## Uncertainty improvement of geometrical thickness and refractive index measurement of a silicon wafer using a femtosecond pulse laser |

Optics Express, Vol. 20, Issue 11, pp. 12184-12190 (2012)

http://dx.doi.org/10.1364/OE.20.012184

Acrobat PDF (930 KB)

### Abstract

We have proposed a modified method to improve the measurement uncertainty of the geometrical thickness and refractive index of a silicon wafer. Because measurement resolution based on Fourier domain analysis depends on the spectral bandwidth of a light source directly, a femtosecond pulse laser having the broad spectral bandwidth of about 100 nm was adopted as a new light source. A phase detection algorithm in Fourier domain was also modified to minimize the effect related to environmental disturbance. Since the wide spectral bandwidth may cause a dispersion effect in the optical parts of the proposed interferometer, it was considered carefully through numerical simulations. In conclusion, the measurement uncertainty of geometrical thickness was estimated to be 48 nm for a double-polished silicon wafer having the geometrical thickness of 320.7 μm, which was an improvement of about 20 times that obtained by the previous method.

© 2012 OSA

## 1. Introduction

1. Y. Zhang, P. Parikh, P. Golubtsov, B. Stephenson, M. Bonsaver, J. Lee, and M. Hoffman, "Wafer shape measurement and its influence on chemical mechanical planarization," in *Proceedings of the First International Symposium on Chemical Mechanical Planarization,* I. Ali and S. Raghavan, eds. (The Electrochemical Society, Pennington, New Jersey, 1997), pp. 91-96.

2. M. Kimura, Y. Saito, H. Daio, and K. Yakushiji, “A New method for the precise measurement of wafer roll off of silicon polished wafer,” Jpn. J. Appl. Phys. **38**(Part 1, No. 1A), 38–39 (1999). [CrossRef]

3. G. Coppola, P. Ferraro, M. Iodice, and S. De Nicola, “Method for measuring the refractive index and the thickness of transparent plates with a lateral-shear, wavelength-scanning interferometer,” Appl. Opt. **42**(19), 3882–3887 (2003). [CrossRef] [PubMed]

5. G. D. Gillen and S. Guha, “Use of Michelson and Fabry-Perot interferometry for independent determination of the refractive index and physical thickness of wafers,” Appl. Opt. **44**(3), 344–347 (2005). [CrossRef] [PubMed]

6. J. Jin, J. W. Kim, C.-S. Kang, J.-A. Kim, and T. B. Eom, “Thickness and refractive index measurement of a silicon wafer based on an optical comb,” Opt. Express **18**(17), 18339–18346 (2010). [CrossRef] [PubMed]

## 2. Basic principle

*L*can be expressed as Eq. (1),where

*I*(

*f*,

*L*) is the intensity of interference spectrum,

*I*

_{0}(

*f*) is the spectrum of the light source,

*f*is the optical frequency,

*c*is the speed of light, and

*φ*(

*f*,

*L*) is the phase to be measured. After taking the DFT (discrete Fourier Transform) of

*I*(

*f*,

*L*), one peak in positive time domain can be found at the position of

*L*/

*c*, which represents the period of the interference signal in the spectral domain. The time resolution in Fourier domain, Δ

*t,*depends on the spectral bandwidth of the light source,

*N∙*Δ

*f*, which is given bywhere Δ

*f*is the sampling interval of the interference spectrum,

*N*is the sampling number. A 10 times larger spectral bandwidth makes the time resolution 10 times higher, which leads to improvement of the combined uncertainty. To get the phase information corresponding to the optical path difference

*L*, only the peak located in positive time domain is selected with a sampling window, and then inverse-Fourier transformed. The phase

*φ*(

*f*,

*L*) can be extracted by taking the imaginary part of the logarithmic function of the inverse Fourier transform result,

*I'*(

*f*,

*L*) like Eq. (3).

*L*can be determined through the linear fit of

*φ*(

*f*,

*L*) as shown in Eq. (4).

## 3. Experiments and uncertainty evaluation

*T*, and refractive index,

*N*, of a silicon wafer using a femtosecond pulse laser having a spectral bandwidth of about 100 nm at the central wavelength of 1550 nm. The mode spacing was extended from 250 MHz to 50 GHz through a Fabry-Perot filter for efficient detection of the individual comb modes using a conventional optical spectrum analyzer (OSA). Figure 2(a) shows a spectrum of the light source in the full range, which is 10 times larger than the spectral bandwidth of a light source, ranging from 1535 nm to 1545 nm, of the previous work [6

6. J. Jin, J. W. Kim, C.-S. Kang, J.-A. Kim, and T. B. Eom, “Thickness and refractive index measurement of a silicon wafer based on an optical comb,” Opt. Express **18**(17), 18339–18346 (2010). [CrossRef] [PubMed]

*L*

_{B}+

*T*+

*L*

_{C}-

*L*

_{A}(≡

*L*

_{1}) was measured in Ray 1 before inserting the silicon wafer. Then two optical path differences, of

*N*·

*T*(≡

*L*

_{2}) and

*L*

_{B}+

*N*·

*T*+

*L*

_{C}-

*L*

_{A}(≡

*L*

_{3}), were obtained in Ray 2 with the silicon wafer in a measurement arm. In this work, the optical thickness of the silicon wafer (

*L*

_{2}) was measured, instead of two additional optical path differences used in the previous work,

*L*

_{B}-

*L*

_{A}and

*L*

_{B}+

*N*·

*T*-

*L*

_{A}. Unlike

*L*

_{1}and

*L*

_{3},

*L*

_{2}is very sensitive to wafer alignment, as in the previous work. But, once the wafer surface is aligned at the right angle to the beam direction, the

*L*

_{2}measurement can be more stable than the individual measurements of

*L*

_{B}-

*L*

_{A}and

*L*

_{B}+

*N*·

*T*-

*L*

_{A}of the previous work because of a common path configuration. As a result, the geometrical thickness,

*T*, and the refractive index,

*N*, of the silicon wafer can be determined from the obtained three optical path differences

*L*

_{1},

*L*

_{2}, and

*L*

_{3}, which are given as

^{15}sampling points and a sampling resolution of 0.003 nm in the spectral range of 1500 nm to 1600 nm in wavelength. As mentioned before, to make the interference of

*L*

_{2}dominant, the optical intensity of the reference arm was adjusted by tilting the reference mirror, M

_{1}in Fig. 1 before inserting the silicon wafer. By reducing the optical intensity of the reference arm, an interference term having information of

*L*

_{2}became dominant to almost 60% of

*L*

_{3}in terms of peak amplitude, while

*L*

_{B}-

*L*

_{A}and

*L*

_{B}+

*N*·

*T*-

*L*

_{A}became relatively weak. The optical intensity of the reference arm was reduced to about 10% of the maximum intensity, which was measured by a conventional optical power meter. In addition, the peak for

*L*

_{2}also could be recognized practically by blocking the reference path and the

*L*

_{C}path in Fig. 1.

*T*and

*N*of the silicon wafer were measured to be 320.699 μm and 3.621, respectively. Also, the standard deviation of

*T*was calculated to be 26.7 nm, which is an improved result by about one order of magnitude in comparison with the precedent task.

^{-6}under general laboratory conditions, which was a minor factor because of the short OPDs. Moreover, for

*L*

_{2}, the uncertainty was not considered because of common path interferences between front and back surfaces of the silicon wafer. Wavelength uncertainty came from the wavelength accuracy of the OSA in use. According to the technical performance, the wavelength uncertainty of the OSA was given as 0.01 nm in the wavelength range of our use.

*T*of 320.7 μm and

*N*of the silicon wafer being a constant value at a center wavelength. In the case of dispersion, the

*N*of the silicon wafer was determined by an experimental equation [8

8. D. F. Edwards and E. Ochoa, “Infrared refractive index of silicon,” Appl. Opt. **19**(24), 4130–4131 (1980). [CrossRef] [PubMed]

*L*

_{2}and

*L*

_{3}in Fourier domain as shown in Fig. 5 . The peaks for

*L*

_{2}and

*L*

_{3}were shifted the same amount, about 2.8 × 10

^{-13}s, which corresponds to 42 μm in the optical path. It was too large to ignore in general cases. However, in this work, the dispersion effect could be canceled out according to Eq. (5). Therefore, it was proved that the proposed method was significantly unaffected by the dispersion effect of the silicon wafer. Finally, the uncertainties for

*L*

_{1},

*L*

_{2}, and

*L*

_{3},

*u*(

*L*

_{1}),

*u*(

*L*

_{2}), and

*u*(

*L*

_{3}), were estimated to be 108 nm, 140 nm, and 208 nm, respectively.

*T*,

*u*(

*T*) can be expressed aswhere,

*i*and

*j*are integer numbers and

*u*(

*L*,

_{i}*L*) is a covariance between

_{j}*L*and

_{i}*L*, which can be estimated by use of Eq. (8) using a correlation coefficient,

_{j}*r*(

*L*,

_{i}*L*).

_{j}*T*of the silicon wafer,

*u*(

*T*) was determined to be 48 nm (

*k*= 1) when measuring a silicon wafer having a

*T*of 320.7 μm. The uncertainty of refractive index

*N*is easily calculated from that of

*T*using Eq. (6).

## 4. Conclusion

## Acknowledgement

## References and links

1. | Y. Zhang, P. Parikh, P. Golubtsov, B. Stephenson, M. Bonsaver, J. Lee, and M. Hoffman, "Wafer shape measurement and its influence on chemical mechanical planarization," in |

2. | M. Kimura, Y. Saito, H. Daio, and K. Yakushiji, “A New method for the precise measurement of wafer roll off of silicon polished wafer,” Jpn. J. Appl. Phys. |

3. | G. Coppola, P. Ferraro, M. Iodice, and S. De Nicola, “Method for measuring the refractive index and the thickness of transparent plates with a lateral-shear, wavelength-scanning interferometer,” Appl. Opt. |

4. | P. Maddaloni, G. Coppola, P. De Natale, S. De Nicola, P. Ferraro, M. Gioffre, and M. Iodice, “Thickness measurement of thin transparent plates with a broad-band wavelength scanning interferometer,” IEEE Photon. Technol. Lett. |

5. | G. D. Gillen and S. Guha, “Use of Michelson and Fabry-Perot interferometry for independent determination of the refractive index and physical thickness of wafers,” Appl. Opt. |

6. | J. Jin, J. W. Kim, C.-S. Kang, J.-A. Kim, and T. B. Eom, “Thickness and refractive index measurement of a silicon wafer based on an optical comb,” Opt. Express |

7. | G. Nam, C.-S. Kang, H.-Y. So, and J. Choi, “An uncertainty evaluation for multiple measurements by GUM, III: using a correlation coefficient,” Accredit. Qual. Assur. |

8. | D. F. Edwards and E. Ochoa, “Infrared refractive index of silicon,” Appl. Opt. |

**OCIS Codes**

(120.0120) Instrumentation, measurement, and metrology : Instrumentation, measurement, and metrology

(120.3180) Instrumentation, measurement, and metrology : Interferometry

(140.3538) Lasers and laser optics : Lasers, pulsed

**ToC Category:**

Instrumentation, Measurement, and Metrology

**History**

Original Manuscript: March 7, 2012

Revised Manuscript: April 19, 2012

Manuscript Accepted: May 2, 2012

Published: May 14, 2012

**Citation**

Saerom Maeng, Jungjae Park, Byungsung O, and Jonghan Jin, "Uncertainty improvement of geometrical thickness and refractive index measurement of a silicon wafer using a femtosecond pulse laser," Opt. Express **20**, 12184-12190 (2012)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-11-12184

Sort: Year | Journal | Reset

### References

- Y. Zhang, P. Parikh, P. Golubtsov, B. Stephenson, M. Bonsaver, J. Lee, and M. Hoffman, "Wafer shape measurement and its influence on chemical mechanical planarization," in Proceedings of the First International Symposium on Chemical Mechanical Planarization, I. Ali and S. Raghavan, eds. (The Electrochemical Society, Pennington, New Jersey, 1997), pp. 91-96.
- M. Kimura, Y. Saito, H. Daio, and K. Yakushiji, “A New method for the precise measurement of wafer roll off of silicon polished wafer,” Jpn. J. Appl. Phys.38(Part 1, No. 1A), 38–39 (1999). [CrossRef]
- G. Coppola, P. Ferraro, M. Iodice, and S. De Nicola, “Method for measuring the refractive index and the thickness of transparent plates with a lateral-shear, wavelength-scanning interferometer,” Appl. Opt.42(19), 3882–3887 (2003). [CrossRef] [PubMed]
- P. Maddaloni, G. Coppola, P. De Natale, S. De Nicola, P. Ferraro, M. Gioffre, and M. Iodice, “Thickness measurement of thin transparent plates with a broad-band wavelength scanning interferometer,” IEEE Photon. Technol. Lett.16(5), 1349–1351 (2004). [CrossRef]
- G. D. Gillen and S. Guha, “Use of Michelson and Fabry-Perot interferometry for independent determination of the refractive index and physical thickness of wafers,” Appl. Opt.44(3), 344–347 (2005). [CrossRef] [PubMed]
- J. Jin, J. W. Kim, C.-S. Kang, J.-A. Kim, and T. B. Eom, “Thickness and refractive index measurement of a silicon wafer based on an optical comb,” Opt. Express18(17), 18339–18346 (2010). [CrossRef] [PubMed]
- G. Nam, C.-S. Kang, H.-Y. So, and J. Choi, “An uncertainty evaluation for multiple measurements by GUM, III: using a correlation coefficient,” Accredit. Qual. Assur.14(1), 43–47 (2009). [CrossRef]
- D. F. Edwards and E. Ochoa, “Infrared refractive index of silicon,” Appl. Opt.19(24), 4130–4131 (1980). [CrossRef] [PubMed]

## Cited By |
Alert me when this paper is cited |

OSA is able to provide readers links to articles that cite this paper by participating in CrossRef's Cited-By Linking service. CrossRef includes content from more than 3000 publishers and societies. In addition to listing OSA journal articles that cite this paper, citing articles from other participating publishers will also be listed.

« Previous Article | Next Article »

OSA is a member of CrossRef.