MICROWAVE PROPERTIES OF LOW LOSS POLYMERS AT CRYOGENIC TEMPERATURES

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1 To be published in IEEE Transactions on Microwave Theory and Techniques MICROWAVE PROPERTIES OF LOW LOSS POLYMERS AT CRYOGENIC TEMPERATURES Mohan V. Jacob a, Janina Mazierska a, Kenneth Leong a and Jerzy Krupka b a Electrical and Computer Engineering, School of Engineering, James Cook University, Townsville, QLD 48, AUSTRALIA b Instytut Mikroelektroniki i Optoelektroniki Politechniki Warszawskiej, Koszykowa 75, -66 Warszawa, POLAND ABSTRACT Dielectric loss tangent and permittivity of Polytetrafuoroethylene (Teflon), high density Polyethylene and cross-linked Polystyrene (Rexolite) were measured at temperature range from 8 K to 84 K and frequency of approximately 8 GHz. The material properties were determined by measurements of the resonant frequency and the Q-factor of a TE mode cylindrical superconducting cavity containing a sample under test. It has been demonstrated that these materials exhibit very low losses at cryogenic temperatures ( -6 for Teflon, 5-5 for HD Polyethylene and -5 for Rexolite). Due to low losses these materials can be useful in construction of various high Q-factor microwave devices for operation at cryogenic temperatures. I. INTRODUCTION Cryogenic electronics is a fast growing branch of modern electronics especially since the discovery of high temperature superconductors that allowed for significant reduction of losses and noise figures in filters and microwave oscillators. Successful design and manufacturing of low loss devices at cryogenic temperatures require careful choice of dielectric materials for their construction. To ensure high Q-factor of an electronic system all the components should exhibit small losses. Single crystal dielectrics like sapphire, YAG, SrLaAlO 4 or quartz [] are the lowest loss materials available. However they are hard to machine into complicated shapes. Hence various plastic materials, easier to machine, have been typically used as supports or other less demanding elements of a cryogenic microwave device or a cryogenic system. Teflon, Polyethylene and Rexolite have been on the market for many years, are inexpensive and easy to fabricate and machine into a desired size and shape. Information on microwave properties of Teflon, Polyethylene and Rexolite exists for wide range of frequencies but at room and higher temperatures only [,3]. There is no data on complex permittivity of these materials available for cryogenic temperatures except for one temperature (77 K) for Teflon and Rexolite [4]. In this paper we present results of accurate measurements of the permittivity and loss tangent of three low loss polymers in the cryogenic temperature range from 5 K to 8 K using the dielectric resonator technique with superconducting thin films. We have been used an accurate multi-frequency measurement and data processing technique (TMQF) [5,6]. Our measurement method accounted for noise, delay due to transmission lines and its frequency dependence, and cross-talk in measurement data. We also took into consideration variations of samples dimensions with temperature to ensure high accuracy. Jacob et al.

2 To be published in IEEE Transactions on Microwave Theory and Techniques II. DIELECTRIC RESONATOR MEASUREMENT METHOD The dielectric resonator with high temperature superconducting plates technique has been used for measurements of low loss polymer materials presented in this paper. A similar technique had been used before in [4], as mentioned in Section, and to measure dielectric properties of single-crystal materials [7,8]. The Dielectric resonator technique with HTS plates is a modification of the metallic dielectric resonator and the cavity perturbation techniques used in the past to characterise dielectric materials [9-4]. The schematic diagram of the dielectric resonator used in this work is shown in Fig.. The dielectric sample to be measured was enclosed in a copper cylindrical cavity between two High Temperature Superconducting films, which allowed for high sensitivity of measurements. As the dielectrics under test exhibit low relative permittivity, the diameter d of the samples was chosen to be sufficiently large to ensure that electromagnetic fields are evanescent in the air region ( k >). In our experiments we used dielectric samples having the aspect ratio (diameter to height) equal to two. ρ Superconducting films Lateral (Cu) wall L Dielectric d a D Fig.. Schematic of a TE mode dielectric resonator. Since the materials under test are isotropic, the TE mode of operation was employed in our measurements. The real part of relative permittivity ε r of a dielectric was determined as the first root of the following transcendental equation [5] using software SUP [6]: kρ J ( kρb) F ( b) + kρ J( kρb) F ( b) = () where: I( k ρ a) F ( ρ) = I ( k ρ ρ) + K ( k ρ ρ) K ( k a) F ( ρ) = I ( k ρ ρ) + K ( k ρ I( k ρ a) ρ) K ( k a) ρ ρ ω ε r ω k ρ = k z, k k z, k z π / L ρ = = c c and ω is the angular frequency (πf), c is velocity of light, ε o is free space permeability, ε r is real relative permittivity of the sample and J, J, I, I, K, K, denote corresponding Bessel and Hankel functions. The loss tangent of a dielectric under test is found on the basis the loss equation of the b Jacob et al.

3 To be published in IEEE Transactions on Microwave Theory and Techniques resonator and measured values of the Q -factor of the resonator, namely: R SS RSM tanδ = () ρe Q AS AM where Q is measured unloaded Q-factor of the entire resonant structure, R SS and R SM are the surface resistance of the superconducting and the metallic parts of the cavity respectively, A S and A M are the geometric factors of the superconducting part and metallic parts of the cavity and ρ e is the electric energy filling factor. Geometric factors A S, A M, and ρ e to be used in () were computed using incremental frequency rules as follows [5]: ω µ ω A S = / 4 L (3) ω µ ω A M = / a (4) ω ε r p e = ε ω (5) r As superconducting plates high quality YBa Cu 3 O 7 thin films (R SS of 4 µω at GHz and 77 K) were used in the measurements. The values of R SS and R M used in () were measured using the sapphire resonator technique at frequency of 5 GHz at varying temperatures from 3 K to 85 K. The method of surface resistance measurements was in principle the same as used by several groups [5,7]. The main advantage (and difference) of our measurements was that the unloaded Q -factor was calculated from the exact equation, namely Q Q + β + ) (6) = L ( β and not from the simplified equation Q o QL = (7) S where S max is the magnitude of the transmission coefficient at the resonant frequency and β and β are coupling coefficients of the resonator to the external circuitry. The loaded Q L -factor and the coupling coefficients were obtained in our case from multi-frequency measurements of S, S and S parameters measured around the resonance using the Transmission Mode Q-Factor Technique [5,6]. The TMQF method accounts for noise, delay due to uncalibrated transmission lines and its frequency dependence, and crosstalk, which occurred in measurement data. Hence our method provided more accurate values of R SS and R M for the Eq. () than the 3dB method used in [4]. Figure shows measured dependence of surface resistance R SS with temperature for the YBCO films used in the dielectric tests. The surface resistance of copper walls of the resonator was measured in the same way. We assess that the maximum overall error in our measurements of the Q o -factor to be %, based on repeatability of the measurement system. This uncertainty results in the most probable error of 4.4% in surface resistances measured with our GHz sapphire resonator [8]. max As measurements of low loss polymers were performed at different frequencies than 5 GHz measured values of R M and R SS were scaled assuming the square root of frequency dependence for the surface resistance of copper and the frequency square law for the surface resistance of superconductors. Jacob et al. 3

4 To be published in IEEE Transactions on Microwave Theory and Techniques 4 Surface Resistance (mω) Fig.. Surface resistance of YBa Cu 3 O 7-δ superconducting films versus temperature used in measurements of polymer samples. III MEASUREMENTS OF MICROWAVE PROPERTIES OF TEFLON, POLYETHYLENE AND REXOLITE All measurements of dielectric properties of Teflon, Rexolite and Polyethylene were performed using the same pair of superconducting YBCO thin films. The dielectrics under test were machined into cylindrical shapes of 7.5 mm height and 5. mm diameter. The copper cavity was of 4 mm in diameter and 7.4 mm in height. The geometrical factors of the resonator used for the dielectric tests are shown in Table I. Table I The geometric factors and energy-filling factor of Dielectric Resonator with Teflon, Polyethylene and Rexolite. Teflon Polyethylene Rexolite A S A M r e The measurement system, which consists of Network Analyser (HP 87C), closed cycle refrigerator (APD DE-4), temperature controller (LTC-), vacuum Dewar and PC is presented in Figure 3. The resonator containing a dielectric sample was cooled from room temperature to around 5 K and then S-parameters (S, S and S ) were measured as a function of increasing temperature using the Network Analyser. The measurement and calculation procedures used for calculations of ε r and tanδwere as described in Section ; with the thermal expansion of dielectrics taken additionally into consideration. The relative permittivity of the sample under test was calculated from the measured resonant frequency using Eq. () and the loss tangent of the material was calculated using () from the measured unloaded Q -factor. Jacob et al. 4

5 CONTROL MONITOR SETUP LTC- TEMPERATURE CONTROLLER LOCAL REM ENTER BACK SPACE CLEAR EXIT INC DEC /- To be published in IEEE Transactions on Microwave Theory and Techniques Network Analyser Microwave Resonator Test Cable GPIB IBM-PC Heater Element Diode Temperature Sensors Temperature Controller Vacuum Dewar Cryopump Helium Compressor Unit Fig. 3. The experimental set-up to measure the Q-factor and f res of dielectric resonator containing the dielectric material to be measured. The Q-factor and resonant frequency of the resonating structure with Teflon was measured at varying temperatures from 8 K to 84 K. The resonant frequency of the Teflon resonator was approximately 89 MHz. Variation of dimensions of Teflon with temperature were taken into consideration in the calculations of the relative permittivity using data provided in [9]. Fig. 4 shows the temperature dependence of the thermal linear expansion coefficient used in () for computations of ε r of Teflon and Polyethylene based on [], and Polystyrene based on [9] Teflon Polyethylene Polystyrene α (/K) Fig. 4. Temperature coefficients of Teflon, Polyethylene and Polystyrene versus temperature based Jacob et al. 5

6 To be published in IEEE Transactions on Microwave Theory and Techniques on [9] and []. The real part of relative permittivity for Teflon, Polyethylene and Rexolite samples measured at differing temperatures is shown in Fig. 5. For Teflon, ε r of. was obtained at temperature of 8 K, and.9 at 84 K. When temperature increased from 8 K to 84 K a decrease in ε r of.48% has been observed taking into consideration variations of dimensions of Teflon at cryogenic temperatures according to [9]. Measurements of the Polyethylene and Rexolite samples resulted in the resonant frequencies of 765 MHz and 6986 MHz respectively. Variations of the dimensions of polyethylene at cryogenic temperatures according to [], as per Fig. 4, were used in calculations of ε r. For Rutile the thermal linear expansion coefficient of Polystyrene [9] was used as no data for Rexolite could be found. ε r of Polyethylene was obtained to be.43 at temperature of 8 K decreasing by.36% to.4 at 84 K. For Rexolite ε r of.6 was measured at 8 K and.6 at 84 K, giving the variation in ε r of.4%. When the thermal expansion phenomenon is not taken into consideration constant values of the real part of relative permittivity (.4,.375 and.57 for Teflon, Polyethylene and Rexolite respectively) versus temperature are obtained. These values are in agreement with results for Teflon (.48) and Rexolite (.54) at 77 K in [4]..6.5 Rexolite Polyethylene Teflon.4 ε r Fig. 5. The real part of permittivity of Teflon (8.9 GHz), Polyethylene (7.65 GHz) and Rexolite (6.986 GHz) versus temperature. Measured dielectric loss tangents of Teflon, Polyethylene and Rexolite (calculated from the Q-factor measurements using Eq. ()) are shown in Fig. 6. The Teflon sample shows the smallest loss of the three materials;.3-6 at 8 K and at 84 K. For Polyethylene we obtained the loss tangents of at 8 K and at temperature of 84 K. Rexolite exhibited tanδof.3-5 and respectively. The loss tangent of Teflon increased nearly.4 times with temperature varying from 8 K to 84 K. For the same temperature change the increase in loss tangent was.9 times and. times for the Polyethylene and the Rexolite samples respectively. Our measurements Jacob et al. 6

7 To be published in IEEE Transactions on Microwave Theory and Techniques give approximately 6% smaller value of tanδfor Teflon and 5% for Rexolite at 77 K as compared to [4] due to more accurate calculation procedure. -3 Rexolite Polyethylene Teflon Loss Tangent Fig. 6. Dielectric loss tangents of Teflon, Polyethylene and Rexolite versus temperature IV. ERROR ANALYSIS OF MEASURED PARAMETERS e r and tand Accuracy of measurements of the real part of relative permittivity presented in this paper depends mostly on accuracy of the resonant frequency measurements using a vector analyser and on uncertainty in dimensions of dielectric samples. Uncertainty in frequency measurements is determined by the phase noise of the HP 87C and with Hz option used for measurements and we assumed it to be negligible. To ensure high accuracy of ε r tests we took into account the thermal expansion phenomenon of materials under test at cryogenic temperatures using values of the thermal expansion coefficients according to [9] and [] in the calculations. However there is a probability that real values of the thermal coefficients of our samples are different than values used, and this might have affected the accuracy of calculations. Also there is a finite accuracy in machining the samples. To assess possible errors in ε r calculation, we performed the error analysis of ε r calculations as a function of uncertainty in dielectric samples dimensions using the software SUP [6]. Figures 7 and 8 show uncertainties in relative permittivities of Teflon, Rexolite and Polyethylene for. and % uncertainties in dimensions respectively. The performed analyses show that the δε r is approximately twice the uncertainty in dimensions for all three polymers and decreases slightly with increase of temperature from 8 K to 85 K. The influence of the uncertainty in dimensions is the highest for Teflon due to the lowest permittivity of this polymer. We assess that uncertainty in dimensions of our samples was.5% (due to machining accuracy and uncertainty in the thermal expansion coefficient) giving measurement errors in ε r of materials under test of approximately.5%. Jacob et al. 7

8 To be published in IEEE Transactions on Microwave Theory and Techniques.4 δε r (%).4 Teflon Polyethylene Rexolite Fig. 7. Most Probable Error in permittivity versus temperature for.% uncertainty in samples dimensions.4 Teflon.3 δε r (%).. Polyethylene Rexolite Fig. 8. Most Probable Error in permittivity versus temperature for % uncertainty in samples dimensions Accuracy in the loss tangent measurements using the dielectric resonator method is associated mainly with errors in the Q -factor measurements. Other contributing factors include errors in surface resistance values of the superconducting plates and the copper walls as well as errors in estimation of geometric factors. To assess the most probable error in the loss tangent of measured dielectric materials, an equation describing the Most Probable Error (MPE) has been derived on the basis of the differential uncertainty method as in []. For a multi-variable function tanδ=f(x, x,...x n ), the relative error tanδ/ tanδcan be expressed as: Jacob et al. 8

9 To be published in IEEE Transactions on Microwave Theory and Techniques n tanδ x k δ tanδ= tanδx k (8) tanδ k= tanδ By considering various parameters influencing tanδ(eq. ()), the Most Probable Error in tanδcan be expressed as: ρ e Q RSS RSS AS + tan + tan + ρ e Q ρ e δ Q AS ρ e δ RSS AS δ tanδ= (9) RSM RSM A + M tan + AM ρ e δ RSM AM To assess errors in our measurements of tanδwe assumed uncertainties in R SS and R SM of 4.5%, and.5% in A S, A M and ρ e as discussed in Section. Calculated errors in measured loss tangent values of Teflon for assumed uncertainties in the Q -factor measurements of %,.5%, % and % are presented in Fig. 9. / δtanδ(%) 5 δq =. % δq =.5 % δq =. % δq =. % Fig. 9. The most probable error in loss tangent of Teflon versus temperature for varying uncertainty in Q. It is interesting to note that the MPE in tanδfor Teflon reaches minimum between 6 K and 7 K. At low temperature region the δtanδincreases due to the very low loss of the material (tanδ<3-6 ) and resulting smaller sensitivity of the measurement system. At temperatures above 7 K, the MPE is high due to the significant increase of the surface resistance of the HTS films close to the critical temperature. Uncertainty in tanδ is determined by the uncertainty in the Q -factor measurements. The system repeatability error in the unloaded Q-factor of our measurement system is assessed to be two percent at most. Therefore the most probable error in measured loss tangent of Teflon is less than 3.5% in the most sensitive temperature zones and less than % for temperatures between 5 K and 7 K. Jacob et al. 9

10 To be published in IEEE Transactions on Microwave Theory and Techniques 4 3 δq =. % δtanδ(%) δq =. % δq =.5 % δq = % Fig.. The most probable error in loss tangent of Polyethylene (dash line) and Rexolite (solid line) versus temperature for varying uncertainty in Q. Figure shows the Most Probable Errors in loss tangent for Polyethylene and Rexolite samples. The δtanδ are smaller and much less temperature dependent than for Teflon. This is attributed to higher loss tangent of these materials as compared to Teflon. The calculated errors in loss tangent for Polyethylene and Rexolite are bigger than errors in the Q -factor measurements by.8% and.65% respectively. Hence we assess that tanδof Polyethylene and Rexolite was measured with errors smaller than 3%. V. CONCLUSIONS We have studied the microwave properties of Teflon, Polyethylene and Rexolite samples at cryogenic temperatures. A comparison of samples dimensions, real part of the relative permittivity and loss tangent of three polymers under test we measured at temperatures of 8 K and 84 K is shown in Table II. Table II Comparison of properties of Teflon, Polyethylene and Rexolite at 8 and 84 K. Temp. (K) Teflon Polyethylene Rexolite 8 Height (mm) 7.39±. 7.4±. 7.46± ±. 7.4±. 7.47±. 8 Diameter 4.73±.4 4.8±.4 4.8±.4 (mm) ±.4 4.8± ±.4 8 Permittivity.±..43±..6±. 84.9±..4±..6±. 8.3±.3 4.8±.4.3±.9 tand ( -5 ) 84.56±. 8.6± ±.7 We have found that at temperature of 8 K Teflon exhibited very low loss tangent of Jacob et al.

11 To be published in IEEE Transactions on Microwave Theory and Techniques approximately -6, polyethylene of and Rexolit of. -4. From the performed error analysis we concluded that measurements of loss tangent were carried out with errors of approximately of % for Polyethylene and Rexolite, and below 3.5% for Teflon. To achieve higher accuracy of measurements of tanδa different method based on the Whispering gallery modes needs to be used. Errors in real part of the relative permittivity measurements were approximately.6% for all three materials. We hope that presented data will assist in applications of Teflon, polyethylene and Rexolite in microwave systems. We would like to point out that although our results differences in microwave properties between different samples of the same materials due to difference in content of impurity. ACKNOWLEDGEMENT This work is done under the financial support of ARC-Large grant. The first author acknowledges the JCU Post Doctoral Fellowship. Jacob et al.

12 REFERENCES [] J. Krupka, K. Derzakowski, M.E. Tobar, J. Hartnett, and R.G. Geyer, Complex permittivity of some ultralow loss dielectric crystals at cryogenic temperatures, Measurement Science and Technology, vol., pp , 999. [] A. von Hippel, Dielectric materials and applications, Artech House, 995. [3] L. Chen, C. K. Ong and B. T. G. Tan, Amendment of cavity perturbation method for permittivity measurement of extremely low-loss dielectrics, IEEE Transactions on Instrumentation and Measurement, vol. 48(6), pp. 3-36, 999. [4] R. Geyer, J. Krupka, Microwave Dielectric Properties of Anisotropic Materials at Cryogenic Temperatures, IEEE Transactions on Instrumentation and Measurements, vol. 44, No., pp , 995. [5] K. Leong, Precise measurements of surface resistance of HTS thin films using a novel method of Q-factor computations for Sapphire Dielectric resonators in the transmission mode, Ph. D. thesis, James Cook University,. [6] K. Leong and J. Mazierska, Accuarate measurements of surface resistance of HTS films using a novel transmission mode Q-factor technique, Journal of Superconductivity (in Press). [7] J. Krupka, R.G. Geyer, M. Kuhn and J. Hinken, "Dielectric properties of single crystal AlO3, LaAlO3, NdGaO3, SrTiO3, and MgO at cryogenic temperatures and Microwave frequencies", IEEE Transactions on Microwave Theory and Technique, vol. 4, pp , 994. [8] C. Zuccaro, I. Ghosh, K. Urban, N. Klein, S. Penn and N. M. Alford, Materials for HTS-shielded dielectric resonators, IEEE Transactions on Applied Superconducitivity, vol 7, pp , 997. [9] Y. Kobayashi, M. Katoh, Microwave measurement of dielectric properties of lowloss materials by the dielectric rod resonator method, IEEE Transactions on Microwave Theory and Techniques, vol. 33, pp , 985. [] J. Krupka, An accurate method for permittivity and loss tangent measurements of low dielectric using TE d dielectric resonators, Proceedings on the 5th International Conference on Dielectric Materials and Applications, pp (988). [] H. Tamura, H. Matsumoto and K.Wakino, Low temperature properties of microwave dielectrics, Japanese Journal of Applied Physics, vol. 8, pp.-3, 989. [] T. Nishikawa, K. Wakino, H. Tamura, H. Tanaka and Y. Ishikawa, Precise measurement method for temperature coefficient of microwave dielectric resonator material, IEEE-MTT-S International Microwave Symposium Digest, no. J-6, pp. 77-8, 987. [3] R. Geyer, Dielectric characterisation and reference materials NIST technical note 338, National Institute of Standards and Technology, 99. [4] J. Konopka and I.Wolf, Dielectric properties of high T c substrates up to 4 GHz, Microwave Symposium 9, vol., pp. 7-3, 99. [5] J. Krupka, M. Klinger, M. Kuhn, A. Baranyak, M. Stiller, J. Hinken and J. Modelski, "Surface resistance measurements of HTS films by means of sapphire dielectric resonators", IEEE Transactions on Applied Superconductivity, vol. 3, No. 3, pp , 993. [6] J. Krupka (Software in Fortran to calculate surface resistance of superconductors or copper, permittivity and loss tangent of dielectric materials, resonant frequency of differing TE and TM modes),. Jacob et al.

13 3 [7] C. Wilker, Z-Y. Shen, V.X. Nguyen, and M.S. Brenner, "A sapphire resonator for microwave characterization of superconducting thin films", IEEE Transactions on Applied Superconductivity, vol. 3, No. 3, pp , 993. [8] J. Mazierska, Dielectric resonator as a possible standard for characterisation of high temperature superconducting films for microwave applications, Journal of Superconductivity, vol (), pp , 997. [9] Y. S. Touloikian, R.K. Kirby, R.E. Taylor and T.Y.R. Lee, "Thermal expansion - Nonmetallic solids, Thermophysical Properties of Matter Data Series, vol. 3, IFI/Plenum, New York, 977. [] D. J. Lacks and G. C. Rutledge, Simulation of the Dependence of Mechanical Properties of Polyethylene, Journal of Physical Chemistry, vol. 98, pp. -3, 994. [] J. Mazierska, J. Krupka and T. Kosciuk, Resonant measurements of surface resistance of high T c superconducting films: How good or bad are they?, Journal of Superconductivity, vol. 8, No. 6, pp , 995. Jacob et al. 3