Surface roughness affects the function of a wide variety of engineering components, including airport runways, highways, ship hulls, and mechanical parts. Perhaps the most demanding applications are in the optics and semiconductor industries. Surface roughness causes scattering and stray light in optical systems and degrades the contrast and sharpness of optical images, so in general, the smoother the surface, the better the component will function.
Figure 1. A surface roughness profile z(x) can be approximated by digital means (z1...zN).
In the optics industry, the terms "surface roughness" and "surface finish" are synonymous. The rms roughness is perhaps the most widely used parameter for specification of the roughness of optical surfaces. For high-performance optics, such as those in lithographic steppers, space optics, and laser gyro systems, the rms roughness is specified in the subnanometer range. If the measured surface topography is represented as a surface profile z(x), the rms roughness (Rq) is defined as the root mean square of the deviations of the surface profile z(x) from the mean line (see figure 1). That is,
where L is the length of the surface profile along the x-direction. The rms roughness is currently designated by Rq or Rq in documentary standards and has often been represented by in the optics literature. 1-3
Because surface profiles z(x) are closely approximated in nearly all modern instruments by a digitized set of points zi, the above formula is replaced in practice by its digital equivalent:
where N is the number of data points in a measured surface profile.
If the measured surface topography is represented as a 3-D topographic image zij, the analogous formula is
where the S indicates that the quantity is averaged over a surface, and Nx x Ny is the number of pixels in the image. For topographic images, Rq and ARq can represent rms roughness. measuring roughness
According to the American Society of Mechanical Engineers (ASME) standard, techniques for the measurement of surface roughness may be classified as profiling methods, area profiling methods, and area averaging methods. Profiling methods probe the peaks and valleys of the surface under test with a high-resolution probe that senses the height of the surface and produces a quantitative surface profile zi. Area profiling methods extend the technique into three dimensions, either by rastering a series of profiles or by some type of quantitative imaging process.
In contrast to these approaches, area averaging methods do not involve measuring the resolved surface topography zij or zi at all. Rather, an area of the surface under test is probed all at once, and the quantity measured by the instrument is related to a statistical parameter of the surface roughness by modeling the probe-surface interaction. An example of an area averaging technique is angle-resolved light scattering (ARLS), which is closely related to the amount of stray light produced by the surface, an important functional property. We will emphasize the profiling and area profiling techniques here.
A number of profiling techniques are capable of measuring surfaces with subnanometer roughness, which is characteristic of the finest optical surfaces. These include stylus-based profiling,4 phase-shifting interferometric microscopy,5 Nomarski profiling,6 and atomic force microscopy.7
Figure 2. This stylus instrument profile of a Si3N4 film on fused silica yields an Rq value of 0.06 nm.8
A stylus-based profiler incorporates a stylus that traverses the surface in direct contact at low force, moving up and down as it rides over the surface peaks and valleys. The instrument converts the vertical motion of the stylus to an electrical signal, often by magnetic or optical displacement-sensing techniques, thus producing the surface profile z(x). The lateral resolution is limited by the lateral dimensions of the stylus tip, which can be as small as 0.1 µm or less. The surface profile of silicon nitride (Si3N4)8 shown in figure 2, for example, has a measured Rq value of 0.06 nm--approximately one-fifth the size of a typical atom in a solid.
Such small values are possible because the stylus tip, roughly 1 µm in diameter, is bearing on and averaging over a number of surface atoms at once, thus producing a smoothing effect in the surface profile. Although this measured profile is dominated by instrument noise, such a small Rq value shows both the quality of the surface and the quality of the instrument. We can tell both that the surface is mighty smooth and that the instrument can measure mighty smooth surfaces.
Figure 3. Phase-shifting interferometric microscope produced this surface topographic image of a silicon carbide surface, yielding a calculated Rq value of 0.12 nm. This method uses averaging and reference surface subtraction techniques described in reference 5.
One disadvantage of the contacting stylus instrument is the potential for surface damage. Optical techniques such as phase-shifting interferometric microscopy avoid this because they are non-contacting. The phase-shifting interferometric microscope (PSIM) has the additional advantage that it is generally based on electronic imaging technology and, hence, is an area profiling technique producing a topography image zij. The PSIM is capable of z-resolution in the 0.1-nm range, particularly if a reasonable degree of signal averaging is used and if the imperfections of the reference surface of the instrument are separated from the measured surface topography (see figure 3).
The Nomarski profiler uses an interferometric height differencing technique to measure surface profiles z(x) over surface lengths up to about 100 mm. The noise resolution for this differencing technique over a roughness sampling length of about 80 µm is in the 0.01-nm range.
Figure 4. 2.5 nm x 2.5 nm AFM image of an individual molecule of sorbic acid on a well-ordered graphite surface clearly shows the atomic corrugations of the graphite.9
The atomic force microscope (AFM) produces the highest lateral resolution of the four techniques. The achievable lateral resolution can be 1 nm or less. Under optimal conditions, individual atoms can be resolved on certain types of surfaces. An AFM operates similar to a stylus instrument. A probe tip mounted on a cantilever contacts the surface. Two modes of contact are typically used: a contact mode with an extremely low contact force, or an intermittent contact mode in which the cantilever is vibrationally excited. Any deflection of the cantilever or change in the mechanical vibration characteristics due to interaction with the surface peaks and valleys produces a signal in a piezoelectric or optical sensor. This signal is held at a null value, usually by displacing the surface in the z-direction to compensate for any sensed z-change. The driving signal for the surface z-displacement yields a topographic image of the surface for a wide range of measurement conditions (see figure 4).9
The AFM is especially useful for measuring optical surfaces with high lateral resolution. This may be important when monitoring surface defects for manufacturing process control. It is also important when studying surfaces designed for short optical wavelengths, such as mirrors working in the extreme ultraviolet (EUV) with a wavelength of 13.6 nm. understanding measurement
Thus, there are several useful profiling or area profiling techniques for studying optical surfaces. AFM is advantageous if high lateral resolution is required. The Nomarski profiler is a convenient way to measure long profiles. Interferometric microscopy is best for measuring topography of a significant area of a surface. The stylus method offers good sensitivity and dynamic range in both the lateral and vertical directions.
One important fact to note is that the rms roughness is not an intrinsic property of the surface. Rather, roughness on a surface is analogous to noise arising in time-series data. The value of rms roughness depends on the bandwidth of surface spatial wavelengths that the instrument can sense.10 That means that two important parameters should be specified for any measurement of a surface parameter such as rms roughness: the lateral resolution, which represents the finest spatial wavelengths that can be measured; and the sampling length, which is approximately equal to the longest spatial wavelengths that are measured. It is important to know the values for these two limiting factors for any measurement condition.
Usually only one of several factors limits the lateral resolution for a particular measurement. These factors include probe-tip size in the case of a stylus or the point-spread function in the case of an optical microscope. Other factors can be the high-frequency response of the instrument, often determined by an analog low-pass filter; a prescribed digital low-pass filter; or in some cases, the lateral data-point spacing, also called the sampling interval or pixel resolution. In fact, you rarely want the pixel resolution to be the quantity that limits the lateral resolution because aliasing can cause high-frequency noise in the signal to seem to appear at low frequencies, thus leading to possible misinterpretation of the measurement data and the instrument sources of error.
The sampling length is also likely to be determined by one of several instrument parameters. These parameters include the profile evaluation length, the high-pass electronic filter cutoff, a prescribed digital high-pass filter, the mathematical extraction of the overall surface shape, the use of a specially shaped reference surface such as a sphere, and mechanical filtering using a skidded probe that functions as a local reference.
It is important to maintain the accuracy of instruments operating at extreme z-magnification ranges. Several types of physical standards may be used for checking the z-calibration of instruments or testing their operation. Step height standards, which feature grooves or raised steps, are readily available down to heights of about 8 nm. For AFMs, several groups, including our own, have been researching the use of single-atom step heights in silicon as calibration standards.11-12 The results look promising.
Figure 5. A topographic image of polished fused silica measured with AFM yields a calculated Rq value of 0.14 nm over a 4 µm x 4 µm field of view.
Smooth-surface standards with rms roughness in the 0.1-nm range are obtainable for checking instrument noise resolution. Smooth periodic standards with uniform roughness are also useful for testing the operation of profiling instruments, but these are not yet readily available in the nanometer roughness range and below. Using an AFM and a PSIM, we tested a prototype13 having a rectangular, periodic-surface profile and obtained rms roughnesses of 0.5 nm (see figure 5) and 0.6 nm, respectively. We have not yet performed uncertainty analyses for these results, but extrapolation from previous data suggests the combined standard uncertainties for the rms values to be roughly 0.1 nm. With their small but highly uniform surface-roughness values, specimens of this type serve a useful purpose as calibration standards for profiling instruments. power spectral density
The rms roughness is only one important parameter calculated from surface profiles and topographic images. Many others have been developed to correlate surface-finish geometry with various functional properties of the surfaces themselves, such as friction and wear of mechanical parts, tire traction or skidding, and resistance to fluid flow. Derived surface parameters include peak-to-valley heights, rms slope, curvature, skewness, and mean spacing of surface irregularities.
Figure 6. A comparison of profiles and PSDs for a sinusoidal profile (left) shows a single peak, while the PSD for a ground surface profile (right) shows a range of spatial frequency data.
Each of these parameters represents an attempt to characterize surface roughness with a single number. Statistical functions, by contrast, describe the surface with a set of numbers so the characterization is more complete. For optical engineers, perhaps the most important function is the power spectral density (PSD) of the surface roughness. This statistical function provides a decomposition of the surface roughness profile into its component spatial wavelengths. For digitized profiles, the PSD is given by
where F represents the spatial frequency (the inverse of spatial wavelength), Δ is the sampling interval, j and k are indices, and i = -1 . For a periodic surface, the PSD function consists of only a fundamental peak and harmonics; for a random surface, the PSD function occupies a range of spatial frequencies (see figure 6). The PSD function is closely related to the amount and distribution of light scattered by a smooth or moderately rough surface. Thus, the PSD is closely related to a key functional property of optical surfaces.
One of the most interesting properties of the PSD is that the area on the graph under the PSD between two spatial frequency limits (for example, f1 and f2) is equal to Rq2 of the original profile when the profile is measured or filtered to have the same spatial frequency limits. The PSDs in figure 6 show why the Rq value measured for the same surface can vary so much. The Rq value calculated between limits f1 and f2 is quite different from the Rq value calculated between limits f2 and f3.
It is important to the user to be confident that the results of a surface measurement with two different types of instruments will agree. This is not easy to demonstrate because different surface-roughness instruments sample a surface over different spatial frequency ranges. The PSD allows comparisons to be made between instruments when their spatial frequency ranges overlap.14 oe
1. ISO 4287 (International Organization for Standardization, Geneva, 1997).
2. ASME B46.1-1995 (American Society of Mechanical Engineers, New York, 1995).
3. ASTM F1811-97 (American Society for Testing and Materials, West Conshohocken, PA, 1997).
4. J. Song, T. Vorburger, Applied Optics 30 (1): 42-50; 1991.
5. K. Creath, J. Wyant, Applied Optics 29 (26): 3823-3827; 1990.
6. B. Wang, S. Marchese-Ragona, et al., Proc. SPIE 3619, 121-127, (1999).
7. T. Vorburger, J. Dagata, et al., in Beam Effects, Surface Topography, and Depth Profiling in Surface Analysis, Czanderna et al., eds. (Plenum Press, New York, 1998), pp. 275-354.
8. J. Bennett, M. Tehrani, et al., Applied Optics 34: 209-212; 1995.
9. D. Rugar, and P. Hansma, Physics Today, 23-30; October 1990.
10. E. Church, G. Sanger, et al., Proc. SPIE 749: 65-73; 1987.
11. R. Dixson, N. Orji, et al., Proc. SPIE 4344: 157-168; 2001.
12. M. Suzuki, S. Aoyama, et al., J. Vac. Sci. Technol. A 14 (3): 1228-1232; 1996.
13. B. Scheer and J. Stover, Proc. SPIE 3141, 78-87, (1997).
14. E. Marx, I. Malik, et al., "Power Spectral Densities: A Multiple Technique Study of Different Si Wafer Surfaces," J. Vac. Sci. Technol. (in press).
While many people measure life in terms of dollars or career successes, Ted Vorburger takes stock of it in terms of roughness and step height measurements. Vorburger leads the National Institute of Standards and Technology (NIST; Gaithersburg, MD) group for surface-finish measurement and calibration techniques. "Our calibrations support the National Measurement system for surface finish," he says. "One of the most fun things of my work is trying to measure the same surface using different techniques and discovering that you get consistent results," with, for example, an AFM and a PSIM.
Vorburger was one of a group who developed the first sinusoidal calibration standards for surfaces. He was also instrumental in developing a calibrated AFM, which is calibrated with respect to the SI unit of length in all three axes, x, y, and z. "We did this by interfacing two laser interferometers to an existing AFM. People have used this technique for calibration in the past, but no one, as far as I know, ever tried this for an AFM," he says. Recently his group began work on a project to measure line edge roughness of semiconductors. "[It] potentially will be a show stopper on the international roadmap for semiconductors," he says.
In or out of work, Vorburger still measures and calibrates. He chaired the American Society of Mechanical Engineers (ASME) Standards Committee B46, Classification and Designation of Surface Qualities. He was the main editor of the ASME B46.1 1995 standard titled Surface Texture. He currently serves as a technical expert on related international working groups.
--Laurie Ann Toupin
Theodore Vorburger, Joseph Fu, Ndubusi Orji
Theodore Vorburger and Joseph Fu are with NIST, Gaithersburg, MD. Ndubuisi Orji is a guest researcher at NIST.