# Permittivity of Undoped Silicon in the Millimeter Wave Range
## 5 Experiment and Results  
The undoped silicon wafer was used for representative measurement. Silicon is a very common material in semiconductor circuits. The provided value of resistivity is 70 k\(\Omega\)-cm. The diameter of the wafer is 300 mm and the thickness is 660 \(\pm\) 10 \(\upmu\)m. Such a size is sufficiently large for measurement. Both sides of the silicon wafer were polished. Measurement was conducted at 25 \({}^{\circ}\)C. The measurement was first conducted on open air. Such a measurement was to remove the background effects, including atmospherical absorption. However, since the atmospherical absorption in the E-band is less than 16 dB/km according to the ITU standard (ITURP.676-11-201609), the total attenuation due to air is less than 0.03 dB given that the path length of this system is smaller than 2 m. In addition, the reference measurement will introduce a phase difference of 2\(\pi\)_Lc/f_, which should be subtracted from the resulting phase. The corrected raw data of S21 are plotted in Figure 7. It is seen that the transmission response is close to a sinusoidal function. To eliminate the noise, the third and fifth order polynomial fittings were used. Using the Tailor expansion, it is found  
\[\left\{ \begin{array}{l}T_{3}(f)\approx a_{0}+a_{1}f+a_{2}f^{2}+a_{3}f^{3}\\ T_{5}(f)\approx a_{0}+a_{1}f+a_{2}f^{2}+a_{3}f^{3}+a_{4}f^{4}+a_{5}f^{5}\end{array}  \right.. \tag{15}\]  
If the fifth order polynomial fitting was used, the difference to the third order is only 0.14 dB in the worst case, and the phase difference is only 1\({}^{\circ}\), as shown in Figure 7. The seventh order can also be used, but no significant difference was observed. This is a piece of evidence that the fifth order fitting provides adequate accuracy. The fitting data were used for permittivity retrieve. In addition, the permittivity versus the frequency is plotted in Figure 8. The real part is plotted in solid line, the imaginary part is plotted in dotted line, and the loss tangent is plotted in dash-dot line. The imaginary part is scaled by 100 and the loss tangent is scaled by 1000. It is seen that the real part of the permittivity is 11.801 at 60 GHz and gradually decreases to 11.702 at 90 GHz for the third order fitting. For the fifth order fitting, it is 11.755 at 60 GHz and slightly  
Figure 6: The flow chart of Newton–Raphson method solving for the permittivity. changes to 11.717. The difference of the third order and fifth order fitting is about 0.5%. Such results are in line with the published data of 11.74 [12], and is slightly smaller than the provided value of 11.9 at 1 MHz. The loss tangent of the intrinsic silicon is on the order of \(10^{-3}\), showing not too much difference between the third and fifth order fitting. However, it has to be mentioned that being constrained by the accuracy of the free space method on the loss factor, it is one order larger than the results in References [14; 15; 16]. It is probably due to this reason that the tendency of loss factor with frequency was not exhibited. To examine the uncertainty due to the magnitude and phase, we followed the error model in Reference [22], (see Equations (27)-(37) in Section III). The magnitude stability of the _VDI_ modules is 0.10 dB, and the phase stability would be 5\({}^{\circ}\). In addition, the fitting model may introduce an extra 0.10 dB magnitude error and 1\({}^{\circ}\) phase error. Feeding these values into the error model would give a maximal uncertainty of 5%, and an average error of 3% can be expected. To investigate the temperature dependent nature of undoped silicon, the temperature was set to 20, 25, and 30 \({}^{\circ}\)C. The test bench was placed in a temperature controllable room. The temperature around the sample was measured using a Fiber Optical Sensor (_OMEGA FOB-102_), with the sensitivity of \(\pm 0.1\)\({}^{\circ}\)C. Using the same procedure and taking the fifth order fitting, the retrieved data are plotted in Figure 9. There is a tendency for the real part to decrease with frequency, though not noticeably.