Basic intent
This
lecture is intended to provide an overview of piezoresistive devices. Some
examples are worked out using this sensing technique.
Piezoresistance
A piezoresistor is basically a device which exhibits a change in resistance when it is strained. There are two components of the piezoresistive effect in most materials - the geometric component and the resistive component.
The geometric component of piezoresistivity basically comes from the fact that a strained element undergoes a change in dimension (Figure 1). These changes in cross sectional area and length affect the resistance of the device.
Figure 1: Geometric change (strain) from applied force
A good example of the geometric effect of piezoresistivity is the liquid strain gauge. It sounds weird, but there was a great many liquid strain gauges in use many years ago. Imagine an elastic tube filled with a conductive fluid, such as mercury (really!). The resistance of the mercury in the tube can be measured with a pair of metal electrodes, one at each end. Since mercury is essentially incompressible, forces applied along the length of the tube stretch it, and also cause the diameter of the tube to be reduced, with the net effect of having the volume remain constant. The resistance of the strain gauge is given by
,
where ρ is resistivity of the mercury, L is length of the conductive fluid, A is the cross-sectional area, and V is the volume. Taking the derivative gives
We define a quantity called the gage factor K as:
Since
,
we have K = 2 for a liquid strain gage.
This means that the fractional change in resistance is twice the fractional change in length. In other words, if a liquid strain gauge is stretched by 1%, its resistance increases by 2%. This is true for all liquid strain gauges, since all that is needed is that the medium be incompressible.
Liquid strain gauges were used in hospitals for measurements of fluctuations in blood pressure. A rubber hose filled with mercury was stretched around a human limb, and the fluctuations in pressure were recorded on strip-chart recorders, and the shape of the pressure pulses could be used to diagnose the condition of the arteries. Such devices have been replaced by solid state strain gauge instruments in modern hospitals, but this example is still interesting to use as an introductory example.
Metal wires can also be used as strain gauges. As is true for the liquid strain gauge, stretching of the wire changes the geometry of the wire in a way which acts to increase the resistance. For a metal wire, we can calculate the gage factor as we did for the liquid gauge, except that we can't assume that the metal is incompressible, and we can't assume that the resistivity is a constant:
Where r is radius of a wire with circular cross-section. Then,
Since
is defined as Poisson's ratio, v, we have
For different metals, this quantity depends on the material properties, and on the details of the conduction mechanism. In general, metals have gage factors between 2 and 4.
Now, since the stress times the area is equal to the force, and the fractional change in resistance is equal to the gage factor times the fractional change in length (the strain), and stress is Young's modulus times the strain, we have
or
So the fractional change in resistance of a strain gauge is proportional to the applied force, and is proportional to the gage factor divided by young's modulus for the material. Clearly, we would prefer to have a large change in resistance to simplify the design of the rest of a sensing instrument, so we generally try to choose small diameters, small young's modulus, and large gage factors when possible. The elastic limits of most materials are below 1%, so we are generally talking about resistance changes which are in the 1% - 0.001% range. Clearly, the measurement of such resistances is not trivial, and we often see resistance bridges designed to produce voltages which can be fed into amplification circuits.
Fig. 2: Thin Film Strain Gage
For many years, there has been an industry associated with the fabrication and marketing of thin metal film strain gauges and the necessary tools and equipment for attaching these gauges and the wires to various mechanical structures. A photograph of a thin film strain gauge is shown in Fig. 2. This particular strain gauge consists of a metal wire patterned so that it is primarily sensitive to elongation in one direction. Strain gauges are available from several vendors, and literally hundreds of patterns of the metal film can be selected, with different patterns providing sensitivity to strain in particular directions.
In recent years, much use has been made of the fact that doped silicon is a conductor which exhibits a gage factor which can be as large as 200, depending on the amount of doping. This creates an opportunity to make strain gages from silicon, and to use them to produce more sensitive devices than would be easy to make in any other material.
Fig. 3: Silicon Strain Gage
Another aspect of the utility of silicon is that recent years have seen the development of a family of etching techniques which allow the fabrication of micromechanical structures from silicon wafers. Generally referred to as Silicon Micromachining, these techniques use the patterning and processing techniques of the electronics industry to define and produce micromechanical structures. An example company manufacturing these microdevices is MicroStrain.
Fig. 4: AFM Thermo-mechanical Data Storage.
Micromachining can be used to fabricate piezoresistive cantilevers for a wide variety of applications. Past research between IBM and Stanford focused on the development of piezoresisitve cantilevers for a data storage applications. In this design, a 100 micron-long piezoresistive cantilever is dragged along a polycarbonate disk at 10 mm/s, bouncing up and down as it passes over sub-micron indentations in the surface of the disk. This idea is essentially a high-performance phonograph needle (for those of you old enough to remember...) The devices shown in figure 4 illustrate cantilevers developed for this data storage application.
A great deal may be said about these techniques, but for now, we simply state that these techniques are capable of producing diaphragms and cantilevers of silicon with thickness of microns and lateral dimensions of hundreds of microns up to millimeters (see Fig. 3). The mechanical properties of these structures are exactly what we would expect from the bulk mechanical characteristics of silicon.
Since these microstructures can have sensitive strain gauges embedded in them, it is easy to see that a number of useful sensing devices can be built. Particular examples include strain gauge-based pressure sensors, where an array of strain gauges can be positioned around the perimeter of a thin diaphragm, and be connected into a bridge configuration to automatically cancel out other noise and drift signals from the gauges. As we discuss the particular sensing devices, we'll see many examples of strain gauge-based microsensors.
Another issue associated with strain gauges is the accuracy of the resistance measurement. Generally, accuracy would be improved by using larger currents and producing larger voltage changes. However, the practical limit to the amount of current that may be used comes about due to power dissipation in the resistive element. For this reason, the technologies for bonding thin film strain gauges has been optimized to maximize the thermal conduction from the thin film to the substrate. Improving the thermal conductance enables the use of more current in the measurement.
Many strain gauges, and particularly doped silicon strain gauges, are sensitive to temperature changes. In some cases, this is a useful effect - especially if your application also needs to measure temperature. Generally, this is not the case, so it is necessary to compensate for this sensitivity. The easiest way to do this is to fabricate reference resistors from the same material, and locate them so that they do not sense the strain signal. A bridge configuration can be easily arranged to retain the strain sensitivity while canceling the temperature sensitivity of an array of strain gauges. Such arrangements are very important, and easily produced, so they are very common.
So, the applications of strain gauges are in sensors where medium-to-large amounts of strain are expected to occur (0.001% - 1%), where very low-cost devices are needed, where miniature silicon devices are necessary, and where signals are expected at frequencies from DC to a few kHz. The frequency limitation comes about because the bonding configuration of these devices generally leads to large stray capacitance, which tends to filter out rapidly varying signals.
Example
Calculation: Piezoresistive Cantilever
Figure 5: Piezoresistive Cantilever
The calculation of the sensitivity of a piezoresistive cantilever is presented here to provide an example of the strain gauge calculations.
As shown in Figure 5, we use a piezoresistive cantilever to sense variations in the shape of a surface which is passed beneath. This technique has been demonstrated as the Atomic Force Microscope (AFM) by several recent graduate students in the group of Cal Quate. In AFM, attractive forces between a sharp tip and a sample surface cause slight cantilever deflections. If the cantilever is thin enough, forces associated with atomic interactions between individual atoms can be measured.
The figure is taken from the thesis of Marco Tortonese, in which fabrication and operation of an AFM based on piezoresistive cantilevers is described in detail.
The load-deflection relationship for a simple cantilever beam is
where
Here, L is the length, T is the thickness, and w is the width. Since F = kZ, we have stiffness :
For a deflection Z, the cantilever has an angle of deflection of approximately
,
and, therefore, a radius of curvature of approximately
The strain in the upper surface of the cantilever is caused by the difference in arc length for the upper and lower surfaces.
The strain is given by
For a typical AFM cantilever (as shown in Fig. 4), we have parameters T = 4µm, L = 100 µm, w = 4 µm, E = 2 x 10^11 N/m2, and F = 10-7 N.
Therefore,
Since doped silicon has a gage factor of about 100, we would expect a change in resistance dR/R of 0.03% for this example.
In fact, the cantilever does not take on a circular deflection, and the strain is largely concentrated at the base. If we place our strain gauge at the base, we can expect a strain enhancement of order 5-10x, thereby increasing the resistance change.
With a decent circuit it is possible to measure resistance changes as small as 1 part in 10^6, so this is indeed a reasonable measurement. It is not simple, but it is possible.
In many cases in AFM, forces as small as 10^-10 N are measured, which requires a careful electrical circuit design. It is this difficulty which allowed this to be the topic of a PhD thesis at Stanford University.