# Permittivity of Undoped Silicon in the Millimeter Wave Range
## 5 Experiment and Results
It is also found that with the increase in temperature, the real part tends to increase with the increase of  
Figure 8: The permittivity in the E-band at 25 \({}^{\circ}\)C.  
Figure 7: Raw data and fitting results of S21. (**a**) Amplitude; (**b**) phase. temperature. This is in line with the results of References [13; 15], where a more broadband temperature range was considered. However, it has to be mentioned that the variation of the permittivity with frequency and temperature is very small, and the overall variation is within \(11.75\pm 0.06\). It implies that the undoped silicon has a very good temperature property, or very small temperature coefficient. This is a very preferential property for electronic systems. A comparison of this work and the published work is shown in Table 2. The extracted data at 25 \({}^{\circ}\)C are plotted in comparison in Figure 10. It is noted that as the silicon samples used are from different vendors, differences in resistivity may exist. However, it is seen that the measured results from different groups are in the range of 11.63-11.76, showing a discrepancy of \(\pm 0.6\%\). The measured results in this work are in agreement with the results in the literature [10; 11; 12; 13; 14], within a discrepancy of 1.0%. It is also seen that the variation of the permittivity of silicon is very small below 450 GHz. In view of this fact, the dispersion of silicon is very weak, if there is any. \begin{table}
\begin{tabular}{c c c c c c c} \hline
**Ref.** & **Freq. (GHz)** & **Method** & **Temp.** & \(\dot{\epsilon}_{r}\) & **tan\(\delta\)** & **Remarks** \\ \hline
[8] & 0.03 & Capacitive method & 77 K & 11.7 \(\pm\) 0.2 & - & High-purity \\  & & & & & & & silicon \\ \hline
[9] & 107.3 & Free space non-focusing & 23 \({}^{\circ}\)C & 11–12 & \(<0.4\) & n-type \\  & & horn system & 27 \({}^{\circ}\)C & 11.68 & \(<2\times 10^{-3}\) & Gen. Diode \\ \hline
[10] & 120–400 & Dispersive Fourier & & & & \\  & & transform spectroscopy & 25 \({}^{\circ}\)C & 11.65–11.71 & \(<2\times 10^{-3}\) & - \\ \hline
[11] & 140 & Dispersive Fourier & & & & \\  & & transform spectroscopy & 25 \({}^{\circ}\)C & 11.65–11.71 & \(<2\times 10^{-3}\) & - \\ \hline
[12] & 30–300 & Fabry–[PERSON] resonator & 30–300 K & 11.74–11.66 & \(<2\times 10^{-4}\) & High-purity \\  & & & & & & silicon \\ \hline
[13] & 6–18 &
\begin{tabular}{c} Cylindrical dielectric \\ resonator \\ \end{tabular} & 10–370 K & 11.46–11.715 & \(\approx 2\times 10^{-3}\) & High-purity \\  & & & & & silicon \\ \hline
[14] & 1–15 &
\begin{tabular}{c} Split post dielectric \\ resonator \\ \end{tabular} & - & 11.65 & \(1.4\times 10^{-4}\) & High-purity \\  & & & & & silicon \\ \hline
[15] & 1–4 &
\begin{tabular}{c} Whispering gallery mode \\ resonator \\ \end{tabular} & 10–350 K & 11.46–11.71 & \(<10^{-4}\) & High-purity \\  & & & & & silicon \\ \hline
[16] & 20–50 & \begin{tabular}{c} Whispering gallery mode \\ resonator \\ \end{tabular} & \
\begin{tabular}{c} Room \\ temperature \\ \end{tabular} & 11.627–11.653 & \(<10^{-4}\) &
\begin{tabular}{c} Relative \\ error. a3\% \\ \end{tabular} \\ \hline
[17] & 9–12 & Waveguide measurement & - & 11.4–12.0 & \(<10^{-3}\) & - \\ \hline This work & 60–90 & Quasi-optical system & 20–30 \({}^{\circ}\)C & 11.717–11.755 & \(\approx 2.1\times 10^{-3}\) &
\begin{tabular}{c} Relative \\ error. a5\% \\ \end{tabular} \\ \hline \end{tabular}
\end{table} 
Table 2: A comparison of the measured permittivity of undoped silicon. Figure 9: The temperature dependent permittivity in the E-band using the fifth fitting.