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IEEE TRANSACTIONS ON INSTRUMENTATION AND MEASUREMENT, VOL. 54, NO. 5, OCTOBER 2005
Differential Open Resonator Method for Permittivity Measurements of Thin Dielectric Film on Substrate Sergey N. Dudorov, Dmitri V. Lioubtchenko, Juha A. Mallat, and Antti V. Räisänen
Abstract—A novel differential method based on the open resonator is developed for permittivity measurement of thin dielectric films on optically dense substrates at millimeter wavelengths. The method is based on measurement of resonant frequency shifts due to appearance of a thin film on upper and lower sides of the substrate. The advantages of the method are that there is no need to know the geometry of the open resonator nor the thicknesses of the film and substrate, though one has to measure separately dielectric properties of the substrate. Index Terms—Open resonator, permittivity, substrate, thin film.
I. INTRODUCTION
T
HE OPEN Fabry–Perot resonator is one of the most precise tools for dielectric property measurement of low-loss materials at millimeter wavelengths. The theory of the open resonator is well developed in, e.g., [1], [2]. Samples consisting of two thick layers are considered in [3]. If one layer is very thin (a few micrometers), the published method is not convenient as it requires quite an accurate knowledge of thicknesses of both layers, permittivity of the substrate, geometry of the resonator, etc. In this paper, we propose a novel method, based on the resonant frequency measurements, when the resonator is first loaded with the substrate only, and then with the substrate covered on one side with the film of interest, and finally with that turned up side down.
Fig. 1. Hemispherical open resonator containing a bilayer sample.
definitions given in [3], phase corrections as
can be rewritten
(3) (4)
II. THEORY
(5)
The basic equations for two dielectric layers can be found in [3] (1) (2) where and are the refractive indices of lower and upper layers, respectively, is the resonant wave number, and are the thicknesses of lower and upper layers, respectively, , (see Fig. 1), is the distance between and are phase corrections. Following the mirrors, and Manuscript received March 22, 2004; revised November 9, 2004. The authors are with MilliLab, Radio Laboratory/Smart and Novel Radios Research Unit (SMARAD), Helsinki University of Technology, FI-02015 TKK, Finland (e-mail:
[email protected]). Digital Object Identifier 10.1109/TIM.2005.853352
where , . From now on, according to Fig. 2, we denote the thicker layer parameters as and , while for the thinner layer, these paramand , . eters are Let us assume that one of the sample layers is very thin comand calculate two resonant pared to the wavelength is the difference of the resfrequency shifts. The first shift onant frequencies when the resonator is loaded with a sample with a thin film on the upper side of the substrate [Fig. 2(a)] and when it is loaded with the substrate only [Fig. 2(c)]. The second one is the difference of the resonant frequencies when the resonator is loaded with the sample with the film on the lower side of the substrate [Fig. 2(b)] and when it is loaded with the substrate only [Fig. 2(c)]. These frequency shifts will be used in the formulas derived next.
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DUDOROV et al.: DIFFERENTIAL OPEN RESONATOR METHOD FOR PERMITTIVITY MEASUREMENTS
Fig. 2.
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Samples to be introduced into the resonator. (a) Film is introduced on the upper side. (b) Film is introduced on the lower side. (c) Substrate only.
A. Lower Layer Is Thin (
,
)
or by introducing a function using (13)
In this case, (1) and (2) can be transformed to
and perturbation
and
(6) (7) (15) where, assuming that terms proportional to are small, we can rewrite relations (3), (4), and (5), using the first two terms of a Taylor expansion
(10)
where the left side equalized to zero corresponds to the case of 0 (substrate only), while the right-hand side of (15) is a perturbation due to presence of a thin film. Let us assume that the relative resonant frequency shift due to presence of a thin film is very small (in order of 10 ), and perdoes not vary considerably due to this. It means turbation is very small compared to that of in the that the slope intersection point. Using the method of small perturbations, we can find the change of the wave number needed to make equal to
. From (6), we can find
(16)
(8) (9)
where that
,
from which we can find (11) (17)
We can rewrite (7) as
(12) and changes of due to presence Phase corrections , , see (9) and (10)] of a thin film [in order of can be neglected (error 2.5%) compared to the terms proportional to . At the next step, we will utilize the Taylor expansion, the fact that the derivative of the tangent function is , and also the equation for the resonator containing a substrate only (13)
However, if , care must be taken when using the Taylor expansion, but we can simplify the equations , if as follows. With 1% accuracy, 0.17 rad (corresponding to 37, which is 0.025 mm for sapphire at 100 GHz). Therefore, we can rewrite (12) as
(18) where is calculated from (11). As before, we introduce a function
and perturbation
After manipulations, we obtain
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(19) (20)
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IEEE TRANSACTIONS ON INSTRUMENTATION AND MEASUREMENT, VOL. 54, NO. 5, OCTOBER 2005
Thus
The resonant wave number shift due to a film on the upper side can be estimated (21) (29)
makes (19) valid. Alternatively (22)
Dividing one frequency difference derived from (29) by the other one given by (17), we obtain (30)
,
B. Upper Layer Is Thin ( In this case, Now we can write
,
,
) ,
,
Similarly to (18), we can simplify (23) and (24)
.
(31)
(23) (24)
(32) where Thus, (24) becomes
, and
.
Furthermore
(33)
(25) . where The Taylor expansion and perturbation method give
(34) (35) Thus (36) or (37)
(26) This can be simplified
(27)
Introducing a function
and perturbation
Here, we have neglected the influence of changes in phase corrections, as they are small compared to the terms proportional to . Subtracting (22) from (37), and using the measured difference in the resonant frequencies of the resonator containing the sample with the film on the lower side and on the upper side, we can calculate the difference in permittivities from the equation
, we write
(28)
(38) . Note, that if the permittivity of the film where is higher than that of the substrate, the resonant frequency is lower when the film is on the upper side of the substrate. Thus, we can measure the difference between permittivities of the film and the substrate. However, these measurements are difficult to carry out
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DUDOROV et al.: DIFFERENTIAL OPEN RESONATOR METHOD FOR PERMITTIVITY MEASUREMENTS
TABLE I RESONANT FREQUENCY SHIFTS FOR 0.5-mm GaAs SUBSTRATE (0.63) WITH 1)=(n 1) = 0.6940 SAPPHIRE THIN LAYERS AND DIFFERENCE FROM (n
0
0
TABLE II RESONANT FREQUENCY SHIFTS FOR 0.6-mm GaAs SUBSTRATES (VERY CLOSE TO 3=4) WITH THIN SAPPHIRE LAYERS AND DIFFERENCE FROM (n 1)=(n 1) = 0.6940
0
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TABLE III EXPERIMENTAL RESULTS OF 6-m PHOTORESIST FILM SAPPHIRE SUBSTRATE
ON
0.33-mm
TABLE IV EXPERIMENTAL RESULTS OF 6-m PHOTORESIST FILM 0.50-mm GaAs SUBSTRATE
ON
0
accurately, because of decreasing of quality factorof the resonator containing such samples, even if the sample is lossless [5]. Alternatively, we can divide (22) to (37), and using the measured difference in the resonant frequencies of the resonator containing the sample with the film on the lower side and on the upper side, we can calculate the difference in permittivities from the equation (39) Equation (39) is more preferable, as the knowledge of the film thickness is unnecessary. III. SIMULATION Developed differential method was compared with the method reported in [3] [(1) and (2)]. The behavior of the resonant frequency of a hemispherical open resonator containing a sample was simulated numerically. The 143.1651-mm distance and the 149.4548-mm curvature radius between mirrors were obtained using the calibration procedure described in [5]. The refractive indices of GaAs and sapphire are taken to be 3.59 and 3.06, respectively, giving a predefined value 0.6940. We compared this value with the . calculated value of The simulation results of the thin sapphire layers on the 0.5-mm and 0.6-mm GaAs substrates at frequencies of 105 GHz are presented in Tables I and II, respectively. Much larger frequency shifts are obtained for a thicker (0.6-mm) GaAs substrates (see Table II). It can be explained by the fact that the upper surface of the sample comes closer to a maximum of the electric field, and therefore, the sensitivity to the thin layer becomes higher. It is possible to detect and measure even a micrometer-thick film in that case. However, it becomes more difficult to find the resonant peaks due to the increase of the plane mirror absorption [5]. IV. EXPERIMENT Two 6- m AZ4562 photoresist films were deposited onto 0.33-mm sapphire and 0.50-mm GaAs substrates. Measurements were performed at different frequencies with the open
resonator (Fig. 1) and an AB Millimetre 5–350-GHz Vector Network Analyzer connected to it. Prior to the measurement, the resonant frequencies of six 0.53-mm sapphire substrates were compared at frequencies close to 80.3 GHz in the resonator in order to find two of them with the closest resonant frequencies. On one of them, a 6- m photoresist film was deposited, while the second wafer was a “reference sample.” Three resonant frequencies of the resonator, loaded with the “reference sample,” the substrate with the film on the upper side, and the substrate with the film on the lower side were measured. The preliminarily measured difference of the resonant frequencies for the “reference sample” and the substrate only was taken into account. After that the resonant frequency shifts and the relations between them were calculated. The permittivity of the film was calculated using (39). The procedure of the resonant frequency measurement was repeated 5–7 times, and the results were averaged. Tables III presents the measurement results and the calculated permittivities of the photoresist film on the sapphire substrate. Measurements were repeated for the GaAs substrate with the same photoresist film (Table IV). V. DISCUSSION The developed differential method allows to estimate the permittivity of the film on the substrate with accuracy within 5% and 2% for 0.6-mm and 0.5-mm substrate thickness, respectively, with film thickness up to 15 m. The above uncertainty is caused at least by the following applied approximations: Taylor expansion, were assumed to be constant, perturbation approach, etc. In the case of thicker films, the accuracy becomes worse due to approximations used in the derivation of (39). Meanwhile, (39) gives a better accuracy for substrates with thicknesses approaching , due to larger frequency shifts to be measured. The theory cannot be used if the substrate thickness is about because of tangential function used in the approximation becomes zero. On the other hand, if , the quality factor of the resonant peak dramatically decreases. Therefore, the desired discrete substrate thicknesses are about , . The main contribution to errors comes from the experiment. This is because of the nonideality of the open resonator (roughness and nonideal flatness of the plane mirror), environmental
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IEEE TRANSACTIONS ON INSTRUMENTATION AND MEASUREMENT, VOL. 54, NO. 5, OCTOBER 2005
instabilities (temperature, humidity), nonideal flatness of the investigated samples, etc. These result in about 10%–15% uncertainty in obtained film permittivities. VI. CONCLUSION Hemispherical open resonator technique is developed for measurements of permittivity of thin films on optically dense substrates. The method is based on measurement of resonant frequency shifts due to appearance of the thin film on the upper and lower side of the substrate. The main advantage of this differential method is that if the permittivity of the substrate is known, it is unnecessary to know the dimensions of the resonator and thicknesses of the substrate and film. Measurements of photoresist films on the 0.5-mm GaAs and 0.33-mm sapphire substrates were carried out. Obtained results show the method can be successfully employed for measurement of dielectric constant of thin films on a substrate. ACKNOWLEDGMENT The authors would like to thank Prof. V. Lyubchenko from the Institute of Radioengineering and Electronics, Russian Academy of Science, for fruitful discussions and providing samples. They would also like to thank Dr. V. Ovchinnikov from the Microelectronics Center of TKK for the help in the film deposition. REFERENCES [1] A. L. Cullen and P. K. Yu, “The accurate measurement of permittivity by means of an open resonator,” in Proc. Royal Soc. London, vol. A325, 1971, pp. 493–509. , “Measurements of permittivity by means of an open resonator. I. [2] Theoretical,” in Proc. Royal Soc. London, vol. A380, 1982, pp. 49–71. [3] S. Wang, L. Hu, and D. Xu, “Open resonator technique for measuring multilayer dielectrics,” Microw. Opt. Technol. Lett., vol. 16, no. 6, pp. 368–371, 1997. [4] B. Komiyama, M. Kiyokawa, and T. Matsui, “Open resonator for precision dielectric measurements in the 100 GHz band,” IEEE Trans. Microw. Theory Tech., vol. 39, no. 10, pp. 1792–1796, 1991. [5] T. Hirvonen, P. Vainikainen, A. Lozowski, and A. V. Räisänen, “Measurement of dielectrics at 100 GHz with an open resonator connected to a network analyzer,” IEEE Trans. Instrum. Meas., vol. 45, no. 4, pp. 780–786, Aug. 1996.
Sergey N. Dudorov was born in the Kirov region, Russia, in May 1975. He received the M.Sc. degree in applied physics and mathematics from the Department of Radio Engineering and Cybernetics, Moscow Institute of Physics and Technology, Moscow, Russia, in June 1998. The topic was “Investigation of dielectric waveguides and devices based on them.” He received the Licentiate degree in June 2001 and the degree of Doctor of Science in Technology in June 2002 at the Radio Engineering Laboratory, Helsinki University of Technology (TKK), Espoo, Finland, in June 2001 and June 2002, respectively. He received the Candidate of Science degree from the Moscow Institute of Physics and Technology. The thesis title is “Rectangular dielectric waveguide and its optimal transition to a metal waveguide.” In November 1998, he joined the Radio Engineering Laboratory of TKK, where he is currently a Postdoctoral Researcher. His research activities are focused on the dielectric property measurements in application to development of new devices for millimeter and microwave applications based on the dielectric waveguides.
Dmitri V. Lioubtchenko was born in Gorky, Russia, in May 1971. He received the B.Sc., M.Sc., and Ph.D. degrees in applied physics and mathematics from the Department of Physical and Quantum Electronics, Moscow Institute of Physics and Technology, Moscow, Russia, in 1993, 1994, and 1998, respectively. From 1994 to 1997, he was a Researcher in the Institute of Radio Engineering and Electronics, Russian Academy of Sciences, Moscow. From 1997 to 1998, he was a Visiting Researcher at The University of Liverpool, Liverpool, U.K. In 1998, he joined the Radio Laboratory of Helsinki University of Technology, Espoo, Finland, where he is currently a Postdoctoral Researcher. His research interests and experience cover various topics including investigations of new materials for millimeter, microwave, and optoelectronic applications, particularly on the development of active and passive dielectric waveguides for the frequency above 100 GHz.
Juha A. Mallat was born in Lahti, Finland, in 1962. He received the Diploma Engineer (M.Sc.) with honors, Lic.Tech., and Dr.Tech. degrees in electrical engineering from the Helsinki University of Technology (TKK), Espoo, Finland, in 1986, 1988, and 1995, respectively. He has been with the TKK Radio Engineering Laboratory (and its Millimeter Wave Group) since 1985, working as a Research Assistant, Senior Teaching Assistant, and Research Associate until 1994. From 1995 to 1996, he was a Project Manager and Coordinator in an education project between TKK and the Turku Institute of Technology. Since 1997, he has been a Senior Scientist with MilliLab (Millimetre Wave Laboratory of Finland—ESA External Laboratory), with the exception of a period of one year during 2001–2002, when he served as a Professor (pro tem) in Radio Engineering at TKK. His research interests and experience cover various topics in radio engineering applications and measurements, especially in millimeter wave frequencies. He has also been involved in building and testing millimeter wave receivers for space applications.
Antti V. Räisänen received the Doctor of Science (Tech.) degree in electrical engineering from the Helsinki University of Technology (TKK), Espoo, Finland, in 1981. In the past, he has held Visiting Scientist positions at the Five College Radio Astronomy Observatory (FCRAO) and the University of Massachusetts, Amherst, at Chalmers University of Technology, Gothenburg, Sweden, at the Department of Physics, University of California, Berkeley, at the Jet Propulsion Laboratory and the California Institute of Technology, Pasadena, and at the Paris Observatory and the University of Paris VI. Currently, he is a Chair Professor of Radio Engineering at TKK and is supervising research in millimeter wave components, antennas, receivers, microwave measurements, etc. at TKK Radio Laboratory, and leads the Smart and Novel Radios Research Unit (SMARAD). He serves also as the Chairman of the Board of Directors of MilliLab (Millimeter Wave Laboratory of Finland—ESA External Laboratory). He has authored and coauthored more than 350 scientific or technical papers and six books, most recently Radio Engineering for Wireless Communication and Sensor Applications, MA: Artech House, 2003. Dr. Räisänen was the Chairman of the IEEE MTT/AP Chapter in Finland from 1987 to 1992. He was the Secretary General of the 12th European Microwave Conference in 1982, and served as the Conference Chair for the 22nd European Microwave Conference in 1992, and for the 2nd ESA Workshop on Millimeter Wave Technology and Applications: Antennas, Circuits and Systems in 1998, and as the Co-Chair for the 3rd ESA Workshop on Millimeter Wave Technology and Applications: Circuits, Systems, and Measurement Techniques in 2003. From 1995 to 97 he served in the Research Council for Natural Sciences and Engineering, the Academy of Finland. In 1997, he was elected the Vice-Rector of TKK for the period of 1997–2000. From 2002 to 2005, he served as an Associate Editor of the IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES.
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