## Determining mean thickness of the oxide layer by mapping the surface of a silicon sphere

Optics Express, Vol. 18, Issue 7, pp. 7331-7339 (2010)

http://dx.doi.org/10.1364/OE.18.007331

Acrobat PDF (1072 KB)

### Abstract

To determine Avogadro constant with a relative uncertainty of better than 2 × 10^{−8}, the mean thickness of the oxide layer grown non-uniformly on the silicon sphere should be determined with about 0.1 nm uncertainty. An effective and flexible mapping strategy is proposed, which is insensitive to the angle resolution of the sphere-rotating mechanism. In this method, a sphere-rotating mechanism is associated with spectroscopic ellipsometer to determine the distribution of the layer, and a weighted mean method based on equal-area projection theory is applied to estimate the mean thickness. The spectroscopic ellipsometer is calibrated by X-ray reflectivity method. Within 12 hours, eight hundred positions on the silicon sphere are measured twice. The mean thickness is determined to be 4.23 nm with an uncertainty of 0.13 nm, which is in the acceptable level for the Avogadro project.

© 2010 OSA

## 1. Introduction

3. P. Becker, H. Bettin, H.-U. Danzebrink, M. Gläser, U. Kuetgens, A. Nicolaus, D. Schiel, P. D. Bièvre, S. Valkiers, and P. Taylor, “Determination of the Avogadro constant via the silicon route,” Metrologia **40**(5), 271–287 (2003). [CrossRef]

5. K. Fujii, M. Tanaka, Y. Nezu, K. Nakayama, H. Fujimoto, P. D. Bièvre, and S. Valkiers, “Determination of the Avogadro constant by accurate measurement of the molar volume of a silicon crystal,” Metrologia **36**(5), 455–464 (1999). [CrossRef]

*et al*[6

6. N. Kuramoto, K. Fujii, Y. Azuma, S. Mizushima, and Y. Toyoshima, “Density determination of silicon spheres using an interferometer with optical frequency tuning,” IEEE Trans. Instrum. Meas. **56**(2), 476–480 (2007). [CrossRef]

*et al*[7

7. P. Becker, H. Friedrich, K. Fujii, W. Giardini, G. Mana, A. Picard, H.-J. Pohl, H. Riemann, and S. Valkiers, “The Avogadro constant determination via enriched silicon-28,” Meas. Sci. Technol. **20**(9), 092002 (2009). [CrossRef]

8. M. J. Kenny, R. P. Netterfield, L. S. Wielunski, and D. Beaglehole, “Surface layer impurities on silicon spheres used in determination of the Avogadro constant,” IEEE Trans. Instrum. Meas. **48**(2), 233–237 (1999). [CrossRef]

*et al*replaced this native oxide layer by a thermal oxide and then measured the thicknesses by XRR directly with 0.2 nm uncertainty. Nevertheless, the determination of the topography of the thermal layer is still necessary. Comparing with SE, XRR is a time-consuming and ineffective method for mapping the surface of the sphere.

## 2. Theories

### 2.1 SE, XRR, and model of the native oxide layer

*ρ*is the ratio of the reflection coefficients of p-polarization

_{2}, the native oxide layer on the silicon sphere usually contains of sub-oxides. Therefore, the mixture of monoxide silicon and dioxide silicon (SiO)

*(SiO*

_{a}_{2})

_{1-}

*(0<*

_{a}*a*<1) is applied to represent this layer, and modeled by the Bruggeman Effective Medium Approximation (BEMA) theory. Considering the roughness of the layer, the optical model for SE measurement is shown in Fig. 1 .

_{2}come from the published data of Aspnes and Edwards [11–13

13. D. E. Aspnes and A. A. Studna, “Dielectric functions and optical parameters of Si, Ge, GaP, GaAs, GaSb, InP, InAs, and InSb from 1.5 to 6.0 eV,” Phys. Rev. B **27**(2), 985–1009 (1983). [CrossRef]

### 2.2 Mapping strategy

16. W. Giardini and J. Ha, “Measurement, characterization and volume determination of approximately spherical objects,” Meas. Sci. Technol. **5**(9), 1048–1052 (1994). [CrossRef]

*cylindrical equal-area projection*, whose coordinate transformation formulas arewhere,

*cylindrical equal-area projection*. In this case, the distance between points is the same, and equal-weighted-mean can be used for estimating the mean thickness. However, the thing is not as simple as it seems. Usually the position changing of the sphere is by means of controlling the spherical angles but not space distance, therefore, the equal-area sampling requires high angle resolution of the sphere-rotating mechanism because the distributed angles are nonlinear, and the required resolution is much more rigorous along with the increase of the sampling points, which makes this method difficult to be applied to large number sampling.

*N*is the total number of sampled points), the estimated mean can be expressed byand the assumption that each point has the equal weight is implicit in Eq. (3). Consequently, the equal-area sampling is necessary, otherwise, the estimated mean will be incorrect. On the other hand, if the distribution of the sampled points is not uniform, the estimated mean should be revised aswhere

*M*is the sampled number along the same latitude direction, and

*N*is that along the same longitude direction).

## 3. Experiment

### 3.1 Experimental Set-up

### 3.2 Calibration by XRR

_{2}film is widely accepted and routine used, the only fitted parameter in the SE measurement is the thickness of the film. Therefore, the results of SE can be considered as “

*independent on the model*”, and the calibration result is much universal. Five single crystal silicon wafers grown SiO

_{2}films with nominal thicknesses of 2 nm, 18 nm, 34 nm, 61 nm and 170 nm were prepared by thermal oxidization. SE (incidence angle fixed at 70 degree) and XRR were performed to measure the film thicknesses separately and the results were compared and matched by a straight line of

## 4. Results

### 4.1 Measurement results

*a*= (3 ± 0.6) %. Then the measured data are fitted again by fixing

*a*= 3.6% as well as

*a*= 2.4%, and the discrepancy 0.03 nm of the mean thickness between these two model is considered as the uncertainty introduced by the optical model.

### 4.2 Uncertainty budget

*N*

_{A}being reduced to about 7 × 10

^{−9}after the error introduce by the oxide layer being corrected.

## 5. Conclusions

*N*

_{A}with relative uncertainty of 2 × 10

^{−8}nowadays. According to the principle of the XRCD method, the mean thickness of the oxide layer should be measured with uncertainty of nearly 0.1 nm. Because the thickness distribution of this layer is not uniform on the surface of the sphere, the work presented here measured the mean thickness by means of combining SE and a sphere-rotating mechanism. A weighted mean method based on equal-area projection theory is performed to avoid the rigorous requirement of the angle resolution of the rotating system, and results of SE are calibrated by XRR. The mean thickness of the oxide layer is determined to be 4.23 nm with uncertainty of 0.13 nm, which is in the range of the acceptable level for the current purpose of the Avogadro project.

## Acknowledgement

## References and links

1. | P. Becker, “History and progress in the accurate determination of the Avogadro constant,” Rep. Prog. Phys. |

2. | P. Becker, P. D. Bièvre, K. Fujii, M. Gläser, B. Inglis, H. Luebbig, and G. Mana, “Considerations on the future redefinitions of the kilogram, the mole and of other units,” Metrologia |

3. | P. Becker, H. Bettin, H.-U. Danzebrink, M. Gläser, U. Kuetgens, A. Nicolaus, D. Schiel, P. D. Bièvre, S. Valkiers, and P. Taylor, “Determination of the Avogadro constant via the silicon route,” Metrologia |

4. | R. A. Nicolaus and K. Fujii, “Primary calibration of the volume of silicon sphere,” Meas. Sci. Technol. |

5. | K. Fujii, M. Tanaka, Y. Nezu, K. Nakayama, H. Fujimoto, P. D. Bièvre, and S. Valkiers, “Determination of the Avogadro constant by accurate measurement of the molar volume of a silicon crystal,” Metrologia |

6. | N. Kuramoto, K. Fujii, Y. Azuma, S. Mizushima, and Y. Toyoshima, “Density determination of silicon spheres using an interferometer with optical frequency tuning,” IEEE Trans. Instrum. Meas. |

7. | P. Becker, H. Friedrich, K. Fujii, W. Giardini, G. Mana, A. Picard, H.-J. Pohl, H. Riemann, and S. Valkiers, “The Avogadro constant determination via enriched silicon-28,” Meas. Sci. Technol. |

8. | M. J. Kenny, R. P. Netterfield, L. S. Wielunski, and D. Beaglehole, “Surface layer impurities on silicon spheres used in determination of the Avogadro constant,” IEEE Trans. Instrum. Meas. |

9. | H. Fujiwara, |

10. | Y. Azuma, J. Fan, I. Kojima, and S. Wei, “Physical structures of SiO2 ultrathin films probed by grazing incidence x-ray reflectivity,” J. Appl. Phys. |

11. | D. F. Edwards, “Silicon(Si),” in |

12. | D. E. Aspnes, “Optical properties of thin films,” Thin Solid Films |

13. | D. E. Aspnes and A. A. Studna, “Dielectric functions and optical parameters of Si, Ge, GaP, GaAs, GaSb, InP, InAs, and InSb from 1.5 to 6.0 eV,” Phys. Rev. B |

14. | D. P. Johnson, “Geometrical considerations in the measurement of the volume of an approximate sphere,” J. Res. Natl. Bur. Stand. A |

15. | G. Mana, “Volume of quasi-spherical solid density standards,” Metrologia |

16. | W. Giardini and J. Ha, “Measurement, characterization and volume determination of approximately spherical objects,” Meas. Sci. Technol. |

17. | J. P. Snyder, |

18. | J. Zhang, Y. Li, and Z. Luo, “A traceable calibration method for spectroscopic ellipsometry,” Acta. Physica. Sinca. |

**OCIS Codes**

(120.3940) Instrumentation, measurement, and metrology : Metrology

(240.0310) Optics at surfaces : Thin films

(350.4600) Other areas of optics : Optical engineering

(240.2130) Optics at surfaces : Ellipsometry and polarimetry

**ToC Category:**

Instrumentation, Measurement, and Metrology

**History**

Original Manuscript: January 20, 2010

Revised Manuscript: March 1, 2010

Manuscript Accepted: March 15, 2010

Published: March 24, 2010

**Citation**

Jitao Zhang, Yan Li, Xuejian Wu, Zhiyong Luo, and Haoyun Wei, "Determining mean thickness of the oxide layer by mapping the surface of a silicon sphere," Opt. Express **18**, 7331-7339 (2010)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-18-7-7331

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### References

- P. Becker, “History and progress in the accurate determination of the Avogadro constant,” Rep. Prog. Phys. 64(12), 1945–2008 (2001). [CrossRef]
- P. Becker, P. D. Bièvre, K. Fujii, M. Gläser, B. Inglis, H. Luebbig, and G. Mana, “Considerations on the future redefinitions of the kilogram, the mole and of other units,” Metrologia 44(1), 1–14 (2007). [CrossRef]
- P. Becker, H. Bettin, H.-U. Danzebrink, M. Gläser, U. Kuetgens, A. Nicolaus, D. Schiel, P. D. Bièvre, S. Valkiers, and P. Taylor, “Determination of the Avogadro constant via the silicon route,” Metrologia 40(5), 271–287 (2003). [CrossRef]
- R. A. Nicolaus and K. Fujii, “Primary calibration of the volume of silicon sphere,” Meas. Sci. Technol. 17(10), 2527–2539 (2006). [CrossRef]
- K. Fujii, M. Tanaka, Y. Nezu, K. Nakayama, H. Fujimoto, P. D. Bièvre, and S. Valkiers, “Determination of the Avogadro constant by accurate measurement of the molar volume of a silicon crystal,” Metrologia 36(5), 455–464 (1999). [CrossRef]
- N. Kuramoto, K. Fujii, Y. Azuma, S. Mizushima, and Y. Toyoshima, “Density determination of silicon spheres using an interferometer with optical frequency tuning,” IEEE Trans. Instrum. Meas. 56(2), 476–480 (2007). [CrossRef]
- P. Becker, H. Friedrich, K. Fujii, W. Giardini, G. Mana, A. Picard, H.-J. Pohl, H. Riemann, and S. Valkiers, “The Avogadro constant determination via enriched silicon-28,” Meas. Sci. Technol. 20(9), 092002 (2009). [CrossRef]
- M. J. Kenny, R. P. Netterfield, L. S. Wielunski, and D. Beaglehole, “Surface layer impurities on silicon spheres used in determination of the Avogadro constant,” IEEE Trans. Instrum. Meas. 48(2), 233–237 (1999). [CrossRef]
- H. Fujiwara, Spectroscopic Ellipsometry: principles and applications (Wiley, 2007).
- Y. Azuma, J. Fan, I. Kojima, and S. Wei, “Physical structures of SiO2 ultrathin films probed by grazing incidence x-ray reflectivity,” J. Appl. Phys. 97(12), 123522 (2005). [CrossRef]
- D. F. Edwards, “Silicon(Si),” in Handbook of Optical Constants of Solid, E. D. Palik ed., (Academic, 1985).
- D. E. Aspnes, “Optical properties of thin films,” Thin Solid Films 89(3), 249–262 (1982). [CrossRef]
- D. E. Aspnes and A. A. Studna, “Dielectric functions and optical parameters of Si, Ge, GaP, GaAs, GaSb, InP, InAs, and InSb from 1.5 to 6.0 eV,” Phys. Rev. B 27(2), 985–1009 (1983). [CrossRef]
- D. P. Johnson, “Geometrical considerations in the measurement of the volume of an approximate sphere,” J. Res. Natl. Bur. Stand. A 78, 41–48 (1974).
- G. Mana, “Volume of quasi-spherical solid density standards,” Metrologia 31(4), 289–300 (1994). [CrossRef]
- W. Giardini and J. Ha, “Measurement, characterization and volume determination of approximately spherical objects,” Meas. Sci. Technol. 5(9), 1048–1052 (1994). [CrossRef]
- J. P. Snyder, Map Projections- a working manual (US Government Printing Office, 1987), pp. 76–81.
- J. Zhang, Y. Li, and Z. Luo, “A traceable calibration method for spectroscopic ellipsometry,” Acta. Physica. Sinca. 59, 186–191 (2010).

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