Permittivity of Undoped Silicon in the Millimeter Wave Range

electronics

Article Permittivity of Undoped Silicon in the Millimeter Wave Range

Xiaofan Yang 1, Xiaoming Liu 2,3,* , Shuo Yu 2, Lu Gan 2,3, Jun Zhou 4,* and Yonghu Zeng 1

1 State Key Laboratory of Complex Electromagnetic Environment Eﬀects on Electronics and Information System, Luoyang 471003, China 2 School of Physics and Electronic Information, Anhui Normal University, Wuhu 241002, China 3 Anhui Provincial Engineering Laboratory on Information Fusion and Control of Intelligent Robot, Wuhu 241002, China 4 Terahertz Research Centre, School of Electronic Science and Engineering, University of Electronic Science and Technology of China, Chengdu 610054, China * Correspondence: [email protected] (X.L.); [email protected] (J.Z.)

Received: 18 July 2019; Accepted: 8 August 2019; Published: 10 August 2019

Abstract: With the rapid development of millimeter wave technology, it is a fundamental requirement to understand the permittivity of materials in this frequency range. This paper describes the dielectric measurement of undoped silicon in the E-band (60–90 GHz) using a free-space quasi-optical system. This system is capable of creating local plane wave, which is desirable for dielectric measurement in the millimeter wave range. Details of the design and performance of the quasi-optical system are presented. The principle of dielectric measurement and retrieval process are described incorporating the theories of wave propagation and scattering parameters. Measured results of a sheet of undoped silicon are in agreement with the published results in the literature, within a discrepancy of 1%. It is also observed that silicon has a small temperature coeﬃcient for permittivity. This work is helpful for understanding the dielectric property of silicon in the millimeter wave range. The method is applicable to other electronic materials as well as liquid samples.

Keywords: millimeter wave; quasi-optics; free space method; undoped silicon; permittivity; temperature

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 fact

Electronics 2019, 8, 886; doi:10.3390/electronics8080886 www.mdpi.com/journal/electronics Electronics 2019, 8, 886 2 of 12 that 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 [8–17]. 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. Afsar 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 Ω 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 Krupka and his colleagues [12–16]. 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 4 of 10− . 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. Plane wave is very desirable in free space measurement. The retrieval of the permittivity incorporates the theories of wave propagation and scattering parameters. Temperature dependent characteristics are examined at 20, 25, and 30 ◦C. The remaining parts are organized as follows: Section2 is devoted to a general description on the permittivity and measurement techniques; Section3 describes the method generating a quasi-plane wave; Section4 is the retrieval method and Section5 is the measurement results; the last part Section6 concludes this work.

2. Permittivity and Measurement Techniques Dielectric permittivity is a macroscopic description of many microscopic processes, such as dipole relaxation, lattice vibration, and electronic polarization. These processes take eﬀect at diﬀerent characteristic frequency ranges, as illustrated in Figure1. Electronics 2019, 8, 886 3 of 12 Electronics 2019 , 8, x FOR PEER REVIEW 3 of 12

εr ( ω ) Polar molecules ε′ ω r ( ) Electronic Polarization Lattice

Ions effect

εr′′ ( ω )

No ions effect

10 3 10 6 10 9 10 12 10 15 (Hz) Microwave Terahertz Visible and purple

FigureFigure 1. 1.An An illustrationillustration ofof frequencyfrequency dependentdependent dielectric dielectric e eﬀffect.ect.

The permittivity is actually a complex quantity. The imaginary part is due to dielectric loss. The permittivity is actually a complex quantity. The imaginary part is due to dielectric loss. One One representative mechanism of loss is the dipole relaxation [1]. When a dipole tries to follow the representative mechanism of loss is the dipole relaxation [1]. When a dipole tries to follow the alternating electric ﬁeld, friction between dipoles would cause loss of energy. Other mechanisms alternating electric field, friction between dipoles would cause loss of energy. Other mechanisms include conductive loss and lattice loss. These mechanisms can be incorporated into a single parameter include conductive loss and lattice loss. These mechanisms can be incorporated into a single the complex permittivity parameter the complex permittivity εr = ε0r jε00 r, (1) − εr= ε r′ − j ε r ′′ , (1) where the real part is the relative permittivity and the imaginary part is referred to as the loss factor. Sincewhere these the real mechanisms part is the fall relative in di ﬀpermittivityerent frequency and the ranges, imaginary the permittivity part is referred exhibits to as athe dependency loss factor. onSince frequency these mechanisms fall in different frequency ranges, the permittivity exhibits a dependency on frequency εr(ω) = ε0r(ω) jε00 r(ω). (2) −

Referring back to Figure1, it has to beεωr( noted) = εω that r′( ) the − j εω ticks r ′′ ( ) . on the axes are not to exact scale, only(2) for illustrative purposes. Referring back to Figure 1, it has to be noted that the ticks on the axes are not to exact scale, only Methods of dielectric measurement shall vary with frequency due to the frequency dependent and for illustrative purposes. electrical size effects. Several methods can be used for dielectric measurement in the millimeter wave Methods of dielectric measurement shall vary with frequency due to the frequency dependent range, such as resonator method, transmission line method, and free space method [18]. The resonator and electrical size effects. Several methods can be used for dielectric measurement in the millimeter technique is an efficient way for low-loss single frequency measurement. The transmission line method wave range, such as resonator method, transmission line method, and free space method [18]. The imposes considerable challenge on sample preparation with the increase of frequency. Therefore, the resonator technique is an efficient way for low-loss single frequency measurement. The transmission free space system is more preferred for measurement spanning over a whole frequency band. line method imposes considerable challenge on sample preparation with the increase of frequency. Free space measurement requires ideally a plane wave system. However, plane wave is merely an Therefore, the free space system is more preferred for measurement spanning over a whole frequency ideal model that can be approximated using the far field of an antenna, but the far field would require a band. space too large to use. Alternatively, we used a quasi-optical (QO) system to create a quasi-plane wave Free space measurement requires ideally a plane wave system. However, plane wave is merely for dielectric measurement. The QO technique is particularly suitable for millimeter wave system due an ideal model that can be approximated using the far field of an antenna, but the far field would to its low-loss and wideband nature. It is also possible for multi-polarization measurement. In addition, require a space too large to use. Alternatively, we used a quasi-optical (QO) system to create a quasi- the size of a QO system is controllable. plane wave for dielectric measurement. The QO technique is particularly suitable for millimeter wave 3.system Generating due toQuasi-Plan its low-loss Wave and wideband nature. It is also possible for multi-polarization measurement. In addition, the size of a QO system is controllable. The QO technique was originally developed for radio astronomy for low-loss and wideband receivers3. Generating [19]. TransferringQuasi-Plan Wave the QO technique to dielectric measurement is a good attempt. The QO theory is based on Gaussian beam description of an electromagnetic wave. The propagation and refocusingThe QO of a technique Gaussian beamwas originally are illustrated developed in Figure for2. Aradio Gaussian astronomy beam canfor low-loss be described and using wideband [ 20] receivers [19]. Transferring the QO technique to dielectric measurement is a good attempt. The QO ! theory is based on Gaussian beam descriptionw r of2 an electromagneticjπr2 wave. The propagation and E(r, z) = exp jkz + jφ , (3) − 2 0 refocusing of a Gaussian beam are illustratedw0 inw Fi−gure −2. λAR Gaussian beam can be described using [20] where w0 is the beam waist located at z = 0, w is the beam radius, R is the radius of curvature and φ0 is w− r2jπ r 2  the phase shift. These parameters can be written as Erz(,)= exp2 −−+ j kz j,φ0  (3) w0  wλ R  Electronics 2019 , 8, x FOR PEER REVIEW 4 of 12

where w0 is the beam waist located at z = 0 , w is the beam radius, R is the radius of curvature and φ is the phase shift. These parameters can be written as Electronics0 2019, 8, 886 4 of 12 1 π w2  R= z + 0  ,  2  (4) z1πλw   R = z +  0 , (4) z  λ  2 0.5 λz   w w 1 , = 0  + 2  0.5 (5) π w  2   λz0    w = w01 +    , (5)  πw2   λz0 tanφ0 = 2 . (6) π wλ0z tan φ0 = . (6) πw2 It is seen from Equation (4) that the radius of curvature0 of the wave front is infinite at the beam waist.It This is seen is froma property Equation of plane (4) that wave. the radius However, of curvature as the beam of the propagates, wave front is the infinite size of at the the beam waist.increases This with is athe property propagation of plane distance wave. due However, to the diffraction as the beam effect propagates, of the electromagnetic the size of the wave, beam graduallyincreases withforming the a propagation spherical wave distance front. due Therefore to the, di aff focusingraction e ffunitect has of the to electromagneticbe used to refocus wave, the beamgradually to a formingbeam waist. a spherical The beam wave transformation front. Therefore, by a focusing unit hascan tobe be analyzed used to refocususing the the ABCD beam matrix.to a beam waist. The beam transformation by a focusing unit can be analyzed using the ABCD matrix.

w− r2jπ r 2  Gaussian beam: Erz(,)= exp2 −−+ j kz j φ0  w0  wλ R 

0.5 2  2  Radius of 1 π w0 Beam λz  R= z +   w= w 1+    wave front: z λ  radius: 0 2 π w0  

f w0 mid f2 w0 in 1 w0 out

Beam waist Beam waist Beam waist Focusing Unit Wave front Focusing Unit Gaussian beam profile

d in dout_mid din_mid dout d

Figure 2. TheThe propagation propagation of of a a Gaussian Gaussian beam beam through through free free spa spacece and and the the refocusing effect eﬀect by focusing units.

In a general case, the transform by a focusing unit of focalfocal lengthlength ff can be calculated by

 dout din/ f 1 dout= 1 + d in f −−1  f =1 + 2 2 2  (din/ f 21) +z2c / f 2 f w− , (7)  ()dfin−1 0 in + zf c  w0 out = 0.5  ( )2+ 2 2 , [ din/ f w1 zc / f ] (7)  − 0 in w0 out = 0.5  df−1 2 + zf2 2  where din, dout, w0 in, and w0 out are the input() distance,in output c distance,  input beam waist, and output beam waist, respectively. Additionally, zc is the confocal distance where din , dout , w0 in , and w0 out are the input distance, output distance, input beam waist, and 2 output beam waist, respectively. Additionally, z πisw the confocal distance z = c 0 in . (8) c λ π w2 z = 0 in . (8) It is seen that given din = f , the output distancec λ dout is always f, independent of the working frequency. Such a feature enables broadband operation. It is seen that given d= f , the output distance d is always f, independent of the working For a special case, if twoin identical focusing units areout used and placed 2f apart, i.e., f1 = f2 = f frequency.and d1 + d 2Such= 2 af feature(see Figure enables2), the broadband output beam operati waiston. will be the same as the input beam waist w0 out = w0 in. Therefore, the transmitting and receiving horns can be fabricated to be the same. Such a special case gives one a symmetrical structure and is referred to as Gaussian telescope. The system in this paper is in the form of Gaussian telescope. Electronics 2019 , 8, x FOR PEER REVIEW 5 of 12

For a special case, if two identical focusing units are used and placed 2 f apart, i.e., f1= f 2 = f and d1+ d 2 = 2 f (see Figure 2), the output beam waist will be the same as the input beam waist

w0 out= w 0 in . Therefore, the transmitting and receiving horns can be fabricated to be the same. Such a special case gives one a symmetrical structure and is referred to as Gaussian telescope. The system in this paper is in the form of Gaussian telescope. For a free space system, low cross polarization, low transmission loss, low reflection, and good bandwidthElectronics 2019 are, 8 ,preferred. 886 To meet these requirements, metallic reflectors can be used as focusing5 of 12 units. Although a dielectric lens can also be used for focusing, the inherent reflection loss and dielectricFor aloss free as space well system, as its lownarrower cross polarization, bandwidth m lowake transmission it less preferred. loss, low To reflection, obtain low and cross good polarization,bandwidth area high-quality preferred. To horn meet antenna these requirements, is always plausible. metallic Corrugated reflectors can horns be usedare good as focusing candidates units. toAlthough the transmitting/receiving a dielectric lens can units also besince used good for focusing, bandwidth the inherentand polarization reflection purity loss and can dielectric be achieved loss as [21].well In as addition,its narrower the bandwidth system has make to beit less capable preferred. of creating To obtain a lowlocal cross plane polarization, wave with a high-qualityacceptable transversehorn antenna size. is Additionally, always plausible. the field Corrugated should be horns close are enough good candidates to a plane to wave the transmitting over a longitudinal/receiving distanceunits since to ensure good bandwidth that thick andsamples polarization can also purity be measured. can be achieved Such requirements [21]. In addition, may thedemand system the has reflectorto be capable to be a of paraboloidal creating a local surface. plane However, wave with th acceptablee ellipsoidal transverse surface is size. more Additionally, common in the a QO field systemshould [20]. be close enough to a plane wave over a longitudinal distance to ensure that thick samples can alsoConsidering be measured. these Such requirements, requirements the may following demand param the reflectoreters were to be used a paraboloidal for the system surface. design However, (see Tablethe ellipsoidal 1). Parameters surface R1,is R more2, and commonθ are illustrated in a QO in system Figure [20 3a,]. and f is the equivalent focal length of M1 andConsidering M2 these requirements, the following parameters were used for the system design (see Table1). Parameters R , R , and θ are illustrated inR Figure R 3a, and f is the equivalent focal length of M1 1 2 f = 1 2 . and M2 R+ R (9) 1R R 2 f = 1 2 . (9) The parameter w is the beam radius along propagation,R1 + R2 as indicated in Figure 3b.

TableTable 1. 1. KeyKey parameters parameters for for the the Quasi-Optical Quasi-Optical System, System, Freque Frequencyncy 90 90 GHz. GHz.

1 2 ParametersParameters ww R R1 RR 2 θ θ f f Tx-HornTx-Horn 8.25 8.25 ------M1M1 33.1933.19 500500 500 500 45 45 250 250 SampleSample 32.1532.15 ------M2M2 33.1933.19 500500 500 500 45 45 250 250 Rx-Horn 8.25 - - - - Rx-Horn 8.25 - - - -

w =32.15 w=33.19 w w=33.19 0 mid w0 =8.25 M1 M2 z z=1000 w0 =8.25 z=250 z=500 z=750

(a) (b)

FigureFigure 3. 3.TheThe photograph photograph of ofthe the quasi-optical quasi-optical system system for for diel dielectricectric measurement. measurement. ( a) ( aThe) The photograph, photograph, (b()b the) the beam beam waist waist as as beam beam propagation. propagation.

TheThe fabricated parameter systemw is the is shown beam radius in Figure along 3a, propagation, and the beam as radius indicated (solid in curves) Figure3 b.and wave front (dashedThe curves) fabricated are plotted system isin shownFigure in3b. Figure The 3beama, and radius the beam at the radius location (solid of curves) M1/M2 and is 33.19 wave mm, front which(dashed requires curves) the are reflectorplotted in to Figure be 132.763b. The mm beam (4 radiusw criterion) at the location in length. of M1 To /M2 accommodate is 33.19 mm, lower which frequencyrequires theof the reflector E-band, to be the 132.76 size mmof the (4 wreflectorcriterion) is infabricated length. Toto accommodatebe 200 mm. At lower the location frequency of of the the E-band, the size of the reflector is fabricated to be 200 mm. At the location of the sample, the beam radius is 32.15 mm, which requires the reflector to be 128.60 mm in length. Again, the length of the sample would be suggested to be 200 mm for low frequency consideration. The dashed lines in Figure3b are calculated wave fronts using a computer program based on Equation (5). Noticeably, the wave front between M1 and M2 is very close to a plane wave. This is a very good property for dielectric measurement since phase error will be minimized. It is calculated that the phase difference between the edge of the reflector to that of the on-axis phase is only within Electronics 2019 , 8, x FOR PEER REVIEW 6 of 12 sample, the beam radius is 32.15 mm, which requires the reflector to be 128.60 mm in length. Again, the length of the sample would be suggested to be 200 mm for low frequency consideration. The dashed lines in Figure 3b are calculated wave fronts using a computer program based on ElectronicsEquation2019 (5)., 8 Noticeably,, 886 the wave front between M1 and M2 is very close to a plane wave. This6 ofis12 a very good property for dielectric measurement since phase error will be minimized. It is calculated that the phase difference between the edge of the reflector to that of the on-axis phase is only within 1010°.◦. The The extenders extenders are are VDIVDI VNAX VNAX standard standard, working, working at waveguide at waveguide band band WR-12, WR-12, corresponding corresponding to 60– to 60–9090 GHz. GHz. The The horns horns are are corrugated corrugated ones ones launching launching Gau Gaussianssian beams. beams. The The Gaussianities Gaussianities of of the the horns are obtained by calculating the power coupling betweenbetween an ideal Gaussian beam and the measured results, giving 98.5%.

4. Retrieval Model Since we created a quasi-plane wave, the analysis cancan be conducted using the plane wave method. Such assumptionassumption wouldwould introduce introduce limited limited measurement measurement error error while while significantly significantly reducing reducing the complexitythe complexity of mathematic of mathematic manipulation. manipulation. Reflection Reflecti onon an on air-medium an air-medium interface interface is an is important an important part inpart electromagnetism. in electromagnetism. The The reflection reflection model, model, however, however, is based is based on anon idealan ideal case case where where the the medium medium is infinitelyis infinitely long. long. To To apply apply the the ideal ideal model model to dielectric to dielectric measurement, measurement, a more a more precise precise model model has to has be to built be basedbuilt based on the on ideal the model. ideal model. This can This be simply can be done simply by done introducing by introducing a second a interface. second interface. The structure The canstructure be described can be described by an air-interface-medium-interface-air by an air-interface-medium-interface-air model, as model, illustrated as illustrated in Figure4 in. Figure 4.

−γ L T T e−γ L T21 T21 e 21 12 R12 −γ L −2γ L T21 R 21 e T21 R 21 T 12 e 2− 3 γ L T21 R 21 T 12 e 3− 4 γ L εr, µ r T21 R 21 T 12 e 2 −γ L (1− R12 ) e −2γ L T = (1− e ) R12 2− 2 γ L R = 1− R12 e 1− R2 e − 2 γ L Interface - 1 Interface - 2 12 Figure 4. Multi-reflectionMulti-reﬂection model.

The interactioninteraction ofof the the electromagnetic electromagnetic wave wave with with matter matter on interfaceson interfaces 1 and 1 and 2 is actually2 is actually the same the assame the as ideal the model.ideal model. Differently, Differently, the reflected the reflected wave wave inside inside the medium the medium will bounce will bounce forward forward and back. and Additionally,back. Additionally, these multi-reflectionthese multi-reflection/transmis/transmission esionffects effects have tohave be includedto be included in calculating in calculating the total the reflectiontotal reflection and transmission and transmission coeffi coefficients.cients. Using Usi theng same the same mathematical mathematical manipulation manipulation of [22 of], the[22], total the reflectiontotal reflection and transmission and transmission coeffi coefficientscients were were found found to be to be

 2γL  (1 e−2γ L)R12  R =(1−− e ) R12  1 R2 e 2γL  R = 122−− 2 γ L 1−−R2 e γL , (10)  (1 R12 )e−  = − 12 ,  T 22 2−γγLL (10) 1 R e−  ()1−−R12 e T = 2− 2 γ L where the reflection of a plane wave on interface 1− 1R is12 e where the reflection of a plane wave on interface 1 is √ εr R12 = − , (11) 1 + √εr 1− ε r R12 = , (11) and the wave propagation coefficient is 1+ εr and the wave propagation coefficient is γ = j2π √εr/λ. (12) . γ= j2 πελr / (12) Such a structure is identical to a two-port microwave network, where the elements of S11, S21 are the reflection and transmission coefficients of port 1, and S22, S12 are the reflection and transmission coefficients of port 1, respectively. Interface 1 and interface 2 are similar to port 1 and port 2 of a two-port microwave network, so that the scattering parameters S11 and S21 measured using a vector network analyzer (VNA) can be correlated to the reflection and transmission coefficients, respectively. Electronics 2019 , 8, x FOR PEER REVIEW 7 of 12

Such a structure is identical to a two-port microwave network, where the elements of S11 , S21 are the reflection and transmission coefficients of port 1, and S22 , S12 are the reflection and transmission coefficients of port 1, respectively. Interface 1 and interface 2 are similar to port 1 and port 2 of a two- port microwave network, so that the scattering parameters S11 and S21 measured using a vector network analyzer (VNA) can be correlated to the reflection and transmission coefficients, respectively. To correlate the transmission coefficient to S21 , a measurement on open air has to be conducted prior to sample measurement. In comparison, to correlate the reflection coefficient to S11 , Electronics 2019, 8, 886 7 of 12 a reference measurement of planar metallic plate has to be used. To this end, a link between the wave propagation in a sample and the scattering parameters measured by a vector network analyzer was

Toestablished. correlate the transmission coefficient to S21, a measurement on open air has to be conducted prior to sampleIn Figure measurement. 5, we present In the comparison, first few terms to correlate in comp thearison reflection to the coe totalfficient reflection/transmission to S11, a reference measurementcoefficients to ofexamine planar metallicthe wave plate interference has to be used.at different To this frequencies. end, a linkbetween It is seen the that wave the propagationfirst two or inthree a sample terms andmake the up scattering most of parametersthe reflection/transmis measured bysion. a vector It is also network found analyzer that at 160 was GHz, established. the first reflectionIn Figure R1 has5, wea phase present of −180°, the first while few the terms secon ind comparison term R2 and to the the third total term reflection R3 are/transmission 0° in phase. coeTherefore,fficients a to destructive examine the interference wave interference is formed. at diff Aterent 80 frequencies. GHz, a constructive It is seen that interference the first two is formed. or three termsAlthough make the up third most term of the is out reflection of phase/transmission. with the first It isterm, also the found amplitude that at 160of R3 GHz, is 13 the dB first smaller reflection than R1R1, hastherefore, a phase it of would180 ,not while contribute the second too term much R2 to and the the interference. third term R3 The are constructive 0 in phase. point Therefore, of the a − ◦ ◦ destructivereflection is interferencethe destructive is formed. point of Attransmission 80 GHz, a and constructive vice versa. interference is formed. Although the thirdRetrieve term is out of of dielectric phase with permittivity the first term, can the be amplitude fulfilled using of R3 isEquation 13 dB smaller (10). For than non-magnetic R1, therefore, itmaterials, would not a simple contribute mathematic too much manipulation to the interference. can be employed The constructive as shown point in Reference of the reflection [23]. The iserror the destructivefunction can point be defined of transmission as and vice versa.

Reflection Transmission 0 0 T1+T 2+T 3

-10 R1 T= ΣTn T1 T1+T 2 T R2 R1+R 2 -10 2 -20 R3 R1+R 2+R 3 -30 -20 Destructive Constructive

Amplitude/dB T Amplitude/dB interference interference 3 Constructive Destructive R= ΣRn interference interference -30 180 180 180 ° 90 90 R3 T 3 90 ° T2 0 0 0° 0° Phase/deg Phase/deg -90 -90 -180 ° R 2 -180 ° -90 ° T -180 R1 -180 1 0 50 100 150 200 250 0 50 100 150 200 250 Freq/GHz Freq/GHz (a) (b)

FigureFigure 5.5. ReflectionReflection/transmission/transmission ofof aa mediummedium havinghavingε εrr == 10 − j0.01.j0.01. ( (aa)) Reflection; Reflection; ( (bb)) transmission. transmission. − Retrieve of dielectric permittivity can be fulfilled using Equation (10). For non-magnetic materials, . a simple mathematic manipulation can be employedE(ε r ) = S21 as − T shown in Reference [23]. The error function(13) can be defined as Or using a weighted error function E(εr) = S T. (13) 21 − . (14) Or using a weighted error functionE(ε r ) = S21 −+ TwS( 11 − R )

E(εr) = S T + w(S R). (14) 21 − 11 − These numerical techniques are available in the literature [22]. In this work, Equation (13) was used as the error function. A ﬂow chart of the Newton–Raphson method for numerical solving is shown in Figure6. Electronics 2019 , 8, x FOR PEER REVIEW 8 of 12

These numerical techniques are available in the literature [22]. In this work, Equation (13) was Electronicsused as 2019the ,error8, 886 function. A flow chart of the Newton–Raphson method for numerical solving8 of 12is shown in Figure 6.

Initialization A first guess value

Using Newton-Raphson

method to calculate F( εr)

Calculating 2 ×2 Jacobian Matrix Next εr

－1 Calculating εr and F(εr) Calculating J ε

No Calculating the inverse of Convergence? Jacobian matrix J －1

Yes

End

Figure 6. The flow ﬂow chart of Newton–Raphson method solving for the permittivity.

5. Experiment and Results The undoped silicon wafer was used for representativerepresentative measurement.measurement. Silicon is a very common material in semiconductor circuits. The provided value of resistivity is 70 kΩ cm. The diameter of the material in semiconductor circuits. The provided value of resistivity is 70 kΩ·cm.· The diameter of the wafer is 300 mm and the thickness is 660 10 µm. Such a size is sufficiently large for measurement. wafer is 300 mm and the thickness is 660 ±± 10 μm. Such a size is sufficiently large for measurement. Both sides sides of of the the silicon silicon wafer wafer were polished. were polished. Measurement Measurement was conducted was conducted at 25 ◦C. The at measurement 25 °C. The wasmeasurement first conducted was first on open conducted air. Such on aopen measurement air. Such wasa measurement to remove the was background to remove e theffects, background including atmosphericaleffects, including absorption. atmospherical However, absorption. since the Howeve atmosphericalr, since the absorption atmospherical in the absorption E-band is lessin the than E- 16band dB /km is less according than to16 the dB/km ITU standard according (ITURP.676-11-201609), to the ITU standard the (ITURP.676-11-201609), total attenuation due to air the is total less thanattenuation 0.03 dB due given to thatair is the less path than length 0.03 dB of thisgiven system that the is smallerpath length than of 2 m.this In system addition, is smaller the reference than 2 measurementm. In addition, will the introduce reference a measurement phase difference will of intr2πoduceLc/ f , a which phase should difference be subtracted of 2π Lc / f from, which the resultingshould be phase. subtracted The correctedfrom the rawresulting data ofphase. S21 areThe plotted corrected in Figure raw data7. It of is S21 seen are that plotted the transmission in Figure 7. responseIt is seen isthat close the to transmission a sinusoidal response function. is To close eliminate to a sinusoidal the noise, function. the third andTo eliminate fifth order the polynomial noise, the fittingsthird and were fifth used. order Using polynomial the Tailor fittings expansion, were used it is. found Using the Tailor expansion, it is found ( ( ) + + 22+ 3 T3Tf3f() ≈+a0 a 012a af1 f +a af2 f + afa 3 f  ≈ 2 3 4 5. . (15) T5( f ) a0 + a1 f + a2 f 2+ a3 f 3+ a4 f 4+ a5 5f (15) Tf5()≈ ≈++ a 012 afaf + af 3 + af 4 + af 5 If the fifth order polynomial fitting was used, the difference to the third order is only 0.14 dB in 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 , as shown in Figure7. The seventh order can also be the worst case, and the phase difference is only◦ 1°, as shown in Figure 7. The seventh order can also used, but no significant difference was observed. This is a piece of evidence that the fifth order fitting be used, but no significant difference was observed. This is a piece of evidence that the fifth order provides adequate accuracy. fitting provides adequate accuracy. The fitting data were used for permittivity retrieve. In addition, the permittivity versus the The fitting data were used for permittivity retrieve. In addition, the permittivity versus the frequency is plotted in Figure8. The real part is plotted in solid line, the imaginary part is plotted in 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 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. 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 ElectronicsElectronics 2019 2019 , ,8 8, ,x x FOR FOR PEER PEER REVIEW REVIEW 99 of of 12 12

Electronics 2019, 8, 886 9 of 12 ItIt isis seenseen thatthat thethe realreal partpart ofof thethe permittivitypermittivity iiss 11.80111.801 atat 6060 GHzGHz andand graduallygradually decreasesdecreases toto 11.70211.702 at at 90 90 GHz GHz for for the the third third order order fitting. fitting. For For t thehe fifth fifth order order fitting, fitting, it it is is 11.755 11.755 at at 60 60 GHz GHz and and slightly slightly changeschangeschanges to to 11.717. 11.717. The The difference didifferencefference ofof thethe the thirdthird third orderorde order andr and and fifth fifth fifth order order order fitting fitting fitting is is aboutis about about 0.5%. 0.5%. 0.5%. Such Such Such results resul resul aretsts arearein lineinin lineline with withwith the the publishedthe publishedpublished data datadata of 11.74 ofof 11.7411.74 [12], [12], and[12], isandand slightly isis slightlyslightly smaller smallersmaller than thethanthan provided thethe providedprovided value value ofvalue 11.9 ofof at 3 11.911.91 MHz. at at 1 1 MHz. TheMHz. loss The The tangent loss loss tangent tangent of the intrinsic of of the the intrinsic intrinsic silicon issi si onliconlicon the is is order on on the the of 10orderorder− , showing of of 10 10 −3−3, ,showing showing not too much not not too too diff much erencemuch differencedifferencebetween the between between third and the the thirdfifth third order and and fifth fifthfitting. order order However, fittin fitting.g. However,it However, has to be it it mentioned has has to to be be mentionedthat mentioned being constrained that that being being constrainedconstrainedby the accuracy by by the the of accuracy accuracy the free of spaceof the the free methodfree space space on metho metho the lossdd on on factor, the the loss loss it is factor, factor, one order it it is is one one larger order order than larger larger the resultsthan than t thehe in resultsresultsReferences inin ReferencesReferences [14–16]. It [14–16].[14–16]. is probably ItIt isis probablyprobably due to this duedue reason ttoo thisthis that reasonreason the tendency thatthat thethe tendency oftendency loss factor ofof lossloss with factorfactor frequency withwith frequencyfrequencywas not exhibited. was was not not exhibited. exhibited.

11 220220 MeasurementMeasurement MeasurementMeasurement 00 3rd3rd order order fitting fitting 200200 3rd3rd order order fitting fitting 5th5th order order fitting fitting 5th5th order order fitting fitting -1-1 180180

-2-2 160160 300300

S21 S21 Amp-dB S21 S21 Amp-dB -3-3 -4.24-4.24 140140 S21 S21 Phase-deg

-0.6 S21 Phase-deg -0.6 -4.26-4.26 212212 106106 -4.28-4.28 211211 120 104104 -4-4 -0.8-0.8 -4.3-4.3 120 210210 -4.32 102102 S21 S21 Amp-dB -4.32 S21 S21 Amp-dB 6060 60.160.1 60.260.2 89.689.6 89.889.8 9090

89.9 89.95 90 S21 Phase-deg 6060 60.560.5 89.9 89.95 90 S21 Phase-deg -5-5 100100 Freq-GHzFreq-GHz Freq-GHzFreq-GHz 6060 Freq-GHzFreq-GHz7070 8080 9090 6060 7070 8080 9090 Freq-GHzFreq-GHz Freq-GHzFreq-GHz ((aa)) ((bb))

FigureFigureFigure 7. 7. Raw Raw data data and and fitting ﬁttingfitting results resultsresults of ofof S21. S21.S21. ( (aa))) Amplitude; Amplitude;Amplitude; ( ((bb))) phase. phase.phase.

1212

11.811.8 [email protected]@60GHz 1010 11.7811.78 33rd rd orderorder

11.76 88 11.76 55th th orderorder [email protected]@60GHz 11.7411.74 [email protected]@60GHz 66 11.7211.72 [email protected]@90GHz 11.711.7 4 4 100100 ××εεr′′′′ 6060 6565 7070 7575 8080 8585 9090 r Freq-GHzFreq-GHz 22 100100 ××2.52.5 ××1010 -2-2 10001000×× tan tan δδ 10001000 ××2.12.1 ××1010 -3-3 00 6060 6565 7070 7575 8080 8585 9090 Freq-GHzFreq-GHz

FigureFigureFigure 8. 8. 8. The TheThe permittivity permittivity permittivity in in in the the the E-band E-band E-band at at at 25 25 25 °C. °C.◦C.

ToToTo examineexamine thethe the uncertaintyuncertainty duedue due toto to thethe the magnitudemagnitude magnitude andand and phase,phase, phase, wewe we followedfollowed followed thethe the errorerror error modelmodel model inin in ReferenceReferenceReference [22], [22], [22], (see (see (see Equations Equations Equations (27)–(37) (27)–(37) (27)–(37) in in in Section Section Section III). III). III). The The The magnitude magnitude magnitude stability stability stability of of of the the the VDI VDI modules modulesmodules is is is 0.100.100.10 dB,dB, dB, andand and thethe the phasephase phase stabilitystability wouldwould bebe 5°.5°. 5◦ . InIn In adad addition,dition,dition, thethe the fittingfitting fitting modelmodel model maymay may introduceintroduce introduce anan an extraextra extra 0.100.100.10 dB dB dB magnitude magnitude magnitude error error error and and and 1° 1° 1 ◦phase phasephase error. error. error. Feeding Feeding Feeding these these these values values values into into into the the the error error error model model model would would would give give give a aa maximalmaximalmaximal uncertainty uncertainty uncertainty of of of 5%, 5%, 5%, and and and an an an average average average error error error of of 3% 3% can can be be expected. expected. ToToTo investigate investigate investigate the the the temperature temperature temperature dependent dependent nature nature of of undoped undoped undoped silicon, silicon, silicon, the the the temperature temperature temperature was was was set set set to to to 20,20,20, 25,25, 25, andand and 3030 30 °C.°C.◦C. TheThe testtest benchbench waswas placedplaced inin a a t temperaturetemperatureemperature controllable controllable room. room. TheThe The temperaturetemperature aroundaroundaround the the sample sample sample was was was measured measured measured using using using a Fiber a a Fiber Fiber Optical Optica Optica Sensorll Sensor Sensor (OMEGA ( (OMEGAOMEGA FOB-102 FOB-102), FOB-102 with the),), sensitivitywith with the the sensitivitysensitivityof 0.1 ◦C. of of Using ±0.1 ±0.1 °C. the°C. Using sameUsingprocedure the the same same procedure andprocedure taking an theandd fifthtaking taking order the the fitting,fifth fifth order order the retrieved fitting, fitting, the the data retrieved retrieved are plotted dat data ina ± areareFigure plottedplotted9. There in in FigureFigure is a tendency 9.9. ThereThere forisis aa the tendencytendency real part forfor to thth decreaseee realreal partpart with toto frequency, decreasedecrease withwith though frequency,frequency, not noticeably. thoughthough notItnot is noticeably.noticeably.also found It It that is is also also with found found the increase that that with with in the temperature,the increase increase in in the temperature, temperature, real part tends the the real real to increase part part tends tends with to to theincrease increase increase wi withth of Electronics 2019, 8, 886 10 of 12 Electronics 2019 , 8, x FOR PEER REVIEW 10 of 12 temperature.the increase of This temperature. is in line with This the is resultsin line ofwith References the results [13 ,of15 ],References where a more [13] broadbandand [15], where temperature a more rangebroadband was considered.temperature However, range was it considered. has to be mentioned However, thatit has the to variationbe mentioned of the that permittivity the variation with of frequencythe permittivity and temperature with frequency is very and small, temperature and the overallis very variationsmall, and is the within overall 11.75 variation0.06. Itis implies within ± that11.75 the ± 0.06. undoped It implies silicon that has the a veryundoped good silicon temperature has a very property, good or temperature very small temperatureproperty, or coeveryffi smallcient. Thistemperature is a very coefficient. preferential This property is a very for preferentia electronic systems.l property for electronic systems.

11.9 @30 °C

11.8

11.7

@20 °C @25 °C 11.6

11.5 60 65 70 75 80 85 90 Freq-GHz Figure 9. The temperature dependent permittivity in the E-bandE-band using the ﬁfthfifth ﬁtting.fitting.

A comparisonTable of this 2. A work comparison and the of publishedthe measured work permittivity is shown of inundope Tabled2 silicon.. The extracted data at 25 C are plotted in comparison in Figure 10. It is noted that as the silicon samples used are from ◦ Freq . Ref. Method Temp. ε′ tan δ Remarks different vendors,(GHz) differences in resistivity may exist. However,r it is seen that the measured results from different groups are in the range of 11.63–11.76, showing a11.7 discrepancy of 0.6%. TheHigh measured-purity [8] 0.03 Capacitive method 77 K ±- results in this work are in agreement with the results in the literature ± 0.2 [10–14], within a discrepancysilicon of Free space non -focusing 1.0%.[9] It is also 107.3 seen that the variation of the permittivity23 °C of silicon 11–12 is very small< 0.4 below 450n-type GHz. silicon In horn system view of this fact, the dispersion of silicon is very weak, if there is any. Dispersive Fourier −3 Gen. Diode [10] 120–400 27 °C 11.68 <2 × 10 transform spectroscopy Silicon

Table 2.DispersiveA comparison Fourier of the measured permittivity of undoped silicon.−3 [11] 140 25 °C 11.65–11.71 <2 × 10 - transform spectroscopy Ref. Freq. (GHz) Method Temp. ε0 r tanδ Remarks −4 High -purity [12] 30–300 Fabry–Perot resonator 30–300 K 11.74–11.66 <2 × 10 High-purity [8] 0.03 Capacitive method 77 K 11.7 0.2 - silicon ± silicon Cylindrical dielectric −3 High -purity [13] 6–18 Free space non-focusing 10–370 K 11.46–11.715 ≈2 × 10 n-type [9] 107.3 resonator 23 C 11–12 < 0.4 silicon horn system ◦ silicon Split post dielectric −4 High -purity [14] 1–15 Dispersive Fourier - 11.65 1.4× 10 Gen. Diode [10] 120–400 resonator 27 C 11.68 < 2 10 3 silicon transform spectroscopy ◦ × − Silicon Whispering gallery mode −4 High -purity [15] 1–4 Dispersive Fourier 10–350 K 11.46–11.71 < 10 [11] 140 resonator 25 C 11.65–11.71 < 2 10 3 - silicon transform spectroscopy ◦ × − Whispering gallery mode Room −4 Relative [16] 20–50 11.627–11.653 < 10 High-purity [12] 30–300resonator Fabry–Perot resonatortemperature 30–300 K 11.74–11.66 < 2 10 4 error: ±3% × − silicon −3 [17] 9–12 Waveguide measurement - 11.4–12.0 < 10 - Cylindrical dielectric 3 High-purity [13] 6–18 10–370 K 11.46–11.715 2 10− This resonator ≈ × −3 siliconRelative 60–90 Quasi-optical system 20–30 °C 11.717–11.755 ≈2.1 × 10 work Split post dielectric 4 High-purityerror: ±5% [14] 1–15 - 11.65 1.4 10− A comparison of this work andresonator the published work is shown in Table 2.× The extractedsilicon data at 25 Whispering gallery mode High-purity [15] 1–4 10–350 K 11.46–11.71 < 10 4 °C are plotted in comparison in resonatorFigure 10. It is noted that as the silicon samples− usedsilicon are from different vendors, differencesWhispering in resistivity gallery mode may exist.Room However, it is seen that the measuredRelative results [16] 20–50 11.627–11.653 < 10 4 resonator temperature − error: 3% from different groups are in the range of 11.63–11.76, showing a discrepancy of ±0.6%. The± measured 3 results in[ 17this] work 9–12 are in agreement Waveguide measurement with the results -in the literature 11.4–12.0 [10–14],< 10− within a -discrepancy This Relative of 1.0%. It is also seen60–90 that the Quasi-opticalvariation systemof the permittivity 20–30 C of11.717–11.755 silicon is very2.1 small10 3 below 450 GHz. In work ◦ ≈ × − error: 5% view of this fact, the dispersion of silicon is very weak, if there is any. ± Electronics 2019, 8, 886 11 of 12 Electronics 2019 , 8, x FOR PEER REVIEW 11 of 12

11.8 This work 25 °C

11.75 [8] [12] [10] 11.7 [14] [13]

11.65

[11] [16] 11.6 0.01 0.1 1 10 100 1000 Freq-GHz Figure 10. A comparison of the measured permittivity of siliconsilicon at didifferentﬀerent frequencyfrequency ranges.ranges.

6. Conclusions The permittivity of undoped silicon was measured usingusing aa quasi-optical system.system. The system was designed to create a quasi-plane wave that was preferablepreferable for the dielectric measurement millimeter wavewave range. range. A multi-layer model was employed employed for for parameter parameter retrieval. retrieval. The process takes into into account multi-path multi-path propagation propagation and and equals equals the reﬂection the refle andction transmission and transmission coeﬃcients coefficients to the scattering to the parametersscattering parameters in a two-port in network.a two-port Undoped network. silicon Undope wasd measuredsilicon was at measured the E-band at (60–90 the E-band GHz). (60–90 It was foundGHz). thatIt was the found undoped that silicon the undoped exhibited silicon stable exhibit permittivityed stable over permittivity the whole band over and the in whole the temperature band and rangein the oftemperature 20–30 ◦C.It range was foundof 20–30 that °C. the It measurementwas found that was the in measurement agreement with was the in providedagreement value withand the resultsprovided in thevalue literature. and results In addition, in the literature. it was demonstrated In addition, that it was silicon demonstrated has very weak that dispersion silicon has below very 450weak GHz. dispersion below 450 GHz.

Author Contributions: Data curation, X.Y. and S.Y.; Investigation, X.L.; Methodology, L.G. and J.Z.; Resources, Author Contributions: Data curation, Xiaofan Yang and Shuo Yu; Investigation, Xiaoming Liu; Methodology, Y.Z.; Writing—original draft, X.L.; Writing—review and editing, X.Y., and J.Z. Lu Gan and Jun Zhou; Resources, Yonghu Zeng; Writing—original draft, Xiaoming Liu; Writing—review and Funding:editing, XiaofanThis work Yang, was and supported Jun Zhou. by the National Natural Science Foundation of China, Grant Numbers: 61871003, 61601472, 61505022, the Open Project of the State Key Laboratory of Complex Electromagnetic EnvironmentFunding: This E workffects was on Electronics supported andby the Information National Natural System Sci underence the Foundation number of of CEMEE2019Z0203B, China, Grant Numbers: and the Natural61871003, Science 61601472, Foundation 61505022, of the Anhui Open province Project underof the theState contract Key Laboratory number of of 1708085QF133. Complex Electromagnetic ConflictsEnvironment of Interest: Effects onThe Electronics authors declare and Information no conflict System of interest. under the number of CEMEE2019Z0203B, and the Natural Science Foundation of Anhui province under the contract number of 1708085QF133.

ReferencesConflicts of Interest: The authors declare no conflict of interest. 1. Sulyman, A.I.; Nassar, A.T.; Samimi, M.K.; MacCartney, G.R.; Rappaport, T.S.; Alsanie, A. Radio propagation Referencespath loss models for 5G cellular networks in the 28 GHZ and 38 GHZ millimeter-wave bands. IEEE Commun. Mag. 1. 2014Sulyman,, 52, 78–86. A.I.; [ Nassar,CrossRef A.T.;] Samimi, M.K.; MacCartney, G.R.; Rappaport, T.S.; Alsanie, A. Radio 2. Balanis,propagation C.A. pathAdvanced loss models Engineering for 5G Electromagnetics cellular networ, 2ndks in ed.; the Wiley: 28 GHZ New and York, 38 GHZ USA, millimeter-wave 2012. bands. 3. Wang,IEEE Commun. Q.; Liu, S.;Mag. Chanussot, 2014 , 52 , J.;78–86. Li, X. Scene classiﬁcation with recurrent attention of VHR remote sensing 2. images.Balanis, IEEEC.A. Advanced Trans. Geosci. Engineering Remot. 2019Electromagnetics, 52, 1155–1167., 2nd [ CrossRefed.; Wiley:] New York, USA, 2012. 4.3. Staecker,Wang, Q.; P. Liu, Microwave S.; Chanussot, industry J.; Li, outlook—Overview. X. Scene classificationIEEE with Trans. recurrent Microw. attention Theory Tech. of VHR2002 remote, 50, 1034–1036. sensing [images.CrossRef IEEE] Trans. Geosci. Remot. 2019 , 52, 1155–1167. 5.4. Zhang,Staecker, L.; P. Zhang, Microwave Q.; Hu, industry C. The inﬂuenceoutlook - ofoverview. dielectric IEEE constant Trans. on Microw. bandwidth Theory of Tech. U-notch 2002 microstrip, 50 , 1034–1036. patch antenna. In Proceedings of the 2010 IEEE International Conference on Ultra-Wideband, Nanjing, China, 5. Zhang, L.; Zhang, Q.; Hu, C. The influence of dielectric constant on bandwidth of U-notch microstrip patch 20–23antenna. September In Proceedings 2010. of the 2010 IEEE International Conference on Ultra-Wideband, Nanjing, China, 6. FCC.20–23Use September of Spectrum 2010. Bands Above 24 GHz For Mobile Radio Services, et al. Report and Order, FCC 16-89. 6. 14FCC. July Use 2016. of Spectrum Available Bands online: Above https: 24//www.federalregister.gov GHz For Mobile Radio Services,/documents et al./2016 Report/07/14 and/2016-16620 Order, FCC/open- 16- commission-meeting-thursday-july-14-201689, 2016-07-14. (https://www.federalregister.gov/do (accessed oncuments/2016/07/14/2016-16620/open-commission- 1 August 2019). meeting-thursday-july-14-2016, accessed on 2019-08-01) 7. Kwon, O.-Y.; Cui, C.; Kim, J.-S.; Park, J.-H.; Song, R.; Kim, B.-S. A Compact Integration of a 77 GHz FMCW Radar System Using CMOS Transmitter and Receiver Adopting On-Chip Monopole Feeder. IEEE Access 2019 , 7, 6746–6757. Electronics 2019, 8, 886 12 of 12

7. Kwon, O.-Y.; Cui, C.; Kim, J.-S.; Park, J.-H.; Song, R.; Kim, B.-S. A Compact Integration of a 77 GHz FMCW Radar System Using CMOS Transmitter and Receiver Adopting On-Chip Monopole Feeder. IEEE Access 2019, 7, 6746–6757. [CrossRef] 8. Dunlap, W.C., Jr.; Watters, R.L. Direct measurement of the dielectric constants of silicon and germanium. Phys. Rev. 1953, 92, 1396–1397. 9. Kinasewitz, R.T.; Senitzky, B. Investigation of the complex permittivity of n-type silicon at millimeter wavelengths. J. Appl. Phys. 1983, 54, 3394. [CrossRef] 10. Afsar, M.N.; Button, K.J. Precise millimeter-wave measurements of complex refractive index, complex

dielectric permittivity and loss tangent of GaAs, Si, SiO2, Al2O3, BeO, macor, and glass. IEEE Trans. Microw. Theory Thic. 1983, 31, 217–223. [CrossRef] 11. Afsar, M.N.; Chi, H. Millimeter wave complex refractive index, complex dielectric permittivity and loss tangent of extra high purity and compensated silicon. Int. J. Infrared Millim. Waves 1994, 15, 1181–1188. [CrossRef] 12. Parshin, V.V.; Heidinger, R.; Andreev, B.A.; Gusev, A.V.; Shmagin, V.B. Silicon as an advanced window material for high power gyrotrons. Int. J. Infrared Millim. Waves 1995, 16, 863–877. [CrossRef] 13. Krupka, J.; Breeze, J.; Centeno, A.; Alford, N.; Claussen, T.; Jensen, L. Measurements of Permittivity, Dielectric Loss Tangent, and Resistivity of Float-Zone Silicon at Microwave Frequencies. IEEE Trans. Microw. Theory Thic. 2006, 54, 3995–4001. [CrossRef] 14. Krupka, J.; Kamiński,P.; Kozłowski, R.; Surma, B.; Dierlamm, A.; Kwestarz, M. Dielectric properties of semi-insulating silicon at microwave frequencies. Appl. Phys. Lett. 2015, 107, 082105. [CrossRef] 15. Krupka, J.; Karcz, W.; Kamiński,P.; Jensen, L. Electrical properties of as-grown and proton-irradiated high purity silicon. Nucl. Instrum. Methods Phys. Res. Sect. B Beam Interact. Mater. Atoms 2016, 380, 76–83. [CrossRef] 16. Krupka, J.; Kaminski, P.; Jensen, L. High Q-Factor Millimeter-Wave Silicon Resonators. IEEE Trans. Microw. Theory Tech. 2016, 64, 1–6. [CrossRef] 17. Ismail, K.; Baba, N.H.; Awang, Z.; Esa, M. Microwave Characterization of Silicon Wafer Using Rectangular Dielectric Waveguide. In Proceedings of the 2006 International RF and Microwave Conference, Putra Jaya, Malaysia, 12–14 September 2006. 18. Kaatze, U. Techniques for measuring the microwave dielectric properties of materials. Metrologia 2010, 47, S91–S113. [CrossRef] 19. Cole, T. IV Quasi-Optical Techniques of Radio Astronomy. Prog. Opt. 1977, 15, 187–244. 20. Goldsmith, P.F. Quasioptical Systems: Gaussian Beam Quasioptical Propagation and Applications; Wiley-IEEE Press: New York, NY, USA, 1998. 21. McKay, J.E.; Robertson, D.A.; Speirs, P.J.; Hunter, R.I.; Wylde, R.J.; Smith, G.M. Compact corrugated feedhorns with high Gaussian coupling efficiency and -60 dB sidelobes. IEEE Trans. Antennas Propag. 2016, 64, 1. [CrossRef] 22. Baker-Jarvis, J.; Vanzura, E.; Kissick, W. Improved technique for determining complex permittivity with the transmission/reflection method. IEEE Trans. Microw. Theory Tech. 1990, 38, 1096–1103. [CrossRef] 23. Liu, X.; Chen, H.-J.; Yang, B.; Chen, X.; Parini, C.; Wen, D. Dielectric Property Measurement of Gold Nanoparticle Dispersions in the Millimeter Wave Range. J. Infrared Millim. Terahertz Waves 2013, 34, 140–151. [CrossRef]

© 2019 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).