How Does It Work?

(Aside: my policy on scientific explanations.)

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Just how does that chip measure heat capacity? Actually, the design is compatible with a variety of measurement techniques; we use the relaxation method. For information on other techniques, see the reference list.

The relaxation method with this device requires us to make four measurements, which are described below: the relaxation time tau, the thermal conductance kappa, the sample mass, and the heat capacity of the addenda. By solving the heat conduction equation for this geometry (I won't do it here), one finds for the heat capacity:

C = (tau)(kappa)

This is the heat capacity of everything in the center of the membrane-- sample, thermometers, heater, and a patch of the membrane itself. So the addenda measurement is required to subtract off everything that isn't the sample. The mass measurement allows you to convert heat capacity (J/K) to specific heat (J/gK).

For the tau and kappa measurements, we use a technique called the ac bridge method.

Here's the ac bridge method in a nutshell: Each of the three thermometers (Pt, and 2 poly-Si or Nb-Si) has a branch on and off the membrane. (The reason for 3 therms is that each one spans a certain temperature range; you can't cover 1-800 K with a single thermometer). The one on the membrane is in direct thermal contact with the sample. If nothing is happening on the device, the thermometers are at the same temperature. The contact pad between the two matched thermometers can be set at zero voltage on a lock-in amplifier (ie, the voltage is measured with ac). Then apply a dc pulse to the heater (long enough for the device to reach a steady state). The electronics on the Si frame will stay at very nearly the same temperature they were at before because the device is mounted on a large copper block that easily transports away the heat delivered by the dc pulse. But the membrane is a very poor thermal conductor, especially over the large distance between sample and frame (0.25 cm). Everything on the membrane gets hot-- sample, heater, thermometer, and the membrane itself. The resistance of the thermometer changes, and the voltage at the center pad no longer reads zero. This is the essence of the ac bridge technique for measuring small changes in resistance.

Tau Measurement

In this case, we are interested in watching the time evolution of the off-null voltage signal so that we know how fast the heat leaks away from the sample. When the dc pulse mentioned in the above paragraph has been on for a while (anywhere from a few ms to several seconds), the heat will start to leak off the membrane through the Si-N itself, as well as the Au-Pd leads, and, at high temperatures, through radiation. Steady state will be reached between heat in and heat out. At this point, the dc pulse is terminated. The sample temperature relaxes back toward the temperature of the Si block, which is observed through the voltage relaxing back toward zero on the lock-in. If you have designed your calorimeter properly, this relaxation is a single exponential drop-off. The main criterion for "proper design" is that the sample and electronics on the membrane must come to equilibrium amongst themselves much faster than the heat can escape from the sample back to the block. Or, more formally, the "internal" time constant for the heat to transfer from the heater to the sample and thermometers must be much faster than the "external" time constant which is measured on the lock-in. This is achieved by (1) having a thin membrane and (2) using a good thermal conductor as a sample, such as a metal. (Insulating samples can be measured if a layer of metal is also evaporated over the sample area to provide thermal conduction; then the metal just becomes part of the addenda).

The exponential relaxation is recorded as data, and the time constant tau is extracted.

Kappa Measurement

In this case, we are interested in the absolute value of the difference in resistance between the 2 thermometers when the heat is on. The thermal conductance measurement is more difficult than the tau measurement because it is an absolute measurement; the correct tau can be measured even if steady state is not achieved with the heat pulse-- the sample always relaxes with the same time constant, no matter how much or little you heated it.

To measure K, we note that by definition K = P/delta(T), where P is the power put into the heater and delta T is the consequent rise in temperature. In our case, we are measuring K for the device itself: how much heat is leaking off the leads and through the membrane each time we put heat through the heater? Clearly it's important to design a device with as small a K as possible so that most of the heat goes to heating up the sample, not leaking off the membrane. This causes some design challenges; see the section at the end on the construction of the device.

P is easy to measure; we just do a 4-wire measurement of the heater voltage after applying a known current: P = IV.

Delta T is harder. What we actually observe is delta V, the change in voltage reading on the lockin. Knowing the current used in the ac bridge, it is not difficult to convert delta V to delta R (see the paper for details). That is the change in resistance between the thermometer on the membrane and the thermometer on the frame. (Astute readers will notice that this requires a metallic sample, and indeed, we always include a separate metallic "conduction layer", even if the sample of interest is insulating). If we knew R(T) for the thermometer, we would immediately have delta T (delta T = delta(R)/[dR/dT]).

The Pt thermometers are remarkably linear, with dR/dT about 2.5 ohm/K. But obviously the NbSi and poly-Si thermometers are far from this simple. So we record R(T) as data, and go back later to fit the function when we have the whole range. I have struggled with this fit, and have decided that the best method is to fit ln(R) vs. ln(T) to a polynomial (usually 3rd or 5th order works well), then convert back to R(T).

Since I want my data-taking to be automated, I have implemented a system by which the computer can guess the approximate K at each temperature and use this to calculate appropriate currents to use in the tau and K measurements. For the Pt, it just takes a linear fit between the previous and present data points, then extrapolates. For the NbSi, I use the function R(T)=[A/T^2] + B. This is good enough to keep the program from going off the deep end and using too much current (thus distorting the measurement), but I still have to go back after the fact and work out the "real" K.

Mass Measurement

The mass is measured by putting a blank chip in the evaporator along with the device. Knowing the area of deposition, the thickness of the film (measured on the blank by profilometry), and the density, we can get the mass.

Addenda Measurement

After the first 3 measurements, not all the work is done. The C that you have is the heat capacity of everything that got hot-- the sample, the leads, the membrane under the sample, the heater, the thermometer. Everything that isn't the sample is the addenda, and must be subtracted.

Subtraction measurements are dangerous in science because a small error can become very important. If the captain plus her ship weigh 1500 lbs + or - 50 lbs and the ship alone weighs 1350 lbs + or - 50 lbs, then the captain weighs 150 lbs + or - 71 lbs. Not a very accurate measurement. Thus, minimizing the addenda is crucial in calorimetry-- another reason for a thin membrane and thin-film electronics (less mass=less heat capacity).

Our addenda is typically about 30-50% of the total heat capacity. That may not sound great if you don't do much calorimetry, but it's really quite good. There are some plots in our paper that show the data with and without the addenda.

The cleanest way of subtracting out the addenda is simply to measure it. Because the silicon nitride membrane is such a poor thermal conductor, a "blank" device cannot be measured (the internal time constant is the same order of magnitude as the external time constant). However, many of our samples are insulators, and require a metal conduction layer anyway. So we deposit the metal, measure the device, then deposit the sample, and measure it again. No guessing involved; the addenda subtraction is exact within the accuracy of the device.

If the sample is air-sensitive or otherwise prone to decaying if left exposed, then there are two options. First, the conduction layer could be deposited on top of the sample, capping it against the air. (The metal still does its job as a thermal conduction layer because the sample is so thin). In this case the addenda cannot be measured independently on the same device, but it can be measured alone on a separate device that had only the metal, not the sample, deposited on it. We find that the difference in measured heat capacity between two different devices is smaller than the inaccuracy of a single device, justifying the use of two devices.

A second option is to make the whole heat capacity measurement in-situ. We have a special UHV chamber with a helium cryostat that fits on top. The device is mounted on the cold finger of the cryostat, sticking down into the chamber. After measuring the addenda, we deposit the sample and measure it immediately, without breaking vacuum. Our UHV chamber/cryostat system can operate from 2 K to 450 K. This is how we were able to measure the heat capacity of K₃C₆₀. You can read more about this material on my fullerene page.

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Email: kimall (at symbol) mindspring.com