Microsystem Technologies

, 15:1879 | Cite as

A nano-metrology system with a two-dimensional combined optical and X-ray interferometer and an atomic force microscope

Technical Paper


A two-dimensional nano-scale measuring system utilizing a two-dimensional combined optical and X-ray interferometer (2D COXI) was developed for the standardization of measurement in the nanometer region. The system consists of a 2D COXI and an atomic force microscope (AFM). The designed, two-dimensional, flexure-stage scans and the cantilever tip probes the nano-structure of the specimen. The calibrated optical interferometers in the 2D COXI were used to measure two-dimensional nano-scale lengths. The accuracy of the optical interferometers was enhanced to enable sub-nanometer measurements. To demonstrate the nano-scale measuring system, we used it to measure the nano-scale pitches of gratings.

1 Introduction

Nanometrological instruments, such as metrological atomic force microscopes (AFMs) and scanning electron microscopes (SEMs), have been widely utilized for the length measurements of atomic-scale structures. These instruments have to be calibrated by materials of standard length, such as a calibrated pitch grating or a step height. Therefore, the calibration of these standard materials is a key technology in nanometrology (Song et al. 2004). The calibration instruments for these standard materials are required to have a high precision and length traceability. A number of laboratories have developed nanometrological AFMs with optical interferometers for the calibration of standard materials (Misumi et al. 2003; Yacoot and Koenders 2003; Schneir et al. 1994).

We have developed a nano-scale measuring system for the calibration of standard materials. The nano-scale measuring system consists of a combined optical and X-ray interferometer (COXI) and an AFM. An optical interferometer is a precise measuring tool and is widely used in dimensional metrology. An X-ray interferometer generates signals with a period of about 0.2 nm and subdivides the signal of the optical interferometer. The two combined interferometers cause the synergy effect in a new measuring tool with a nonlinearity-free scale in the nanometer region. The COXI was developed recently by several national metrological institutes (Basile et al. 2000). It provides a system for measuring one-dimensional displacements with sub-nanometer sensitivities.

We developed a two-dimensional COXI (2D COXI), which combines an X-ray interferometer and two optical interferometers. An AFM instrument was combined with the 2D COXI for the realization of two-dimensional measurement. The 2D COXI can provide the basis for two-dimensional sub-nanometer positioning, the calibration of nano-sensors, and the length measurements of nano-structures. We designed a new scanner for a metrological AFM that can minimize parasitic errors by minimizing the crosstalk in the 100 μm-range stroke (Lee et al. 2007). The nano-scale measuring system that is developed in this study is expected to be a nearly nonlinear-free instrument for an optical interferometer and to measure a length of about 100 nm pitches or line widths with the traceability of standard lengths. We have demonstrated the performance of the instrument by measuring the pitches of one- and two-dimensional grating structures and estimating the uncertainty in measurement.

2 Configuration of the system

As shown in Fig. 1, the nano-scale displacement measuring system consists of a 2D COXI, which provides a sub-nanometer scale, and an atomic force microscope, which probes nano-scale objects. The combination of these two instruments yields precision measurements at the nanometer level. The measuring system provides a nearly nonlinear-free scale after the calibration of an optical interferometer through an X-ray interferometer.
Fig. 1

Photograph of a nano-scale measuring system

2.1 2D COXI

The 2D COXI consists of an X-ray and two optical interferometers. Two optical interferometers were connected with an X-ray interferometer and the XY flexure stage of an atomic force microscope. In principle, the construction of a 2D COXI requires two set of interferometers, i.e., a two-dimensional monolithic X-ray interferometer that is connected to two optical interferometers. However, a two-dimensional monolithic X-ray interferometer has a complex structure. Hence, the process for manufacturing it is time-consuming. To avoid building in an excessive amount of complexity in the instrument, the monolithic X-ray interferometer was linked to two optical interferometers.

In the linked system of interferometers, the X-ray interferometer calibrates the optical interferometers and then, the calibrated optical interferometers are used for nano-scale measurements of displacements. The calibration of an optical interferometer can be carried out in situ and is nearly real-time for two-dimensional measurement. However, the calibrated optical interferometer has a residual nonlinear error in the measurement of displacements. A perfect, nonlinear-free, optical interferometer requires the direct measurement of a COXI. (In the near future, we will modify the structure of the X-ray interferometer for a nonlinear-free system.) The 2D COXI system, which will be modified, will not adopt the calibrating method for an optical interferometer; instead, it will adopt the direct measurement of the COXI.

2.1.1 X-ray interferometer

The X-ray interferometer, which was made from a nearly perfect Si (220) crystal, is a major instrument for precision measurement in a nano-scale measuring system (Basile et al. 1994; Cavangnero et al. 2004; Yacoot and Downs 2000; Yacoot and Cross 2003; Becker et al. 1981). The periodic signal that is generated from an X-ray interferometer is about 0.192 nm. The X-ray interferometer consists of three thin lamellas: a beam splitter (S); a mirror (M); and an analyzer (A). Figure 2 shows a photograph of the X-ray interferometer. A monolithic X-ray interferometer with a single parallel spring structure was manufactured for translating the analyzer lamella. An optical interferometer beam was reflected to one side of the analyzer lamella. To avoid an Abbe error in the X-ray interferometer, a mirror was attached on the side of the scanning part of the X-ray interferometer. The two incident beams were aligned so that their heights were within 1 mm from the vertical center of the mirror. However, the reflections of the two incident beams are 7.2 mm apart on the horizontal plane of mirror, i.e., they are symmetrically opposed to each other and are each 3.6 mm away from the horizontal center of the mirror. The symmetrical separation of the two optical beams can be compensated through the difference between the displacements of the two beams. The value of the lattice parameter of the Si (220) crystal was that measured by Becker et al. (Becker et al. 1981) and Becker (Becker 2001) because the X-ray interferometer was made of the same crystal—sourced from Wacker GmbH in Germany—that Becker et al. (Becker et al. 1981) and Becker (Becker 2001) had used in their measurements (the corrected lattice parameter was 0.192 015 497 ± 1.2 × 10−8 nm in air (Basile et al. 2000)). The radiation source for the X-ray interferometer was a water-cooled molybdenum X-ray tube. The power of the radiation source was typically 2.0 kW. The X-ray signals were generated at the rate of five signals per second. The calibration of an optical interferometer took 5 min. This produced a thermal drift of about 0.3 nm. A linear equation was fitted to the data in order to remove the drift.
Fig. 2

Photograph of an X-ray interferometer that shows a ray tracing. S, splitter; M, mirror; A, analyzer

2.1.2 The optical interferometer

The optical interferometer employed in the 2D COXI of the nano-scale measuring system is a conventional homodyne-type interferometer, as shown in Fig. 3. The optical source of the interferometer is a commercial laser, which is a He–Ne laser (05 STP 903 of Melles Griot) with the frequency stabilized. The frequency stability of the laser is 0.04 ppm, which gives rise to an uncertainty of 4 × 10−3 nm in measurements in the 100 μm range. The optical interferometer signals are interpolated as 512 electronic subdivisions and converted into an encoder signal, which yields a resolution of 0.6 nm. For the calibration of the optical interferometer, the movable part of the X-ray interferometer (the analyzer lamella) scans two optical beams using the piezoelectric actuators. The 3,296 X-ray signals that correspond to the two optical signals are obtained during the calibration. The measured nonlinear errors are sine-fitted and compensated through simple calculations (Eom et al. 2001, 2002).
Fig. 3

Schematic of the experimental setup of the 2D COXI for the measurement of two-dimensional displacements. The X-axis represents horizontal movement and the Y-axis represents vertical movement of the XY flexure stage. OI, optical isolator; BS, beam splitter; PBS, polarizing beam splitter; W, quarter wave plate; P, polarizer; PD, Photo detector; M, mirror; Pr, prism

Figure 6 of Lee et al. (2007) shows the nonlinear errors in two cycles of the optical signals. The measurement range of the X-ray interferometer is 633 nm. The obtained maximum amplitudes of the nonlinear errors of the X and Y axes are ±1 nm due to the changes in the refractive index of air and the temperature gradient. They are slightly different from each other due to the varying photo detectors and associated electronics. Discontinuities are slightly apparent at the point of intersection (316.5 nm) due to the stitching routine that joins together the data from the two separate optical signals. The amplitudes of the residual nonlinearities of both axes of the optical interferometers were less than ±0.3 nm, as shown in Fig. 7 of Lee et al. (2007). The residuals of the nonlinear error arise from the electronic noise, vibration, air fluctuation, and stitching routine of the two optical signals.

2.2 Atomic force microscope

The atomic force microscope, which is combined with the 2D COXI, consists of a commercial AFM head (Park Systems Corp. XE-150) and includes a Z-axis coarse stage, a fine scanner, and a structural frame. The fine scanner consists of a monolithic flexure hinge guide that includes a motion-amplifying mechanism. The scanner self-compensates for the thermal expansion. A double compound linear motion guide minimizes the parasitic motion error and the orthogonally decoupled XY motion. The natural frequencies are calculated as 127 and 220 Hz. The full range of the fine scanner is 120.8 μm for the X-axis and 130 μm for the Y-axis. The vertical resolution of the AFM system, which determines the performance of the AFM image, is measured as 0.8 nm peak-to-peak. The vertical resolution is measured as the gap between the tip and the sample using the optical lever of the AFM head at the null state of the cantilever with no XY scanning. As for the horizontal resolution, the feedback noise of the 2D COXI that is used in the fine scanner is measured as 0.6 nm peak-to-peak. In order to assess the crosstalk of the planar scanner, the rotational parasitic motion error is measured. The maximum yawing motion is measured as 2 arcsec for the X-axis and 1.5 arcsec for the Y-axis in a motion of range 100 μm.

3 Application of the nano-scale measuring system

For the application of the 2D nano-scale measuring system that combines a 2D COXI and an AFM, the periodic nano-scale pitch of gratings was measured by the system. The pitch sizes of samples were 200 and 700 nm for one-dimensional and two-dimensional measurements, respectively. These samples were fabricated by LG Electronics Ltd. using optical interferometer lithography on the silicon crystal. The sample size was 3 mm × 4 mm and the range of measurement at each point was approximately 73 μm for one-dimensional gratings and 85 μm for two-dimensional gratings. The scan speed of the stage was 0.5 μm/s. The sampling frequency of data acquisition of the optical interferometer was controlled by the sampling interval, which was approximately 1 nm. The spring constant and the resonant frequency of the cantilever of the AFM were about 30 N/m and 300 kHz (Park Systems Corp., AR5-NCHR), respectively.

The determination of the peak position of the pitch, the rotation angle, and the angle of inclination of the sample should be considered in the computation of the value of the pitch. The peak position was determined as the center position of the profile of each pitch. The pitch value was taken to be the mean pitch-value, which was calculated simply from the probing distance and the number of pitches. 381 and 122 pitch values were obtained for one line scan at one point for 1D and 2D gratings, respectively.

The angle of inclination is the profile difference between the X-axis of the direction of scanning and the centerline of the profile of pitches. The angle of rotation is obtained by comparing the rotated grooves of the sample with the Y-axis. The mean pitch-value can be derived from the corrected angles and the temperature deviation of the sample (Misumi et al. 2003; Schneir et al. 1994).
$$ p_{\text{m}} = L \cdot C_{\text{s}} \cdot {{C_{\text{t}} } \mathord{\left/ {\vphantom {{C_{\text{t}} } N}} \right. \kern-\nulldelimiterspace} N}. $$

In Eq. (1), pm is the mean pitch-value of the sample, L is the total displacement of the first and last peak positions, Cs is the corrected angle of inclination and rotation, Ct is the corrected value due to the temperature deviation of the sample, and N is the total number of measured pitches.

Figure 4 shows the results of the measurements, which indicate the repeatability of the nano-scale measuring system. The standard deviations of the measurement repeatability of samples were about 0.011 nm for one-dimensional and 0.06 nm in the X-axis and 0.09 nm in the Y-axis for two-dimensional gratings, respectively. The standard deviation of the repeatability indicates the performance of the system. These values were about 0.01% of the corresponding pitch sizes of 193 and 700 nm. They are good enough to confirm that samples can be measured with high stability.
Fig. 4

The pitch values of one-dimensional and two-dimensional pitches. (a) Repeatability of a one-dimensional pitch. (b) Repeatability of a two-dimensional pitch

4 Uncertainty

The uncertainty in measurement of the nano-scale measuring system was associated with the calibration of the optical interferometer through an X-ray interferometer and with the pitch measurement through the system. The uncertainties were estimated according to the guidelines in the ISO document, ‘Guide to the Expression of Uncertainty in Measurement.’ They were multiplied by a coverage factor of k = 2. Table 1 summarizes the uncertainties for the 2D COXI.
Table 1

Uncertainty budget of the system

Source of uncertainty

Value of standard uncertainty


Degree of freedom

Component of combined standard uncertainty (nm)

Optical interferometer

 Wavelength variation

4 × 10−8


7.72 × 10−6 × p

 Refractive index of air

0.6 nm



 Nonlinearity of interferometer

0.16 nm




 Thermal expansion

0.2 nm





0.6 nm



X-ray interferometer

 Lattice parameter

1.2 × 10−8


1.80 × 10−5

 Variation in atmospheric pressure

1,300 Nm−2


1.30 × 10−6

 Temperature fluctuation




0.70 × 10−5

 Noise of X-ray signal

10 pm




 Abbe error

1 × 10−7 arcsec/nm




Atomic force microscope


0.1 nm




 Abbe error

2 × 10−5 arcsec/nm



5.54 × 10−5 × p

 Cosine error



1.93 × 10−6 × p

Combined standard uncertainty U = √((0.42)2 + (6.51 × 10−5 × p)2) nm

p is nominal pitch value

In the optical interferometer, the major sources of uncertainty were the resolution of the optical interferometer and the changing refractive index of air. The air turbulence was minimized by the placement of an aluminum curve in the path of the optical beam and by the placement of the system in a chamber. The resolution of the optical interferometer can be enhanced by the reduction of electrical noises and vibration. The uncertainty that was derived from the nonlinear error was estimated from the calibration through the X-ray interferometer.

In the X-ray interferometer, the uncertainties that were associated with the silicon lattice parameter were referred to the PTB report because the silicon crystal ingot for the X-ray interferometer that was used in this research came from the same silicon crystal maker, Wacker Ltd, which PTB used. The lattice parameter error over a range of movement of 633 nm of the analyzer lamella was 1.8 × 10−5 nm. The uncertainty that was derived from the noise on one X-ray signal was determined as the position error of the analyzer lamella. The uncertainty that was derived from the Abbe error was estimated from the Abbe offset and the angle of rotation of the stage. The vertical Abbe offset between the contact points of the two interferometer beams and the center of the analyzer lamella was within 1 mm. The horizontal Abbe offset was 3.6 mm from the center of the lamella. The pitching and yawing rotation of the analyzer lamella was smaller than 0.01″ in the range of displacements of 633 nm.

The horizontal plane of the mirror gave rise to a symmetrical difference in measurement of approximately 0.6 nm between the two optical interferometers due to the horizontal rotation of the mirror. The horizontal difference in measurement could be compensated. Thus, the Abbe error was estimated only for the vertical Abbe offset. The total uncertainty that was related to the X-ray interferometer was 0.05 nm.

In an AFM, the major source of uncertainty is the Abbe error of the fine scanner. The uncertainty that was estimated from the Abbe error was within 0.01 nm with an Abbe offset of 1 mm and 2-arcsec parasitic rotations in a pitch measurement of 200 nm. However, the uncertainty will be large when the range of displacement is long, for example, 6 nm in measurement ranges of 100 μm. Therefore, in order to reduce the Abbe error, the accurate alignment of the optical beam and small rotation of the fine scanner are required. For the estimation of the uncertainty that is associated with the orthogonality of the XY stage, the orthogonality of an L-type mirror for an optical interferometer was measured as a deviation of 30 arcsec from the right angle. The cosine error in optical alignment was estimated as 1.5 arcmin.

5 Conclusion

We have developed a two-dimensional nano-scale measuring system that combines a 2D COXI and an AFM. The 2D COXI provides two-dimensional nano-scale measurements with the traceability of standard lengths. The accuracy of the optical interferometer that was calibrated through an X-ray interferometer was improved to the sub-nanometer level. The calibration of the optical interferometers was carried out in situ. The performance of the nano-scale measuring system was demonstrated in the measurement of the nano-scale pitches of gratings. The results revealed a 0.01% value for the repeatability of measurement. The application of the nano-scale measuring system will be expanded to sub-100 nm measurements of the line width or pitch.

The calibrated optical interferometer had a residual of the nonlinear error in measurement due to some limitation in the compensation and structure of the 2D COXI in the present system. To overcome the limitation in measurement, the type of X-ray interferometer should be modified, in particular, the analyzer in the X-ray interferometer. One optical interferometer had better correspond to single-analyzing part. Therefore, the X-ray interferometer should have two moving parts for reflecting two optical beams, i.e., it should have two moving mirrors with a double-spring structure. This type of X-ray interferometer will enhance the accuracy in measurements of the 2D COXI because of the perfect, nonlinear-free characteristic of an optical interferometer. Moreover, in the near future, we will adopt a frog-jump method for fast and accurate measurements through an X-ray interferometer (Bergamin et al. 1993, 1997).



This work was supported by the Korea Science and Engineering Foundation (KOSEF) grant funded by the Korea government (MEST) (No. 20090082696).


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Copyright information

© Springer-Verlag 2009

Authors and Affiliations

  1. 1.Manufacturing Engineering CenterSamsung Electro-Mechanics Co., LtdSuwonRepublic of Korea
  2. 2.Department of Mechanical EngineeringKorea UniversitySeoulRepublic of Korea
  3. 3.School of Mechanical EngineeringYeungnam UniversityGyeongsan-siRepublic of Korea

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