TECHNICAL PHYSICS VOLUME 44, NUMBER 4 APRIL 1999 Modeling and calculation of the capacitance of a planar capacitor containing a ferroelectric thin film O. G. Vendik, S. P. Zubko, and M. A. Nikol’ski St. Petersburg State Electrical Engineering University, 197376 St. Petersburg, Russia ~Submitted December 24, 1997! Zh. Tekh. Fiz. 69, 1–7 ~April 1999! The capacitance of a two-layer planar capacitor containing a thin layer of SrTiO3 is calculated by conformal mapping using the partial capacitanc e method. Simple formulas are obtained for approximation calculation of the capacitances of individual components of a planar structure, and their limits of applicability are determined. A relation for the capacitance of a planar capacitor is derived which takes account of the size effect in a ferroelectric film within the context of the partial capacitance method. The calculated result is compared with the experimentally measured capacitance. © 1999 American Institute of Physics. @S1063-7842~99!00104-X# INTRODUCTION implemented in the works of Kochanov;10,11 however, its Recently interest has been rekindled in the practical use current name was gradually formed only in the English- 12–16 of controllable ferroelectric components in microwave language literature. The partial capacitance method is technology.1–4 This has been largely the result of advances in based on the introduction of zero boundary conditions microwave cryoelectronics at liquid-nitrogen temperatures for the normal components of the field at the boundary stimulated by developments in microwave circuit elements of the media — the ‘‘magnetic wall’’ — and subsequent based on high-temperature superconductors.5 The practical correction of the value of the dielectric constant of each of application of devices working at liquid-nitrogen tempera- the media. The use of this method to calculate complicated tures makes it possible to actively employ the ferroelectric planar structures requires an estimate of its accuracy. Such 11 nonlinearity, since the dielectric losses at this temperature an estimate was first made for a coplanar transmission line are much lower than at room temperature.6 The large value by comparing with a calculation performed on an analog of the dielectric constant of ferroelectrics has led to the ne- computer: the error did not exceed 1–2%. Later the method 12 cessity of using these materials in the form of thin films with was included in a handbook and used in calculations as 13–15 thickness on the order of one to several microns when used a priori reliable without an estimate of possible errors. as components of microwave devices.6 Thus, the use of fer- A comparison was made in Ref. 16 of the calculated and roelectrics in microwave devices is based on a planar tech- experimentally measured group velocity of the wave in a nology in which the main design elements are a planar and a coplanar line. The revealed error did not exceed 5%, which capacitor, a coplanar or gap transmission line. To utilize can most probably be assigned to errors in the measure- these elements toward this end, it is necessary to develop ments. For a two-layer planar structure containing a thin film mathematical models of these elements which can form the with a large dielectric constant ~«5300–10 000!, and a thick basis of a system of computer-aided design ~CAD! of micro- substrate («510– 25), the applicability of the partial capaci- 17 wave devices utilizing ferroelectric materials. The develop- tance method was subjected to a special analysis, which ment of such models is the goal of the present work. made it possible to determine the limits of applicability of The dependence of the dielectric constant of bulk and the method. Corresponding quantitative estimates are given film samples of ferroelectric materials on the temperature below. and electric field strength is well described on the basis of a validated phenomenological model.7,8 In addition, models CONFORMAL MAPPING OF A PLANAR STRUCTURE are needed which allow one to calculate the capacitance of a planar structure formed by a ferroelectric thin film on an The distribution of the electric field in planar capacitors insulating substrate with metal ~superconducting! planar is nonuniform. electrodes with allowance for the fringing fields in the am- To calculate the capacitance of a planar capacitor, it is bient free space ~air!. Such a calculation can be based on the customary to use conformal mapping. The use of conformal method of conformal mapping, which allows one to trans- mapping based on the Christoffel–Schwarz transformation9 form the field in a planar structure into the field of a sand- makes it possible to transform the structures shown in Fig. 1a wich capacitor.9 The planar structure in question consists of and 1b into a rectangle ~Fig. 1c!. The capacitance of a sand- three component parts: the ferroelectric film, substrate, and wich capacitor without fringing fields ~Fig. 1c! is easily cal- air. A separate calculation of the capacitance of each of the culated: component parts with subsequent summation constitutes the «0«rK~k8! basis of the partial capacitance method. This method was C5 , k85A12k2, ~1! first 2K~k! 1063-7842/99/44(4)/7/$15.00 349 © 1999 American Institute of Physics 350 Tech. Phys. 44 (4), April 1999 Vendik et al. FIG. 1. Conformal mapping of the planar structure: a — two planar electrodes in free space; b — result of mapping into a sandwich capacitor without fringing fields. where K(k) is the total elliptic integral of the first kind, k is formed by the capacitance of the stray fields in the ambient the modulus of the elliptic integral, «0 is the permittivity of space ~air! with dielectric constant «151, the capacitance of free space, and «r is the dielectric constant of the medium in the ferroelectric layer of thickness h2 with dielectric constant which the electric field is concentrated. «2, and the capacitance of the substrate of thickness h3 with For the structures shown in Fig. 1a and 1b we have dielectric constant «3. The calculation was based on the par- tial capacitance method.10–16 We represent the composite s ps pl ka5 , kb5tanh /tanh . ~2!,~3! layered capacitor ~Fig. 2! as three simple planar capacitors l S 4h D S 4hD with homogeneous filler ~Fig. 3! connected in parallel, and The ratio K(k8)/K(k) can be represented by the follow- seek its capacitance as the sum of the three partial capaci- ing approximate form ~Ref. 18, pp. 38 and 39!: tances 11 12k2 0.25 21 ~ ! 2 p ln 2 for k <0.5, C5C11C21C3 , ~5! F 12~12k2!0.25G K~k8! F~k!5 5 21 K~k! 11k0.5 p ln 2 for k2>0.5. where C1 , C2, and C3 are the capacitances of the component 5 F S 12k0.5DG parts of the planar capacitor, which are the fringing field in air, the ferroelectric film, and the substrate. ~4! The error of the above approximate form does not ex- ceed 1025 ~Ref. 18!. Note that the above formulas give the capacitance of planar structures per unit length in the direc- tion perpendicular to the plane of the figure. FORMULAS FOR CALCULATING THE CAPACITANCE OF THE COMPONENT PARTS OF A PLANAR STRUCTURE Consider a planar capacitor ~Fig. 2!. Its capacitance is FIG. 2. Diagram of a planar capacitor consisting of an insulating substrate, FIG. 3. Separation of a planar capacitor ~Fig. 2! into partial capacitances a thin dielectric layer with a large dielectric constant and conducting elec- ~the procedure underlying the present calculation!: a — electrodes in ‘‘air’’; trodes separated by a gap of width s. b — thin dielectric layer, c — thick dielectric layer. Tech. Phys. 44 (4), April 1999 Vendik et al. 351 Here the dielectric constants of the media are decreased possible to formulate the following restriction on the appli- in accordance with the equalities cability of the method, which vouchsafes the applicability of the concept of a ‘‘magnetic wall.’’ «*5« 2« , «*5« 21. ~6! 2 2 3 3 3 The method is applicable if the inequality These changes in the values of the dielectric constants of s<10h ~11! the layers constitute the main idea of the partial capacitance 2 2 method, in which each layer is screened from the other layers is satisfied for «2 /«3.10 . by a ‘‘magnetic wall’’ which does not transmit the normal This is in good agreement with results17 calculated for a components of the electric field vectors. The mutual influ- two-layer planar structure without assuming the presence of ence of the various layers on the field distribution in each of a magnetic wall. Calculation shows that if inequality ~11! is them is taken into account by the changes in the values of the satisfied the error in the calculation of the total capacitance dielectric constant of each of the layers represented in the of the planar structure arising as a consequence of the above case under consideration by relations ~6!. approximations will not exceed 61%. Now let us calculate each capacitance separately taking We may also compare the results obtained using the ex- the above into account. act formulas obtained by conformal mapping, ~7!–~9!, and 1. The capacitance of the air gap ~Fig. 3a! using the simplified formulas ~10!–~109!. We consider a specific realization of a planar capacitor s in which we vary the width of the gap s over wide limits for k15 , C15«0F~k1!•w. ~7! l the remaining dimensions fixed ~these are given in Table I!. 2. The capacitance of the ferroelectric layer bounded by Figures 4a–4c plot the values of each of the partial ca- the ‘‘magnetic wall’’ ~Fig. 3b! pacitances calculated according to formulas ~5!–~7! and ~8!– ~10!.