As makers of semiconductors look toward future technologies such as extreme ultraviolet lithography (EUVL), their ability to precisely measure material dimensions and characteristics becomes increasingly important. Along with reflective EUVL optics and mask substrates for EUVL, other components and systems, such as atomic clocks, metrology instrumentation, and space optics, require precise alignment and a high level of dimensional stability. One important characteristic is the linear coefficient of thermal expansion (CTE).
CTE is a measure of the susceptibility of a material to thermally induced changes in linear dimension. In critical applications, engineers minimize thermally induced dimensional changes by using ultra-low CTE materials, such as ULE and Zerodur. These materials have a CTE of approximately 0 ± 20 ppb K-1; by comparison, steel has a value of about 10 ppm K-1. A high-accuracy determination of the CTE of such materials is essential for process control, for providing input to the design of such systems, and for error compensation. Future applications such as EUVL photomasks require the development of instrumentation capable of determining absolute CTE with uncertainty less than 1 ppb K-1.
CTE is defined as the fractional change in length per unit change in temperature.1 The exact value for the CTE at any temperature T is given by the slope of the fractional length change versus temperature curve at that point. Fundamentally, we can determine the CTE of a material by measuring the change in dimension and the accompanying change in temperature. Thus, given the change in dimension ΔL of a sample from L1 to L2 and the corresponding change in temperature ΔT from T1 to T2, the mean CTE, αm, over the temperature range is given by
where L is the length of the sample at a given reference temperature. The dimensionless quantity ΔL /L is referred to as the dilatation (or dilation).
Figure 1. In an ideal situation, only the expansion of the sample would be measured (a), but in reality, changes in the frame contribute to changes in the measurement loop (b).
Dilatometers, the instruments for measuring the change in length of the sample, are analogous to the common indicating micrometer or caliper. Most dilatometers have three main components (see figure 1). The first is a means for determining the ends of the sample. This takes the form of the two anvils in a micrometer; in a dilatometer, this may take the form of mechanical contacts, light incident upon reflective end-faces, or capacitance gages, for example. The second component is a measurement system for detecting and measuring the change in length of the sample. In a practical system, this may be implemented using a linear variable differential transformer, laser interferometer, or other method. The third component is a structural system or frame that positions the above components relative to the sample and serves as a return path for the metrology loop. This component typically comprises the instrument structure or a second measurement path.
In addition, there must be a method for changing the temperature of the sample, such as a furnace or cooler, and the requisite metrology for monitoring this change in temperature. Analysis shows that for the ultra-high accuracy measurement of CTE, the dominant contributor to the uncertainty in the measurement is the uncertainty associated with the measurement of the change in length of the sample, which makes accurate assessment of that quantity paramount. error sources
The interaction of the various components and the overall behavior of a dilatometer may be understood with the aid of the metrology loop. This conceptual construct takes the form of a closed loop that passes through all components of the system that have an effect on the measured change in length of the sample. Changes in length along any part of this loop manifest themselves in the measurement as apparent changes in the length of the sample and are indistinguishable from the dimensional changes of the sample. For example, in a length determination with a common micrometer, deformations of the C-frame of the micrometer are indistinguishable from length changes in the part being measured. In a perfect situation, the indicator measures only the expansion of the sample. Realistically, the measurement is a convolution of the dimensional changes of all components in the system.
The largest source of uncertainty in the measurement often comes from the thermal deformation of the instrument's frame. The design challenge is to confine the temperature changes to the sample while isolating the rest of the metrology loop. In practice, thermal coupling between the energy source and the frame is an issue that can result in spurious displacements. The frame may take the form of a physical structure or an optical reference in an interferometer system. Typically, we stabilize the frame by some combination of environmental control and use of materials with significantly lower CTE (often several orders of magnitude smaller) than that of the sample, such as ULE, Zerodur, or Invar. When we're measuring these ultra-low expansion materials themselves, however, stringent environmental control of the frame is the only option.
In high-accuracy interferometric systems, we may replace the reference frame with an equivalent optical path. We monitor changes in this optical paththat is, dimensional changes of the reference frameas part of the overall measurement. Since optical path length is the product of refractive index and physical path length, changes in either show up as dimensional changes. In order to minimize the effects of changes in refractive index, we typically perform such measurements in vacuum, effectively producing a reference frame that is minimally affected by changes in the environment and is equivalent to a frame with low CTE. techniques
All the current high-accuracy techniques are interferometric. A more detailed review of absolute high-accuracy interferometric techniques and a general review of measurement techniques may be found in the papers by Badami and Linder2 and James,3 respectively. Each method has advantages and disadvantages, and each yields a different level of uncertainty in CTE determinations (see table). Single-sided, with sample and base plate
This is the configuration of choice for a large number of instruments. Users attach the sample to a base plate and make measurements directly off the front face of the sample (with an auxiliary mirror in some cases) and indirectly off the rear face via the base plate. Two dial indicators show the differential nature of the measurement. This arrangement is designed to be immune to gross motions of the sample along the line of action of the indicators; thus, in principle, the difference between the readings of the two dial indicators represents only the dimensional changes of the sample. The routing of the metrology loop indicates that any dimensional changes of the frameany changes that result in a change in relative position between the indicatorswill be interpreted as a dimensional change in the sample.
One real-world interferometer implementation of this configuration replaces the two dial indicators with the measurement and reference arms of a Michelson-type interferometer.4 This is an example of partially replacing the reference frame with an equivalent optical-path length. The sample arrangement is symmetric, with the two reference beams straddling the sample. Double-ended, without auxiliary optics
This configuration is a relatively uncommon implementation. Two dial indicators show measurements off either end of the sample, with the two sensors linked through the frame. The frame then becomes part of the metrology loop, making any changes in the frame dimensions indistinguishable from changes in sample length.
A practical implementation consists of two separate Michelson interferometers with separate reference arms, referenced to one another through an artifact made of ultra-low expansion material.5 This approach minimizes the absolute expansion of the artifact, and hence its uncertainty contribution, by exploiting the fact that for a given temperature change, the expansion scales with the length: minimizing the length of the reference artifact minimizes its expansion. This method considerably shortens and stabilizes the metrology loop, which now passes through the short artifact rather than the instrument frame. Double-ended, with auxiliary optics
Figure 2. A general double-ended measurement system with auxiliary optics (a) can be performed using a Fabry-Perot etalon (b).
This configuration represents an inversion of the previous one (see figure 2). As in other configurations, we cannot distinguish dimensional changes in the frame from changes of the sample. To perform this type of measurement, the user constructs a high-finesse Fabry-Perot etalon with the sample as the spacer.6 Mirrors with high-reflectance coatings are optically connected to the ends of the sample. The cavity is probed by a laser with its frequency locked to a transmission maximum of the cavity. The user determines the frequency of the probe laser by beating it against a frequency standard. The change in resonance frequency of the etalon measures the change in optical-path length.
This method of displacement measurement is capable of sensing dilatation of about 10-11, which is a much greater accuracy than that offered by the other three methods. In addition, unlike the other methods, it has no separate reference arm. Theoretically, only dimensional changes of the spacer cause changes in the observed path length. In practice, of course, any effect that produces an optical-path length change contributes to the observed displacement.
Current state-of-the art instrumentation appears to require a fivefold improvement to meet the needs of applications such as EUVL. In reality, the situation is more complex. Estimates of uncertainty are typically derived from a measurement of the statistical variation (scatter) in the measurement. It is worth noting that many effects, such as the phase behavior of reflective optics, thermal influences on optical contacts, and deformation of the optics used to generate the reference optical length, can be highly repeatable. They can, therefore, be the source of significant systematic errors without having a deleterious effect on the scatter in the measurement.
As technology improves and the precision of measurement increases, systematic errors will most likely dominate the determination of CTE at this level of accuracy. The challenge for developers of ultra-high accuracy CTE metrology lies in identifying and quantifying systematic errors. oe
The authors would like to acknowledge Dave Young for the production of the figures.
1. American Society of Testing and Materials ASTM E 289, pp. 166-174 (1995).
2. V. Badami and M. Linder, Proc. SPIE Vol. 4688, pp. 469-480 (2001).
3. J. James, J. Spittle, et al., Meas. Sci. Technol., 12, pp. R1-R15 (2001).
4. S. Bennett, J. Phys. E: Sci. Instrum., 10, pp. 525-530 (1977).
5. E. Wolff and R. Savedra, Rev. Sci. Instrum., 56(7), pp. 1313-1319 (1985).
6. S. Jacobs, J. Bradford, et al., Applied Optics, 9(11), pp. 2477-2480 (1970).
Vivek Badami and Michael Linder
Vivek Badami is a senior research scientist at Corning Tropel, Fairport, NY, and Michael Linder is metrology manager at Corning Inc., Corning, NY.