QCM mini-FAQ

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Where can I find links to companies that sell QCMs and groups that do research with QCMs?

Here.

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What is a QCM, physically?

Physically, a QCM consists of a thin, usually round, slice of crystalline quartz with an electrode on each side. The slice is cut in a particular orientation called an "AT-cut".

The electrode can be made of any metal, but gold is the most common choice because it does not oxidize in air. If gold is used, a thin "undercoat" of chromium is usually put onto the quartz first. This is because gold by itself does not stick all that strongly to quartz. However, chromium sticks well to both gold and quartz. So using a chromium undercoat improves adhesion.

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How does a QCM work?

If the two electrodes are put at different potentials, an electric field results across the QCM, i.e. in the "Y direction". Because of the piezoelectric properties of quartz, such an electric field in the "y direction" couples to shear motion "around" the z-axis, and vice versa. The end result is that shear waves in the quartz, in which the mechanical displacement is in the "x" direction, also called the electric axis, are coupled to voltage between the electrodes.

What are QCMs used for?

Almost all QCMs are used as sensitive detectors of mass deposited on them. This added mass decreases the resonant frequency of the QCM. By measuring the decrease in the resonant frequency of the QCM, and knowing something about the physics of the QCM you can calculate the added mass per unit area on the QCM. Because frequency changes can be measured to very high precision, QCMs are very sensitive. They can measure amounts of deposited material with an average thickness of less than a single atomic layer. Hence the "microbalance" part of their name.

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Who uses QCMs?

Three places where QCMs are used are in vacuum systems, in electrochemical cells, and in "sniffers" (which are sometimes called "e-noses"). There are also several miscellaneous uses.

In vacuum systems, one usually wants to know the mass deposited on a sample by a sputtering or thermal evaporation system. In that case the QCM is put near the sample, in similar conditions. If the deposition rate at the QCM is not the same as that at the sample, a correction factor ("tooling factor") is needed. Then one can monitor the amount of material put on the sample. Devices that do this are called thickness monitors or deposition monitors. If the frequency change of the QCM is used to control the deposition system so as to deposit a predetermined amount of material on the sample the device is typically called a thickness controller or a rate controller. These devices are widely used by the optical coating industry.

With the discovery that QCMs can be made to oscillate in liquids, it became practical to use them to measure the amount of material electrochemically deposited on the QCM. It is also possible to simultaneously measure the charge transferred to or from the electrode. I believe that most of these systems are used for research. However, IBM has used electrochemical QCMs to monitor the thickness of thin film read-write heads for hard drives.

QCMs have been used to make "sniffers". This is done by having an array of QCMs, each topped with a different thin film which absorbs a particular set of chemicals. When these chemicals are present in the environment they are absorbed, increasing the mass of the QCM and decreasing its resonant frequency. The pattern of which sensor's frequency decrease gives information about what chemicals are present in the environment. People who build these things sometimes say they have invented an "electronic nose" or e-nose. Most of these are research gadgets, but there are companies that sell QCM sniffers for monitoring air pollution. Several other technologies are used to build e-noses.

In a related use, it is possible to attach antibodies to the top of a QCM that will bind to, say, a single protein. That makes the QCM an extremely sensitive detector of this protein.

The Pathfinder mission on Mars used a QCM to measure how fast dust settles onto surfaces on Mars.

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What is a simple useful approximate equation for the frequency shift of a QCM as a function of added mass?

The following equation is for a QCM in vacuum. First consider an unloaded QCM. Let L be the thickness of a QCM. Let v be the velocity of shear waves in the QCM in the direction perpendicular to the faces (y-direction). Then the shear waves bounce back and forth between the two faces of the QCM. The boundary condition at these faces is that there is no shear force on the top and bottom surfaces (because nothing is there to exert a shear force). So both the top and bottom function as "open ends". The lowest frequency standing wave has one node, and half a wavelength fits within L, so that its wavelength equals 2L. So its period T is

T = 2L/v.

The nth standing wave has n nodes, and n+1/2 wavelengths fit within L so that its period T is

T = 2L/(n+1)v .

What about an overlayer? In general the frequency shift will depend upon the elastic properties of the overlayer, as well as its density and thickness. However the simplest situation is that in which the overlayer has the same properties as quartz. Then the increase in the period of mode n is Delta T where

Delta T = 2 Delta L / (n+1)v.

So Delta T/T = Delta L/L .

We then notice that there is still a condition of no shear at the top of the overlayer. Thus if the overlayer is thin, it will not be stretched very much. Thus for thin enough overlayers the elastic properties will not be very important and Delta T is proportional to the added mass per unit area. Thus we rewrite our Delta T for a quartz overlayer in terms of the added mass per unit area of a quartz overlayer, and hope that the resulting equation works fairly well for all overlayers that are not too thick. We get

Delta T /T =~ (D Delta L)/(D_q L)),

where D is the density of the overlayer and D_q is the density of quartz. This relationship is generally okay up to Delta T / T = 10% in thickness monitors. Actually, it is better than the original approximate relationship found by Sauerbrey, according to which the decrease in frequency was proportional to the added mass per unit area.

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So what's the next step in describing how QCMs work?

That would be the "mechanical model".

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Okay, what physical assumptions go into this mechanical model? And what stuff gets left out?

The piezoelectric effect is left out of the mechanical model. One considers the displacements from equilibrium of the quartz and of whatever overlayers may be present. The overlayers are assumed to be isotropic.) The mechanical model considers shear waves only. They are assumed to vary only in the direction perpendicular to the interfaces of the quartz crystal (the y direction); i.e. the problem is considered to be one dimensional. In the quartz, and also in each overlayer, at any given frequency, there are two such shear wave solutions. If the frequency is such that shear waves propagate in the y direction then one of the solutions will travel in the +y direction and the other in the -y direction. And if there is attenuation, one will attenuate in the +y-direction while the other will attenuate in the -y direction. In what follows, I will call the +y direction "up".

Boundary conditions must be imposed at the top and bottom of the stack, and at the interfaces between layers. At interfaces one would normally impose two boundary conditions. One would be a "no-slip" boundary condition. The second condition is continuity of horizontal stress.

The no-slip condition simply says that, if two layers are adjacent, the displacement of the top of the lower layer equals the displacement of the bottom of the upper layer. That is, one layer does not slip over the other. (It has been questioned whether the no-slip condition is always true.)

The continuity of stress across the boundary is just Newton's Third Law of Motion in disguise. It says that, if two layers A and B are adjacent, the force per unit area which layer A exerts on layer B equals the force per unit area which layer B exerts on layer A.

The boundary condition to be used at the top of the stack depends on whether the uppermost layer of the stack is considered to be bounded by vacuum or to extend upward forever. If it extends upwards forever (this could be appropriate for a QMB with one side immersed in a fluid), and the top layer is at all lossy, the boundary condition is that the displacement is finite at y equals positive infinity, meaning that the solution in the uppermost layer must attenuate in the +y direction. If the uppermost layer is considered to extend forever, and is not lossy, the boundary condition to be used is that the solution in the uppermost layer must propagate upward. If the uppermost layer is bounded on top by vacuum, the boundary condition is that its interface with vacuum must be "zero-stress" because the vacuum cannot exert a force on it.

The same considerations apply for the boundary condition at the bottom, with appropriate changes in signs. If the bottom layer is bounded by vacuum, the condition at its bottom must be the zero stress condition.

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This sounds complicated. Where can I find a program to do these calculations for me?

Here's a free program that calculates electrical admittance per unit area at any frequency. It runs on PC compatible computers (386 SX or higher) in the DOS box, or under DOS (3.1 or higher). It also works on Windows XP. It has not been tested on Vista or on Windows 7. A help file is included. This version handles up to 100 layers on top of the quartz. QMB program & help file. Zipped, 74 kB (updated September 29, 2003)

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Here's the source code. It's in C, and zipped (10.4kB). These files are provided on an as is basis. I do not accept responsibility for any problems that may be caused by their use or misuse.

Can you give me some references about QCMs?

• 1. "Linear Piezoelectric Plate Vibrations" by H. F. Tiersten. (Plenum, New York, 1969) Treats the mathematics of piezoelectricity in detail. A valuable resource for theoretical work on QCMs. Difficult to read. Does not mention QCMs.

  2. "IEEE Standard on Piezoelectricity", copyright 1978 by the Institute of Electrical and Electronics Engineers, Inc. ANSI/IEEE Std 176-1978. IEEE Transactions on Sonics and Ultrasonics, SU-31, No. 2, Part II, March 1984. 55 pages. Similar notation to Tiersten, but easier to read.

  3. "Introduction to Quartz Crystal Design" by Virgil E. Bottom. (van Nostrand Reinhold, New York, 1986) Moderately Mathematical. Has a few mistakes in derivations. Well written. Discusses quartz and numerous piezoelectric devices in detail. Gives information important to experimentalists. Does not mention QCMs.

  4. G. Sauerbrey. Z. Phys., 155 206 (1959) The original paper proposing QCMs.

  5. C. Lu and D. Lewis. J. Appl. Phys., 43 4383 (1972) This paper was not the first to correctly analyze QCMs with a single solid overlayer in the "mechanical model", but it was the first to reduce the results to a simple transcendental formula.

  6. E. Benes. Discusses QCMs based on a transmission line analogy. Piezoelectric coupling is included using an analogy to transformers.

  7. C. E. Reed, K. K. Kanazawa and J. H. Kaufman. J. Appl. Phys., 68 1993 (1990) Discusses QCMs from a physical point of view including piezoelectric coupling.

  8. Z. Lin and M. D. Ward, Analytical Chem., 67 685 (1995). This paper provides experimental evidence that a QCM operating in a liquid can emit sound waves into the liquid, producing extra losses.

  9. Many additional references, especially on the use of QCMs by chemists, can be found in Georgia Arbuckle's qcm bibliography

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<To get my e-mail address, replace "at" with "@" & remove the spaces:

christopher e reed at cs . com>

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