S18_Investigating the Dielectric Loss Tangent and Resistivity of Crystalline Silicon

Measurements of Absorption of Crystalline Silicon (c-Si)

Wenqian Sun, Calvin Zachman

Adviser: Prof. Shaul Hanany

Abstract

We report on measurements of absorption of crystalline Silicon. Signals of frequency around 110 GHz were made normally incident on a c-Si sample, whose resistivity was, according to manufacturer's data, greater than 500 Ω • m. Dielectric loss tangent, tan 𝛿, of the sample was determined by calculating the ratio between output power measured without the sample and output power measured with the sample in place. A major source of systematic error in the measurements was standing waves. Two effective ways were found to reduce standing-wave effects. After diminishing the standing waves, we obtained tan 𝛿 = 0.0023 ± 0.0009, which lay in the accepted range given in literature [5].

1 Introduction

Cold stellar bodies emit electromagnetic radiation in the microwave frequency range. In experimental astrophysics, this range is often referred to by wavelength as the millimeter-submillimeter (MSM) band. Observations in the MSM band are critical to the understanding of our universe, providing experimental data on the physics of the big bang, stellar formation and the evolution of galaxies, all of which can be studied through observations of cosmic microwave background radiation (CMB). Any probe into the early universe requires the detection and analysis of CMB signals. While these signals occupy all of interstellar space, they are incredibly faint and susceptible to loss as they pass through the optical equipment necessary for their detection [1]. From the study of electromagnetism, it is known that as electromagnetic radiation propagates through a medium it loses energy. The extent to which a dielectric material dissipates electromagnetic energy is characterized by the so called dielectric loss which is a function of the frequency of incident radiation in addition to the resistivity and permittivity of the material. This energy loss is present in all materials, but becomes a growing concern in the domain of low amplitude signal analysis such as in the detection of cosmic microwave background radiation.

Materials with a high index of refraction, such as silicon, display low absorptive loss across the MSM band and as a result promise to provide greater observing efficiency in measurements of CMB radiation.

This project is aimed at measuring absorption of c-Si as well as setting up equipment for possible future studies, such as investigating the relation between dielectric loss tangent and electrical resistivity of c-Si.

Results of this series of studies will hopefully provide people with a better understanding of c-Si's utility in the construction of optical lenses for future use in observations of the early universe.

2 Theory and Background

When an electromagnetic wave propagates through a material, it will lose energy. The amount of energy loss can be characterized by the electromagnetic absorption coefficient of the material, α. α is also termed power loss per unit distance. As this name suggests, α is defined by

where l denotes the thickness of the material, P_in the input power, and P_out the output power. Fig. 1 illustrates what we mean by the input and output power.

Permittivity of a material, ε := ε' - iε'', is often assumed to be real, when we believe that ε'' is negligible. In general, the magnitude of ε'' depends on bound charges and dipole relaxation within the material. It can give rise to electromagnetic energy loss, which is hardly distinguishable from the energy loss by conduction. Microwave engineers commonly employ dielectric loss tangent of a material (denoted by tan 𝛿), the ratio between ε'' and ε', to describe the absorption of microwave radiation by the sample. The relation between α and tan 𝛿 is given by

where λ₀ is the wave's wavelength in free space, and n and tan 𝛿 are the index of refraction and dielectric loss tangent of the material, respectively [2]. Combining the above two equations, we obtain

where

In this experiment, we will first measure the output power when a c-Si sample is not in place, P_{out, w/os}. After that, we place the sample in the optical path and then measure the output power, P_{out, w/s}, again. We assume that P_{out, w/os} is approximately equal to P_incident. Expressing β in terms of P_{out, w/os} and P_{out, w/s} ,we get

Hence, we have

3 Experimental Setup

Electromagnetic waves of frequency 109.7 GHz are produced by a Gunn diode, G. The resulting beam propagates spherically with a Gaussian intensity profile. Next, the spherical wavefronts generated by the diode will be collimated by an off-axis parabolic mirror M₁ and directed onto a sample of crystalline silicon of thickness l. A portion of the signal is transmitted through the sample and reaches a second parabolic mirror M₂ where the planar waves are converted back into spherical wavefronts and analyzed using a spectrum analyzer, S. Fig. 2 shows a schematic of the experimental setup, and Fig. 3 shows the real setup.

4 Result

When taking measurements of P_{out, w/os} and P_{out, w/s}, we occasionally discovered P_{out, w/s} > P_{out, w/os}, which clearly violated conservation of energy. We hypothesized that it were standing waves in the setup that gave rise to this phenomenon. In order to detect standing waves in the setup, we decided to change the receiver's position in the optical path and then measure the respective output power. For starters, a point near M₂'s focal point (no more than 5 mm) was chosen to be z₀ = 0, at which the receiver was initially placed. After that, we moved the receiver in the z direction (see Fig. 2) back and forth within the range -6 mm to 6 mm and measured the output power with the spectrum analyzer. Fig. 4 and 5 show a plot of P_{out, w/os} versus z and a plot of P_{out, w/s} versus z, respectively.

From Fig. 4 and 5 we observed that the wavelength of the oscillation was around 1.5 mm, which was approximately equal to one-half of the wavelength of the signal generated by G, 2.7 mm. (Recall that the frequency of the signal was 109.7 GHz.) This phenomenon proves the existence of standing waves in the setup, since we know

In order to accurately measure the absorption of the c-Si sample, it was indispensable to reduce standing-wave effects.

To avoid the defocusing effect (see Fig. 5), we placed the c-Si sample in a sample holder that was mounted on a translational stage to change its position in the optical path (instead of moving the receiver). Also, we used an aperture, whose diameter was 13.01 mm ± 0.05 mm, to filter out noise signals. We tried different setups to reduce standing-wave effects. Amongst these setups, we found two that were remarkably successful.

A sheet of eccosorb was initially placed between G and M₁ to reduce the intensity of the wave generated by G. After we realized that what we were really measuring was the standing wave between the receiver and the surfaces of the sample, we placed the sheet between S and M₂. Like before, a point between the two mirrors was first chosen to be y₀ = 0, at which the sample holder was initially placed. After that, we moved the sample holder in the y direction (see Fig. 2) back and forth within the range 0 mm to 3.5 mm and measured the output power. We took measurements both with and without the sample in place. Fig. 6 shows the resulting plot.

From Fig. 6 we observed that the amplitude of the standing wave significantly decreased. We then computed the arithmetic means of the two data sets, respectively. Using the sixth equation in Theory and Background, we obtained tan 𝛿 = 0.0017 ± 0.0009.

In general, standing waves are more likely to exist between parallel surfaces. Therefore, we hypothesized that we was able to reduce standing-wave effects by simply rotating the sample. Like before, we first took measurements without the sample in place. After that, we rotated the sample by 10.18° and took the same measurements again. Fig. 7 shows the resulting plot.

From Fig. 7 we observed that standing-wave effects diminished. We computed the arithmetic means of the two data sets, respectively. Also, we applied Snell's law to calculate the path length that the wave travelled through the sample, l'. We obtained l' = 6.008 mm, which differed from l by only 0.1%. This calculation implied that the sixth equation in Theory and Background was still valid. Employing this equation (we still used l in the equation), we got tan 𝛿 = 0.0017 ± 0.0009.

5 Conclusion

We discovered that standing waves existed in the setup. The amplitudes of these standing waves were comparable with the power loss due to absorption. Therefore, in order to accurately measure the absorption of the c-Si sample, it was indispensable to reduce standing-wave effects. We found that two setups were effective to reduce the standing-wave effects, namely placing a sheet of eccosorb between the receiver and the second parabolic mirror and rotating the c-Si sample. We obtained tan 𝛿 = 0.0017 ± 0.0009 and tan 𝛿 = 0.0023 ± 0.0009 by using the former and the latter setup, respectively. These two results lay in the accepted range given by the literature, 5.00E-6 to 1.70E-3 [5].

Future experimentation should seek to quantify the frequency dependence of the loss tangent. In addition, measurements of the resistivity would increase the ease of comparison with findings of other groups and provide better indication as to the validity of the experimental setup used here. Finally, the expected reflection was calculated using electromagnetic theory. For the purposes of this experiment this was acceptable. However, the calculations made several assumptions, the validity of which could be called into question for the tilted sample in particular. Future work could include an experimental determination of the reflection.

References

[1] Shaul Hanany, Karl Young, et al. Broadband Millimeter-Wave Anti-Reflection Coatings on Silicon Using Pyramidal Sub-Wavelength Structures. (Academic, Minnesota, 2017).

[2] Eugene Hecht. Optics. (Academic, Massachusetts, 2002).

[3] Paul F. Goldsmith. Quasioptical Systems: Gaussian Beam Quasioptical Propagation and Applications. (Academic, New York, 1998).

[4] Kevin Newman. An Introduction to Off-Axis Parabolic Mirrors. (Academic, Arizona, 2013).

[5] James Lamb. Miscellaneous Data on Materials for Millimetre and Submillimetre Optics. (Academic, France, 1996).

Acknowledgment

我们感谢 Mr. Karl Young 提供所有的帮助。

我们感谢 Prof. Hanany 当我们遇到困难时的鼓励 (特别是当我们打破 multiplier 时)。

我们感谢 Prof. Dan Dahlberg for his "life is good"。我们感谢 Kevin Booth 和 Kurt Wick 这些年来为 MXP学生 提供的服务和帮助。

Wenqian will always remember your kindness.