers of such waves, each inducing a unique type of synchronous motion of the atoms, act simultaneously to carry heat along a material.
Physicists know that waves sometimes act like particles and vice versa, so they've given these vibrations the particle-like name phonon (reminiscent of "electron" but named after the Greek root phon for sound.) For heat flow in the macroworld, since each phonon is just one among a sea of many others, an individual phonon's contribution alters the total only imperceptibly.
But in the nanoworld, this "phonon sea" is actually rather finite, quantum effects rule, and the heat conduction can become radically different. When an object becomes extremely small, only a limited number of phonons remain active and play a significant role in heat flow within it. In fact, in small devices at temperatures close to absolute zero, most types of motion become almost completely "frozen out," and heat must then be carried by only the several remaining types of wavelike motions that persist.
It has recently become apparent that, in this regime, a strict limit exists to the amount of heat that can be conducted in a small structure or device. Although never before observed, this maximum value is actually a fundamental law of nature, independent of composition or material. It stipulates that the only way thermal conductance can be increased in a very small device is simply to make the conductor larger.
The Nature paper reports that this fundamental limiting value, called the quantum of thermal conductance, can be observed by using tiny devices with specially patterned features only 100 billionths of a meter across (about 300 atoms wide). To carry out this work, Keith Schwab, a postdoctoral fellow in Roukes's group, developed special devices from silicon nitride with assistance from research staff member Erik Henriksen. The work was carried out in the group's nanofabrication and ultralow-temperature laboratories in Pasadena, in collaboration with University of Utah research professor, John Worlock, a visiting associate at Caltech.
The Roukes team has demonstrated that the maximum possible value of energy transported per wavelike motion (phonon mode) is a number composed of only fundamental physical constants and absolute temperature itself. (The relation is given by the product of pi squared, Boltzmann's constant squared, and absolute temperature, over three times Planck's constant.)
Numerically, at an ambient temperature of one kelvin, this quantized conductance roughly translates into a temperature rise of one kelvin upon the application of only a thousandth of a billionth of a watt of power (its precise value is 9.4 x 10^-13 W/K).]
Their new result has important implications for nanotechnology as well as for the transmission of information. Moore's Law, a popularized rule-of-thumb, can be used to loosely describe the continuous decrease in size of the individual building blocks (the transistors) that populate, now in the tens of millions, the integrated circuits forming today's powerful computer chips.
In the unrelenting technological drive toward increased function and decreased size, these individual transistor components have been scaled downward in size to a realm where the underlying physics of their operation can change. In the most extreme cases at the smallest scales, conventional operation may completely break down.
One example is the so-called "power dissipation problem" stemming from the fact that when each individual transistor on a microchip is turned on, each gives off a little heat. This accumulates to become a very significant problem when millions of such transistors, each in effect a microscopic heat generator, are placed in close proximity.
"This will become especially serious for future molecular-scale devices," says Roukes. "No matter how small it is, you always have to put a finite amount of power into a device to turn it on. In this quantum regime, when only a limited number of modes are capable of transferring heat, it will be crucial to take this fundamental limitation into account."
Separate theoretical studies carried out elsewhere indicate that this quantum of thermal conductance is universal, and independent of whether the heat is carried by electrons, phonons, or any other mechanism. "It would seem there is no way of escaping this fundamental law of nature," says Roukes.
These other studies indicate that the maximum thermal conductance, observed in this work, is linked to the maximum rate that information can flow into a device having a single quantum "channel." This surprising connection between information theory and thermodynamics is a manifestation of a deep connection between information and entropy.
"As we engineer smaller and higher speed computational elements, we will also encounter this fundamental quantum limitation in the rate of information flow," Schwab says.
The group's three-year effort followed upon work of Thomas Tighe, a previous postdoctoral fellow in the group, and culminated in new techniques for creating the miniature devices studied. At the heart of each device is an isolated heat reservoir, which the researchers term a "phonon cavity." It resembles a miniature plate freely suspended by four narrow beams. Each beam acts as a quasi one-dimensional "phonon waveguide" for heat flow, and it is precisely this reduced-dimensional flow that is the focus of the researchers' measurements.
On top of the cavity, Schwab and Henriksen patterned two small, separate patches of thin-film gold, described by Roukes as "puddles of electrons." In the course of a measurement, one of these is heated by passing a very small electrical current through it. Electrical connections allowing this current to flow were made using superconducting leads (patterned on top of the phonon waveguides).
This insures that heat is deposited only within the resistive gold film and, therefore, transferred only to the phonon cavity. To escape from the suspended device, the heat must eventually flow through the phonon waveguides. Since the waveguides' thermal conductance is weak, the phonon cavity temperature ultimately rises to a new, and hotter, steady-state level that directly reflects the thermal conductance of the phonon waveguides.
Measurement of the current-induced temperature rise within the small devices is a significant challenge in its own right, and required both ingenuity and the investment of a significant portion of the researchers' efforts. Most available thermometry techniques applicable at the nanoscale are electrical, and thus involve power levels that greatly exceed that used by the researchers in their measurements.
"The power level we used to carry out these experiments, about a femtowatt, is equivalent to the power your eye would receive from a 100-watt light bulb at a distance of about 60 miles," says Schwab. Instead of the standard electrical methods, the researchers coupled the second "electron puddle" to extremely sensitive dc SQUID (superconducting quantum interference device) circuitry.
This allowed them to observe the feeble current fluctuations that have a magnitude directly proportional to the absolute temperature of the nanoscale device. This so-called Johnson/Nyquist noise, which is also the origin of the electrical noise causing background hiss in audio systems, here plays a pivotal role by allowing the local temperature of the phonon cavity to be measured without perturbing the ultraminiature device.
In the end, because the researchers know the precise amount of heat deposited, and can directly measure the absolute temperature reached by the phonon cavity in response to it, they can directly measure the thermal conductance of the narrow beams acting as phonon waveguides. Simply stated, the ratio of the heat flowing through the waveguides to the rise in cavity temperature is the phonon thermal conductance of the quasi one-dimensional waveguides.
This work was carried out over the past three years within the research laboratories of Caltech Professor of Physics, Michael Roukes. Schwab, formerly a Sherman Fairchild Distinguished Postdoctoral Scholar within Roukes' group, is the principal author of the paper.
Schwab's life as a young postdoctoral scientist, and his role in the efforts to observe the quantum of thermal conductance, are the subjects of an upcoming documentary film by independent filmmaker Toni Sherwood. The title of the film is The Uncertainty Principle: Making of an American Scientist.
Coauthors of the paper are John Worlock, visiting associate at Caltech and research professor of physics at the University of Utah, a long time collaborator with Professor Roukes; and former research staff member Erik Henriksen.
