Differential Open Resonator Method for Permittivity Measurements ...

DUDOROV et al.: DIFFERENTIAL OPEN RESONATOR METHOD FOR PERMITTIVITY MEASUREMENTS 1919TABLE IRESONANT FREQUENCY SHIFTS FOR 0.5-mm GaAs SUBSTRATE (0.63) WITHSAPPHIRE THIN LAYERS AND DIFFERENCE FROM (n 01)=(n 01) = 0.6940TABLE IIIEXPERIMENTAL RESULTS OF 6-m PHOTORESIST FILM ON 0.33-mmSAPPHIRE SUBSTRATETABLE IIRESONANT FREQUENCY SHIFTS FOR 0.6-mm GaAs SUBSTRATES (VERY CLOSETO 3=4) WITH THIN SAPPHIRE LAYERS AND DIFFERENCE FROM(n 0 1)=(n 0 1) = 0.6940TABLE IVEXPERIMENTAL RESULTS OF 6-m PHOTORESIST FILM ON0.50-mm GaAs SUBSTRATEaccurately,becauseofdecreasingofqualityfactoroftheresonatorcontaining such samples, even if the sample is lossless [5].Alternatively, we can divide (22) to (37), and using the measureddifference in the resonant frequencies of the resonatorcontaining the sample with the film on the lower side and onthe upper side, we can calculate the difference in permittivitiesfrom the equation(39)Equation (39) is more preferable, as the knowledge of the filmthickness is unnecessary.III. SIMULATIONDeveloped differential method was compared with themethod reported in [3] [(1) and (2)]. The behavior of the resonantfrequency of a hemispherical open resonator containing asample was simulated numerically. The 143.1651-mm distancebetween mirrors and the 149.4548-mm curvature radiuswere obtained using the calibration procedure describedin [5]. The refractive indices of GaAs and sapphire are takento be 3.59 and 3.06, respectively, giving a predefined value0.6940. We compared this value with thecalculated value of .The simulation results of the thin sapphire layers onthe 0.5-mm and 0.6-mm GaAs substrates at frequencies of105 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 explainedby the fact that the upper surface of the sample comes closer toa maximum of the electric field, and therefore, the sensitivityto the thin layer becomes higher. It is possible to detect andmeasure even a micrometer-thick film in that case. However,it becomes more difficult to find the resonant peaks due to theincrease of the plane mirror absorption [5].IV. EXPERIMENTTwo 6- m AZ4562 photoresist films were deposited onto0.33-mm sapphire and 0.50-mm GaAs substrates. Measurementswere performed at different frequencies with the openresonator (Fig. 1) and an AB Millimetre 5–350-GHz VectorNetwork Analyzer connected to it.Prior to the measurement, the resonant frequencies of six0.53-mm sapphire substrates were compared at frequenciesclose to 80.3 GHz in the resonator in order to find two ofthem with the closest resonant frequencies. On one of them, a6- m photoresist film was deposited, while the second waferwas a “reference sample.” Three resonant frequencies of theresonator, loaded with the “reference sample,” the substratewith the film on the upper side, and the substrate with the filmon the lower side were measured.The preliminarily measured difference of the resonant frequenciesfor the “reference sample” and the substrate only wastaken into account. After that the resonant frequency shifts andthe relations between them were calculated. The permittivity ofthe film was calculated using (39). The procedure of the resonantfrequency measurement was repeated 5–7 times, and theresults were averaged. Tables III presents the measurement resultsand the calculated permittivities of the photoresist film onthe sapphire substrate.Measurements were repeated for the GaAs substrate with thesame photoresist film (Table IV).V. DISCUSSIONThe developed differential method allows to estimate the permittivityof 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 iscaused at least by the following applied approximations: Taylorexpansion, were assumed to be constant, perturbationapproach, etc. In the case of thicker films, the accuracy becomesworse due to approximations used in the derivation of(39). Meanwhile, (39) gives a better accuracy for substrates withthicknesses approaching ,due to larger frequency shifts to be measured.The theory cannot be used if the substrate thickness is aboutbecause of tangential function used in the approximationbecomes zero. On the other hand, if, the quality factor of the resonant peak dramatically decreases.Therefore, the desired discrete substrate thicknesses areabout , .The main contribution to errors comes from the experiment.This is because of the nonideality of the open resonator (roughnessand nonideal flatness of the plane mirror), environmentalAuthorized licensed use limited to: COLORADO SCHOOL OF MINES. Downloaded on December 31, 2009 at 12:43 from IEEE Xplore. Restrictions apply.