# Permittivity of Undoped Silicon in the Millimeter Wave Range
## 1 Introduction  
Dielectric permittivity plays a fundamental role in describing the interaction of electromagnetic waves with matter. The boundary conditions, wave propagation in matter, and wave reflection/transmission on interfaces all involve this parameter. In practical applications, the processes of link budget (wave attenuation), channel characterization (wave dispersion), and multi-path effect (reflection/transmission) are representative examples of wave interaction with media [1]. Understanding these phenomena is a fundamental requirement in communication system design. With the rising of millimeter wave technology, for example, 5G communication [2], millimeter wave radar and sensing [3], accurate characterization of electronic materials for these applications is of fundamental significance. Although the dielectric properties have been widely investigated in the microwave range or below, the work in the millimeter wave range is much less thorough. The reason is that it has been a long period that the operation frequency of communication systems has been limited to the low frequency range [4]. However, problems such as frequency shift and increased insertion loss are often observed in the millimeter wave circuit design due to inaccurate permittivity [5]. In view of these facts, it is necessary to investigate the dielectric property of materials in the millimeter wave range. Particularly, we restricted our study in the E-band (60-90 GHz), mainly due to the factthat many applications fall in the range, such as the planned unlicensed 64-71 GHz band for 5G communication in the USA [6] and 77 GHz vehicle radar [7]. In addition, we selected silicon as the representative materials for measurement since it is one of the most important semiconductor materials in electronic systems. There are many publications focused on the complex permittivity of silicon [8; 9; 10; 11; 12; 13; 14; 15; 16; 17]. Reference [8] used a capacitive method, which was suitable for measurement in low frequency range. Reference [9] investigated the n-type silicon at 107.3 GHz using a non-focusing free-space system. [PERSON] and his colleagues [10; 11] measured several semiconductor samples using dispersive Fourier transform spectroscopy (DFTS) and presented data over 100-450 GHz [10; 11]. It was found that the permittivity slightly increased with frequency from 100 to 180 GHz and then decreased with frequency. In contrast, the loss tangent decreased with the frequency monotonically. However, for silicon of resistivity 11,000 \(\Omega\)-cm, the loss tangent increases with frequency (see Figure 3 in Reference [11]). The Fabry-Perot resonator was also utilized for silicon characterization over the frequency range of 30-300 GHz [12]. Additionally, it was found that the permittivity had a slight tendency of linear decreasing over the frequency range. An extensive investigation was conducted by [PERSON] and his colleagues [12; 13; 14; 15; 16]. These measurements used resonator techniques at various frequency ranges below 50 GHz. It was demonstrated that the loss tangent decreased with increasing frequency, and in most cases on the order of \(10^{-4}\). Temperature effects were also systematically investigated over the range of 10-370 K. The real part showed very stable properties over the investigated frequency and temperature ranges, in the worst case in the range of 11.46-11.71. These findings provide valuable reference to the permittivity of silicon. In these studies the E-band is covered in [12], where it was stated that the refractive index decreased linearly with frequency. Although it is in a small magnitude, such a statement is slightly different from other publications. Considering that many applications have been developed in the E-band, it is best to conduct the measurement in this band. In consideration of these facts, this paper introduces a broadband method for dielectric measurement on semiconductor materials, with measurement conducted on undoped silicon. A detailed description on the condition of creating quasi-plane wave is given.