Anisotropic complex permittivity measurements of ... - AIP Publishing

Anisotropic complex permittivity measurements of mono-crystalline rutile between 10 and 300 K Michael Edmund Tobar, Jerzy Krupka, Eugene Nicolay Ivanov, and Richard Alex Woode Department of Physics, University of Western Australia, Nedlands, 6907 Western Australia, Australia

~Received 29 April 1997; accepted for publication 24 October 1997! The dielectric properties of a single crystal rutile (TiO2) resonator have been measured using whispering gallery modes. Q factors and resonant frequencies were measured from 300 to 10 K. Q factors as high as 104 , 105 , and 107 were obtained at 300, 80, and 10 K, respectively. Using the whispering gallery mode technique we have determined accurately the loss tangent and dielectric constant of monocrystalline rutile and obtained much more sensitive measurements than previously reported. We show that rutile exhibits anisotropy in both the loss tangent and permittivity over the range from 10 to 300 K. © 1998 American Institute of Physics. @S0021-8979~98!03103-X#

I. INTRODUCTION

Low-loss single crystal dielectrics have become important materials in constructing high-Q resonant cavities and low-noise microwave oscillators at microwave frequencies.1–4 Materials such as sapphire have enabled the construction of resonators with Q values of greater than 105 at room temperature, 107 at 77 K, and 109 at 4 K.1,4–6 Sapphire is a uniaxial anisotropic material with a permittivity parallel and perpendicular to the c axis of order 10. To make use of the low-loss tangent of sapphire high order whispering gallery modes must be excited. These modes have high values of azimuthal mode number (m.4) that allow the radiation and conductor losses to be kept to a minimum. The field density for a whispering gallery modes travels mainly around the perimeter of the cylinder with most of the energy confined to the resonator due to total internal reflection. However, this means the resonator must be larger than the conventional TE01d mode, and for operation at X-band crystals of the order 3–5 cm in diameter are common.7 To reduce the dimensions low-loss high permittivity materials are required. Recently single crystal rutile has been studied as a resonator at microwave frequencies.8 Rutile has been identified as an important low-loss material as its permittivity can be more than an order of magnitude higher than sapphire allowing the construction of smaller resonators. Also, rutile exhibits a negative temperature coefficient of permittivity while sapphire exhibits a positive coefficient of permittivity. It has been shown that a combination of these materials can be used to construct a composite microwave resonator with a zero temperature coefficient of frequency and a ultrahigh Q factor, either using lower order modes shielded by a HTS cavity9 or by using whispering gallery modes.10 Previous measurements of rutile using low order dielectric modes reveal a loss tangent perpendicular to the crystal c axis of order 102411,12 at room temperature, 1.231025 ~Ref. 12! to 331026 ~Ref. 8! at 77 K and 1.431027 ~Ref. 8! at 4 K. In this article we measure the properties of rutile by implementing the whispering gallery mode method. This method has been shown to be extremely useful when measuring low loss materials such as sapphire.13 Our measurements agree with the measurements in Ref. 8 only above 40 K. Below this temperature we have measured a loss tangent 1604

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of up to five times smaller. Probably the measurements in Ref. 8 were not limited by the dielectric losses of rutile below 40 K. In addition, we determined that both the loss -tangent and the permittivity are anisotropic by making a direct measurement of the tensor components using whispering gallery modes. II. METHODOLOGY

A rutile cylindrical puck form Single Crystal Technology BV, Netherlands of 1 cm height and 2 cm diameter was measured. A schematic of the resonator is shown in Fig. 1. The fixture was mounted in a vacuum can inside a cryogenic Dewar. Platinum and germanium thermometers thermally contacted to the copper cavity were used to read out and control the temperature depending on the temperature. Measurements down to 50 K were done in a liquid nitrogen environment, temperatures below 77 K were obtained by pumping on the cryogenic fluid. To make measurements down to 10 K liquid helium was used. Mode frequencies of the resonator were measured between 2.5 and 5.5 GHz. They were excited by 1 mm diam loop probes as shown in Fig. 1. The resonator was analyzed

FIG. 1. Fixture used to determine the dielectric properties of rutile at microwave frequencies. The rutile crystal is 20 mm in diameter, 10 mm high with a 5 mm diam hole in the center. The crystal c axis is aligned along the axial direction of the cylinder. The rutile is supported by copper posts with indium squashed in the center to provide a good thermal contact. The inner diameter of the copper cavity is 35 mm with an inner height of 25 mm. The walls of the copper cavity are 7 mm thick and the holes that house the probes are 3 mm in diameter. The probes are co-axial cable with 1 mm diam loops used to couple to the dielectric resonances.

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resonant frequencies that exhibit a quasi-TE ~H mode! and a quasi-TM ~E mode! mode structure were identified. Consequently, a system of two nonlinear determinant equations were solved;

J

F 1 ~ f ~ H ! , e' , e i ! 50 . F 2 ~ f ~ E ! , e' , e i ! 50

FIG. 2. A stable swept frequency source from 2 to 12 GHz was created by mixing the 8662A synthesiser with a 8673H synthesiser. The rutile resonator was measured in transmission with very low coupling on the input ports. The upper or lower side band was transmitted by the cavity and measured using a detector and a cathode-ray oscilloscope ~CRO! in conjunction with a frequency counter.

in transmission with small values of coupling of less than 0.01 on both ports. A stable swept frequency was created using a 8673H Hewlett–Packard ~HP! synthesizer ~2–12.4 GHz! mixed with a 8662A HP synthesizer ~10 kHz–1.2 GHz! as shown in Fig. 2. When measuring a resonance either the upper or the lower side band was transmitted and the other was filtered by the rutile resonator. To determine which sideband was being measured a microwave frequency counter was used to read out the frequency. The resonant frequencies were determined by the frequency of maximum transmission, and the loaded Q factor was determined by measuring the half power point bandwidth. Because of the low cavity coupling we assumed the loaded Q factor was equal to the unloaded Q factor. In general all the measured modes were hybrid modes and had both an axial electric field dependence and an axial magnetic field dependence. It is common to denote a mode with a dominant axial electric field dependence as an E mode ~quasi-TM! and a dominant magnetic field dependence as a H-mode ~quasi-TE!. To calculate the permittivity we used the whispering gallery mode technique. This method has been shown to be more accurate for measuring the principal tensor components of uniaxial anisotropic materials compared to low order mode techniques.13 To calculate the tensor components of permittivity two whispering gallery mode

~1!

Here f (H) and f (E) are the measured resonant frequencies for a quasi-TE (H) and a quasi-TM (E) whispering gallery mode, and e' and e i are the real parts of the permittivity tensor components perpendicular and parallel to the anisotropy axis. The eigenvalue equations represented by F 1 ,F 2 result from the application of the mode matching technique. In most cases ~when the resonant system can be treated as a planar symmetric! these are separate equations for symmetric and antisymmetric whispering gallery mode families. The size of the matrix leading to the eigenvalue equations depends on the number of terms used in the field expansion series. Once the permittivities were calculated the electric energy filling factors for both the E and H modes parallel, P (E,H) , and perpendicular, p (E,H) , were calculated numeriei e' cally by using the following incremental frequency rule:14

] f ~E! ] e' ] f ~E! p ~eEi ! 52 ]ei ] f ~H! H! p ~e' 52 ] e' ] f ~H! p ~eHi ! 52 ]ei E! p ~e' 52

e' f ~E! ei f ~E! e' f ~H! ei f ~H!

6

~2!

,

Following this we used the two simultaneous equations given by ~3! to solve for the dielectric loss tangent parallel, tan di , and perpendicular, tan d' , to the c axis:

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~E! ~E! ~E! Q ~21 E ! 5 p e' tan d' 1 p e i tan d i 1R S /G . 21 ~H! ~H! Q ~ H ! 5 p e' tan d' 1 p e i tan d i 1R S /G ~ H !

~3!

Here R s is the surface resistance of the cavity enclosing the dielectric resonator and G is the geometric factor of the mode. It is important to note that one of the modes must

TABLE I. Measured and computed resonant frequencies at 77.2 K assuming e' 5106.67 and e i 5230.89 and solving for Eq. ~1!. On the second iteration we assumed Eq. ~1! was exact for the S2 6 and N1 11 modes. The simultaneous equations were solved to calculate the permittivity to be e' 5107.13 and e i 5231.26. Mode number

N1 ~Comp!

N1 ~Meas!

3 4 5 6 7 8 9 10 11 12

2530.51 2834.71 3148.33 3468.03 3791.76 4118.04 4453.08 4777.10 5107.40

2536.11 2836.91 3148.84 3467.33 3790.20 4115.98 4443.67 4772.56 5102.54

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N2 ~Comp!

N2 ~Meas!

S1 ~Comp!

S1 ~Meas!

S2 ~Comp!

S2 ~Meas!

2992.00 3372.00 3748.00 4120.90 4495.60 4860.20

3043.21 3387.93 3751.30 4121.45 4488.79

3146.2 3584.2 3843.2 4088.4 4342.4 4606.0 4878.1

3155.06 3583.30 3842.95 4086.91 4342.15 4606.06

3464.00 3738.00 4222.20 4719.03 5132.85 5471.95

3445.10 3734.90 4214.20 4710.09 5128.63

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FIG. 3. Frequency vs azimuthal mode number m for the N1, N2, S1, and S2 mode families.

be an E mode while the other must be a H mode, otherwise the two equations would be close to redundant, causing large inaccuracies in the calculation of the anisotropy in loss tangent. For whispering galler ~WG! modes (m.4) the geometric factor, G, is significantly large so that the effect of the cavity can be ignored when compared to the loss tangents. Specific examples confirming this effect are given in Sec. III. The Q factors and frequencies of an E and a H whispering gallery mode may be measured to determine the loss tangent and permittivity of rutile as a function of temperature, T, from Eq. ~1!–~3!. The permittivity dependence is another important parameter that is necessary to determine. In general the dimension change in the sample must be considered if an accurate permittivity determination is to be made. It is convenient to write a set of simultaneous equations that relate the temperature coefficients of permittivity perpendicular, a e*' , and parallel, a * e i , to the c axis, to the temperature coefficient of frequency of the resonator, ( ] f / ] T)/ f , and the temperature * , and parallel, coefficients of expansion perpendicular, a D a L* , to the c axis.

FIG. 5. Q factor and filling factor vs azimuthal mode number for the N2 mode family.

] f ~H! ]D ] f ~E! p ~DE ! 5 ]D ] f ~H! p ~LH ! 5 ]L ] f ~H! p ~LE ! 5 ]L

p ~DH ! 5

D f ~H! D f ~E! L ~H! f L f ~E!

6

,

~5!

where D is the diameter of the cylinder and L is the length. III. MODE NOMENCLATURE

Here we define the radial, p (E,H) , and axial, p (E,H) , field D L energy filling factors as

It is common to designate modes in a dielectric resonator as E mn p1 d ~quasi-TM! or H mn p1 d ~quasi-TE!. Here m is the number of azimuthal variations, n is the number of radial variations, and p is the number of axial variations. E modes have the majority of the magnetic field in the transverse plane perpendicular to the cylindrical axis, while H modes have the majority of the electric field in the transverse plane. It is common to group the modes into ‘‘mode families’’ which have the same number of axial ( p) and radial (n) field variations with different azimuthal field variations (m). For example, the fundamental E mode family is designated as E m 1 d . This type of nomenclature is okay when we analyze the lowest frequency mode families in isotropic or slightly anisotropic materials. For instance, sapphire exhibits a ratio e' / e i of 1.22 to 1.23 between the parallel and perpendicular tensor components of permittivity from 300 to 10 K.15 It has been shown that the six lowest order mode families can be

FIG. 4. Q factor and filling factor vs azimuthal mode number for the N1 mode family.

FIG. 6. Q factor and filling factor vs azimuthal mode number for the S1 mode family.

1 1 1 ~H! p a * 1 p ~ H ! a * 52 ~ H ! 2 e' e' 2 e i e i f 1 ~E! 1 1 p a * 1 p ~ E ! a * 52 ~ E ! 2 e' e' 2 e i e i f

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] f ~H! * 2p ~LH ! a L* 2p ~DH ! a D ]T . ] f ~E! ~E! ~E! * * 2p D a D 2p L a L ]T ~4!


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FIG. 9. Q factor vs temperature for the N1 11 and S2 6 modes. FIG. 7. Q factor and filling factor vs azimuthal mode number for the S2 mode family.

The first two mode families of the N and S type modes at 77 K have been measured and identified between 2.5 and 5.5 GHz. The comparisons are given in Table I and plotted in Fig. 3. The Q factors and filling factors are plotted in Figs. 4–7.

From Figs. 4 and 5, the N1 and N2 modes can be identified as the E m 1 d and the E m 1 21 d mode families, respectively, as ( p e' , p e i ). These two mode families behave as expected for increasing mode number m. As m increases the mode becomes more confined and more whispering gallerylike with a corresponding increase in p e i and decrease in p e' . Thus, we observe a general increase in the Q factor as radiation loss is minimized and the modes approach a TM WG mode. From Fig. 6 we note that the S1 mode family starts of as a H mode family ( p e' . p e i ) for small m (m ,5). However, at larger values of m the mode family becomes hybrid in nature (p e' ; p e i ). By m58 the mode family becomes an E mode family ( p e' , p e i ). The S2 mode Q factors and filling factors are shown in Fig. 7. This mode family is clearly made up of H modes up to m56. However, at m57 the mode is hybrid ( p e' ; p e i ), and by m58 the family becomes an E mode family. The Q factor curves shown in Figs. 6 and 7 can be explained by two competing facts. For low azimuthal mode numbers the Q factor is degraded due to the low confinement. Once m is sufficiently high (m.5), from comparison of the results in Figs. 6 and 7, it is clear that H modes have a higher Q factor than the E modes. This is a clear indication of anisotropy in the loss tangent. The anisotropy of the permittivity of rutile has been measured previously to various degrees of accuracy over the range 4–300 K.8,11,12 Figure 8 shows the temperature dependence of frequency of the N1 11 and S2 6 modes, which have been identified as an E and an H mode, respectively. It is clear from the frequency shift that there is anisotropy in the permittivity. Figure 9 shows the Q factors of both these

FIG. 8. Frequency vs temperature for the N1 11 and S2 6 modes.

FIG. 10. Filling factors vs temperature for the N1 11 and S2 6 modes as calculated from Eq. ~2!.

considered as fundamentally E or H in a 5 cm diam sapphire crystal from 2 to 12 GHz.16 However, our measurements presented in this article reveal that this is not true for rutile. Rutile exhibits a ratio e i / e' of 1.9 to 2.25 between the parallel and perpendicular tensor components from 300 to 10 K. This is quite different from sapphire, and accordingly the mode families are different in nature. Because the permittivity perpendicular to the c axis is less than the permittivity parallel, most hybrid modes exist as E-like modes, and H modes are harder to excite. Because of this fact, the modes tend to be more hybrid in nature and sometimes it is hard to say whether the mode is E or H. Thus, in this article we will use the nomenclature NX m or SX m as described by Krupka et al.16 Here N refers to a nonsymmetric magnetic field distribution along the axial direction ~symmetric electric field!, while S refers to a symmetric magnetic field distribution along the axial direction ~nonsymmetric electric field!. The number X refers to the order in frequency, i.e., X51 refers to the fundamental ~or lowest frequency! mode family while X52 refers to the next highest frequency mode family. As before, the mode number m refers to the number of azimuthal variations in the field pattern. IV. EXPERIMENTAL RESULTS


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FIG. 13. Loss tangent components vs temperature as calculated from Eq. ~3! using the N1 11 and the S2 6 modes. FIG. 11. Permittivity vs temperature as calculated from Eq. ~1! using the N1 11 and the S2 6 modes.

modes as a function of frequency. One can also see clearly that there exists anisotropy as a function of temperature in the loss tangent as well. V. DETERMINATION OF DIELECTRIC PROPERTIES

The dominant source of frequency shift and loss in the rutile resonator is due to the temperature dependence of the complex permittivity as long as m is larger than five and the temperature is above 10 K. Below 10 K the dominant source of the frequency shift was due to paramagnetic impurities internal to the dielectric and contraction effects of the resonator.17 In addition, the losses in the rutile became so small that the copper cavity had a measurable effect even on the whispering gallery modes. The behavior in this regime is more complicated and requires a much more detailed analysis beyond the scope of this article. From the properties of the N1 11 and the S2 6 modes shown in Figs. 8–10 we can calculate the dielectric constant and loss tangent as a function of temperature using Eq. ~1!– ~5!. The filling factors are calculated at room temperature ~77 and 4 K! and do not change much with temperature, a linear fit is sufficient over this range. Combining this calculation with Eq. ~1!, ~4!, and ~3!, we have plotted the permittivity, temperature coefficients, and loss tangent as a function of temperature in Fig. 11–13, respectively. We note here that the permittivity parallel to the c axis cannot be accurately measured using standard resonant cavity techniques. For example, the data published in Ref. 11 has inaccuracies of up to 25% parallel to the c axis and up to 9% perpendicular to it.

In another case the anisotropy of rutile was measured using a quasioptical technique at 200 GHz,8 due to the inadequacies of the standard resonant cavity techniques at microwave frequencies. Figure 13 clearly shows the anisotropy in the loss tangent for rutile. Our sample of rutile came from Single Crystal Technologies BV. Klein et al.8 measured the properties of a sample from the same supplier using a TE01d mode. Our measurements perpendicular to the c axis agree well with the measurements of Klein et al. down to 40 K. However below 40 K we measured a much lower loss tangent perpendicular to the c axis. The measurements by Klein et al. were most likely limited by the loss in the microwave cavity due to the smaller values of G in Eq. ~3! for the TE01d mode. Because whispering gallery modes have a much larger value of G this error was eliminated. To accurately calculate the temperature coefficients of permittivity we must know the temperature coefficients of contraction as well. These values have been published previously down to 100 K18 and below 100 K they have been measured by White.19 Comparing our results with Ref. 18 and 19 we note that the contraction effect is two orders of magnitude smaller than the permittivity effect. Thus we may ignore these values and still obtain 1% accuracy. VI. DISCUSSION

The whispering gallery mode method has been used to measure the dielectric properties of rutile over the temperature range of 10–300 K. We have shown that the loss tangent of rutile is anisotropic and exhibits very low losses at microwave frequencies, although not as low a loss as sapphire, it’s TABLE II. Measured loss tangent of rutile and sapphire perpendicular and parallel to the crystal c axis. It is evident that the loss is anisotropic for both crystals. However rutile exhibits less loss perpendicular to the c axis while sapphire exhibits less loss parallel to the c axis. Loss tangent

FIG. 12. Temperature coefficients of permittivity vs temperature as calculated from Eq. ~4! using the N1 11 and the S2 6 modes. 1608


parallel to c axis 290 K 77 K 4 K perpendicular to c axis 290 K 77 K 4 K

Sapphire

Rutile

531026 231028 7310210

131024 131025 5.431028

831026 531028 9310210

831025 431026 2.631028

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permittivity is over an order of magnitude larger. Table II summarizes measurements made at University of Western Australia UWA of the dielectric loss tangent of sapphire and rutile. In addition, rutile has a negative temperature coefficient of permittivity while sapphire has a positive coefficient. The magnitude of the coefficient for rutile is one to two orders of magnitude greater than sapphire. The dielectric properties of rutile make it an ideal material for compensating the frequency temperature dependence of a high-Q sapphire resonator. It has been shown that it is possible to compensate the temperature dependence of a room temperature and liquid nitrogen cooled sapphire resonator with a thin slice of rutile dielectric, while maintain a Q factor of order 105 and 107 , respectively.10,20 This type of resonator can be used in the construction of temperature stable microwave oscillators operating at temperatures above liquid helium temperature, including room temperature. ACKNOWLEDGMENTS

Thanks are due to Dr. Andre Luiten and Dr. Tony Mann for many interesting discussions about the results of this work. This work was supported by the Australian Research Council, and the participation of A/Professor Jerzy Krupka was made possible due to the support of the Department of Industry Science and Tourism. 1

M. E. Tobar, A. J. Giles, S. Edwards, and J. Searls, IEEE Trans. Ultrason. Ferroelectr. Freq. Control 391 ~1994!. M. E. Tobar, E. N. Ivanov, R. A. Woode, J. H. Searls, and A. G. Mann, IEEE Microwave Guid. Wave Lett. 5, 108 ~1995!. 3 E. N. Ivanov, M. E. Tobar, and R. A. Woode, IEEE Microwave Guid. Wave Lett. 6 ~1996!. 2


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R. A. Woode, M. E. Tobar, E. N. Ivanov, and D. G. Blair, IEEE Trans. Ultrason. Ferroelectr. Freq. Control 43, 936 ~1996!. 5 V. B. Braginsky, V. S. Ilchenko, and Kh. S. Bagdassarov, Phys. Lett. A 120, 300 ~1987!. 6 A. G. Mann, A. N. Luiten, and D. G. Blair, Electron. Lett. 29, 879 ~1993!. 7 M. E. Tobar and A. G. Mann, IEEE Trans. Microwave Theory Tech. 39, 2077 ~1991!. 8 N. Klein, C. Zuccaro, U. Dahne, H. Schulz, and N. Tellmann, J. Appl. Phys. 78, ~1995!. 9 N. Klein, A. Scholen, N. Tellmann, C. Zuccaro, and K. W. Urban, IEEE Trans. MTT, 44, ~1996!. 10 M. E. Tobar, J. Krupka, R. A. Woode, and E. N. Ivanov, in Dielectric Frequency-Temperature Compensation of High Quality Sapphire Dielectric Resonators, in Proceedings of the 1996 IEEE Frequency Control Symposium, ~IEEE! pp. 799–806. 11 A. Okaya, and L. F. Barash, in The Dielectric Microwave Resonator, Proceeding of the IRE, 1962 pp. 2081–2092. 12 A. Okaya, The Rutile Microwave Resonator, Proceeding of the IRE, 1960, p. 1921. 13 J. Krupka, K. Derzakowski, A. Abramowicz, M. E. Tobar, and R. G. Geyer, ‘‘Complex permittivity measurements of extremely low loss dielectric materials using whispering gallery modes,’’ IEEE MTT-S Int. Microwave Symp. Dig., 1997. 14 Y. Kobayashi, Y. Aoki, and Y. Kabe, IEEE MTT-S Int. Microwave Symp. Dig., St. Louis, NO, 1985, pp. 281–284. 15 R. Shelby and J. Fontanella, J. Phys. Chem. Solids 41, 69 ~1980!. 16 J. Krupka, D. Cros, A. N. Luiten, and M. E. Tobar, Electron. Lett. 32, 670 ~1996!. 17 A. N. Luiten, M. E. Tobar, J. Krupka, R. A. Woode, E. N. Ivanov, and A. G. Mann, J. Phys. D: Applied Physics ~to be published!. 18 Y. S. Touloukian et al., in Thermophysical Properties of Matter ~IFI/ Plenum, New York, 1970!, Vols. 2 and 4. 19 G. K. White ~private communication, 1996!. 20 M. E. Tobar, J. Krupka, J. G. Hartnett, E. N. Ivanov, and R. A. Woode, Sapphire-rutile Frequency-Temperature Compensated Whispering Gallery Microwave Resonators, 1997 IEEE International Frequency Control Symposium ~IEEE!.

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