Proceedings of the 40th European Microwave Conference
Switchable FBAR Based on Paraelectric Films A. Mikhailov, A. Prudan, S. Ptashnik, T. Samoilova and A. Kozyrev Department of Electronics, St. Petersburg State Electrotechnical University (LETI), Saint Petersburg, 197376, Russia
[email protected] [email protected] Abstract– In present paper the calculation of eigen acoustic modes of a multilayer microwave capacitor structure with thin ferroelectric films in a paraelectric state is carried out. For the estimation of the efficiency of bulk acoustic waves excitation by ferroelectric films the system of electromechanical equations is solved and a spatial distribution of mechanical displacements in the structure is presented. Finally, the switchability of eigen acoustic modes is shown and the novel switchable film bulk acoustic resonator (FBAR) is discussed.
I. INTRODUCTION Multilayer structures, comprising thin films of bariumstrontium titanate (BST) in the paraelectric (centrosymmetric) state are commonly used for microwave tunable applications due to the dc bias dependent capacitance of the structure. However, if a dc bias is applied the abnormal frequency behavior of both capacitance and loss tangent is observed [1-4]. The abnormal frequency behavior of dielectric properties is caused by the resonance excitation of bulk acoustic waves by microwave signal. Although this phenomena can be useful for development of novel class of microwave devices like tunable thin-film bulk acoustic resonators, oscillators, filters, etc. [3-7], it is negative in point of the conventional microwave application of these ferroelectric (FE) capacitor structures. If a dc bias is applied to the multilayer structure with FE film, the central symmetry of the BST crystal structure is broken due to electrostriction, and piezoelectricity is induced [6]. The multilayer structure in the microwave frequency range seems like high Q-factor acoustic resonator with hypersound source, because the BST film generates bulk acoustic waves of microwave frequency. Existing to date analytical technique for analysis of this phenomena is based on calculation of a complex impedance of a multilayer structure with one piezoelectric film [8]. However, it is not suitable for multilayer structures with two ferroelectric films. II. CALCULATION OF EIGEN ACOUSTIC MODES OF MULTILAYER
resonator. In what follows, we consider this thin film capacitor as a five layer acoustic resonator with mechanically free surfaces. 0
x
Fig. 1 One dimensional schematic layout of the multilayer capacitance structure with two ferroelectric films. Layer thicknesses: top and bottom electrodes – 500 nm, FE films – 225 nm, middle electrode – 50 nm
The problem of determining the eigen modes of a multilayer acoustic resonator was solved in terms of acoustic displacements excited at the free boundary of the structure. A solution of the system of equations describing the continuity of mechanical stresses (σ) and displacements (η) at the interfaces determines the spectrum of eigen frequencies fi of the acoustic modes and provides data on the distribution of the amplitude ηm(x) of standing waves with the frequencies fi. Figure 2 shows the profiles of ηm(x) for the first two modes with odd (curve 1) and even (curve 2) number of standing wave nodes. The numerical analysis has been performed for a resonator representing a symmetric structure with the parameters of layers given in the Table I, where ρ is the density, c is the elastic modulus, Q is the acoustic figure of merit, tanδ0 is the dielectric loss tangent, G is the coefficient of electrostriction, and β is the coefficient of dielectric nonlinearity.
STRUCTURE
In this work we concentrate our efforts on the multilayer (parallel-plate) capacitor structure based on Ba0.3Sr0.7TiO3 films. Typical 1D-layout of a parallel-plate structure used is illustrated in figure 1. We have studied a thin film capacitor structure, in which (i) the substrate dimensions are significantly greater than the hypersound wavelength and (ii) a wave reflected from the free surface of the substrate decays and does not influence on the distribution of plane wave amplitudes in the five layer Pt/BST/Pt/BST/Pt acoustic
978-2-87487-016-3 © 2010 EuMA
Top electrode FE film Middle electrode FE film Bottom electrode
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TABLE I MATERIALS PARAMETERS
Parameter ρ, 103 kg/m3 c, 1011 N/m Q (1 GHz) tanδ0 (1 GHz) G, 1010 m/F β, 109 m5/C2F
Pt 21.3 3.34 150 -
Material BST 4.4 3.21 200 0.035 1.08 2.69
28-30 September 2010, Paris, France
where PDC is the ferroelectric layer polarization by the control field and ε 0ε =
((ε ε (0) ) 0
−1
2 + 3β PDC
)
−1
is the permittivity of
the ferroelectric layer in the presence of this field. As it follows from Eq.(3), mathematically, piezoelectric modulus can have both the same and opposite sign due to different polarities of constant electric field in ferroelectric layers. The calculation of the field of hypersonic displacements in a five layer resonator is based on the solution of a system of equations describing the continuity of displacements (η) and mechanical stresses (σ) at the interfaces (x = xj) between adjacent layers: ⎧ηi ( x j ) = ηi +1 ( x j ) (4) , i = 1...5 . ⎨ ⎩σ i ( x j ) = σ i +1 ( x j ) Fig. 2 Spatial profiles of the modulus of the amplitude ηm(x) of the standing wave of mechanical displacements in a five layer acoustic resonator for the first (1) and second (2) eigen modes. Vertical lines indicate the boundaries between layers
III. ESTIMATION OF EFFICIENCY OF BULK ACOUSTIC WAVES EXCITATION BY FERROELECTRIC FILMS
At the next stage, we have solved the problem on an excitation of the resonator eigen modes by the microwave field in the capacitor. The acoustic wave in a ferroelectric film with the piezoelectric modulus e ≠ 0 coexists with an electric field of the wave type. The dependence of the potential φ of this electric field on the coordinate and time is determined by solving the wave equation for the mechanical displacements η jointly with the Poisson equation, 2 ⎧ ∂ 2η D ∂ η ρ = с ⎪ , (1) ⎪ ∂x 2 ∂t 2 ⎨ ⎪ ∂ ⎛ −ε ε ∂ϕ + e ∂η ⎞ = 0 ⎜ 0 ⎟ ⎪⎩ ∂ x ⎝
∂x
∂x ⎠
where ε0ε is the permittivity of the ferroelectric film and cD = c + e2/ε0ε is the elastic modulus. A solution of the system of equations (1) has the following form: ⎧η ( x, t ) = Ae j (ω t + kx ) + Be j (ωt − kx ) ⎪ , (2) e D (t ) ⎨ − + ϕ ( x , t )= η ( x , t ) x b ⎪ ε 0ε ε 0ε ⎩ where ω and k are the wave frequency and wave vector, respectively; D(t) is the homogeneous component of the electric induction; and b is an arbitrary constant. The structure of the φ(x, t) function shows that a homogeneous microwave field in the ferroelectric films is capable of exciting an acoustic mode in the resonator. The piezoelectric modulus, e, of the ferroelectric layers occurring in a paraelectric state is nonzero only in the presence of a constant electric field. A phenomenological theory of ferroelectrics with second order phase transition establishes the following relationship between the control field and the piezoelectric modulus: (3) e = 2ε 0ε GPDC ,
In the model under consideration, dissipation of the acoustic and electromagnetic energy in the component layers is taken into account using a complex representation of the permittivity [ε = ε'(1 – jtanδ0)] and the elastic modulus [cDi = cD0(1 + j/Qi)] with the frequency dependent parameter Qi. Numerical solution of the system of equations (4) determines the complex amplitudes (Ai, Bi) of waves in various parts of the Pt/BST/Pt/BST/Pt multilayer resonator. Figure 3a shows the spatial distributions of the amplitude of standing acoustic waves excited by the microwave field at two resonance frequencies that corresponds to odd (curve 1) and even (curve 2) modes under the condition of the same polarities of control DC field in BST films. Figure 3b shows the spatial distributions of the amplitude of standing acoustic waves excited at two resonance frequencies that corresponds to odd (curve 1) and even (curve 2) modes under the condition of the opposite polarities of control DC field in BST films. The amplitudes ηm in figure 3 are normalized to a parameter (having the dimensionality of length) η0 = UAC/2GPDC, where UAC is the microwave voltage amplitude across the capacitor. An analysis of the results shows that if the control DC field of the same polarity is applied to the both ferroelectric films, the homogeneous microwave field in the capacitor ensures the excitation of the odd eigen modes only. The action of a microwave field with an even mode frequency (Fig. 3a, curve 2) leads to the appearance of a standing wave with an amplitude profile not corresponding to the given eigen mode. Thus, in the symmetric Pt/BST/Pt/BST/Pt structure even modes are not excited by the microwave field under the condition of the same signs of BST film piezoelectric modulus. From the other hand, if the control DC field of the opposite polarity is applied to the ferroelectric films the microwave field ensures the excitation of the even eigen modes only. The action of a microwave field with an odd mode frequency (Fig. 3b, curve 1) leads to the appearance of a standing wave with an amplitude profile not corresponding to the given eigen mode. Thus, under the condition of the opposite signs of BST film piezoelectric modulus odd modes are not excited by the microwave field.
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(a)
Fig. 5 Transmission characteristics of 3-pole bandpass filter based on the multilayer resonators with BST films for the same (solid lines) and opposite signs (dashed lines) of DC field polarities in BST films
(b) Fig. 3 Spatial profiles of the modulus of amplitude ηm of the standing wave of mechanical displacements in a five layer acoustic resonator excited at the frequencies of acoustic eigen modes f1 (1) and f2 (2) under the condition of the same (a) and the opposite (b) signs of piezoelectric modulus of BST films
Fig. 4 Frequency dependence of a real part of an electrical impedance of a multilayer resonator for the same (solid lines) and opposite signs (dashed lines) of DC field polarities in BST films. Data are normalized to their maximal values
This conclusion is confirmed by the analysis of the electric response of the capacitor to the applied microwave voltage. The electric response of the acoustic resonator with ferroelectric film manifests itself as an anomalous frequency dependence of electrical impedance (Z) at the frequency of eigen acoustic resonance (Fig. 4). Figure 5 illustrated the modeled reflection and transmission characteristics of 3-pole pass-band filter based on the multilayer resonators with two BST films. From figures 4 and 5 it follows that switching the DC field polarity in BST films provides the novel technique of FBAR and filter tuning. IV. CONCLUSIONS We have solved the problem of the description of the resonance excitation of acoustic eigen modes in a multilayer capacitor structure based on two ferroelectric films by a microwave electric field. It was established that a microwave voltage applied to the capacitor with a symmetric structure ensures the effective excitation of hypersound at the frequencies of odd eigen modes of the multilayer resonator if the DC field polarities were the same in both FE layers. And if the DC fields of the opposite polarities were applied to FE layers the microwave signal ensures the effective excitation of hypersound at the frequencies of even eigen modes. This phenomena gives us an opportunity to design a novel switchable FBAR based on two BST films being in paraelectric state. ACKNOWLEDGMENT This study was supported within the framework of the Federal Program “Scientific and scientific-educational personnel of innovative Russia (2009–2013)”.
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