IEEE Transactions on Ultrasonics, Ferroelectrics, and Frequency Control ,

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Electro-Thermo-Mechanical Model for Bulk Acoustic Wave Resonators Eduard Rocas, Member, IEEE, Carlos Collado, Senior Member, IEEE, Jordi Mateu, Senior Member, IEEE, Nathan D. Orloff, Robert Aigner, and James C. Booth Abstract—We present the electro-thermo-mechanical constitutive relations, expanded up to the third order, for a BAW resonator. The relations obtained are implemented into a circuit model, which is validated with extensive linear and nonlinear measurements. The mathematical analysis, along with the modeling, allows us to identify the dominant terms, which are the material temperature derivatives and two intrinsic nonlinear terms, and explain, for the first time, all observable effects in a BAW resonator by use of a unified physical description. Moreover, the terms that are responsible for the second-harmonic generation and the frequency shift with dc voltage are shown to be the same.

I. Introduction

B

ulk acoustic wave (BAW) technology is meeting its market expectations with a promising future in the field of microwave filters for mobile applications. Several advantages, including miniaturization, power handling and multiple frequency operation, make BAW filters preferable among other existent technologies [1]–[4]. However, in spite of the current success of this technology, there is still a lack of a complete understanding of all observable effects in BAW resonators, which could eventually be crucial in pushing this technology beyond its current limits. In particular, harmonic generation resulting from intrinsic effects, intermodulation distortion, and detuning caused by temperature changes are subjects of active research. The intrinsic nonlinear behavior of the piezoelectric phenomena causes harmonic generation, intermodulation distortion, and frequency detuning with an applied dc voltage. Although recent work has shed light on the possible origin of these nonlinearities [5], none is conclusive,

Manuscript received March 26, 2013; accepted July 3, 2013. This work was supported in part by the Spanish Ministry of Science and Innovation under grants TEC-2009-13897-C03-01/TCM and MAT201129269-C03-02, and by the ENIAC ARTEMOS European Project (EUI2010-04252). Work partially supported by the U.S. government, not subject to U.S. copyright. E. Rocas, C. Collado, and J. Mateu are with the Department of Signal Theory and Communications, Universitat Politècnica de Catalunya (UPC), Barcelona, Spain (e-mail: [email protected]). J. Mateu is also with Centre Tecnologic de Telecomunicacions de Catalunya (CTTC), Castelldefels, Spain. N. D. Orloff is with the National Institute of Standards and Technology (NIST), Gaithersburg, MD. R. Aigner is with TriQuint Semiconductor, Orlando, FL. J. C. Booth is with the National Institute of Standards and Technology (NIST), Boulder, CO. DOI http://dx.doi.org/10.1109/TUFFC.2013.2835

0885–3010/$25.00

and none successfully models all of those undesired effects with a unified nonlinear description. On the other hand, temperature dependency has always been a matter of special attention for BAW resonators. Temperature variation induces changes in the material properties and dimensions that should ideally be compensated for, generally by use of additional compensation layers, if the drift of the resonant frequency with temperature is to be minimized [6]. An accurate understanding and modeling of the material temperature derivatives is crucial to properly design the compensation layers. Several authors have addressed this problem [6], [7], but none of them has taken into account the specific clamping conditions in solidly mounted resonators (SMR), which impact the effective vertical coefficient of thermal expansion. In addition, temperature concerns involve not only the frequency drift in BAW devices, but also the temperature rise resulting from self-heating, which can limit the power handling of, for example, film bulk acoustic resonators (FBAR) [8], and/or contribute to the thirdorder intermodulation distortion [9]. Reference [10] was the first distributed nonlinear modeling approach to cover intrinsic effects through the integration of nonlinearities in different domains of the circuit model. A similar approach is followed in [5] and [11], but with a grouping together of the nonlinear sources. On the other hand, our previous work [9] proposed a phenomenological approach to cover intrinsic and thermal effects. In this article, we propose a rigorous electro-thermomechanical study, based on the expansion of the Gibbs electrical energy function, which is capable of explaining most of the observable effects within a BAW resonator with a unified physical description. We derived a circuit model from the constitutive equations obtained and validated it with linear and nonlinear measurements on BAW resonators, which allows for the identification of the dominant terms that produce the frequency shift with temperature and with an applied dc voltage, as well as describing the generation of harmonics and intermodulation distortion. II. Constitutive Relations in a BAW Resonator The working mechanism of a BAW resonator relies on several physical principles that can be explained with the electro-thermo-mechanical constitutive equations and Newton’s second law of motion. The first group of equa-

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∆TNL =

Fig. 1. The Heckmann diagram shows the electro-thermo-mechanical relations in a crystal, where T, S, E, D, θ, and σ are stress, strain, electric field, electric displacement, temperature, and entropy, respectively.

tions describes the energy coupling between fields associated with each physical domain (electrical, mechanical, and thermal, as shown in the Heckmann diagram, Fig. 1), whereas Newton’s second law describes the relation between particle velocity and stress in a medium. A rigorous nonlinear analysis of these constitutive relations allows for modeling of most of the experimental observables in BAW resonators. We start by obtaining the nonlinear electro-thermo-mechanical relations for a piezoelectric material, which can be simplified for nonpiezoelectric dielectrics and metals. We make a choice of independent variables to be strain S, electric field E, and temperature θ, to obtain equations describing the stress T, and electric displacement D, which have a direct representation in the Mason model [12]. Moreover, this selection of independent variables enables the field derivatives of the material properties to be easily changed when performing measurements, such as the dependence of the material properties on electric field and temperature. Appendix I describes the procedure we have followed to obtain the nonlinear constitutive equations by use of a third-order Taylor-series expansion of the relations between fields plotted in Fig. 1, in a uniaxial notation. The procedure detailed there assumes no approximations other than the series truncation and results in the following equations for the dependent variables T, D, and σ:

T = c E θS − e θE − τ Eθ + ∆TNL (1)



D = e θS + ε S θE + ρ Sθ + ∆D NL (2)



σ = τ ES + ρ SE + r ESθ + ∆σ NL, (3)

where the nonlinear contributions, with all the constants defined in Appendix I, are described by

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1 Eθ 2 (c S − ϕ 3E 2 − ϕ 4θ 2) + ϕ 5SE + ϕ 6S θ − ϕ 7E θ 2 2 1 + (c 3E θS 3 − e 3θ,EE 3 − χ 5θ 3) − χ 6SE θ  6 1 + (χ 7SE 2 + χ 8S θ 2 − χ 9S 2E − χ10S 2θ 2 − χ11E θ 2 − χ12E 2θ),

(4) 1 ∆D NL = (ε 2S θE 2 − ϕ 5S 2 + ϕ1θ 2) + ϕ 3SE + ϕ 7S θ + ϕ 2E θ 2 1 + (ε 3S θE 3 + χ 9S 3 + χ1θ 3) + χ12SE θ  6 1 + (χ 4SE 2 + χ11S θ 2 − χ 7S 2E + χ 6S 2θ 2 + χ 2E θ 2 + χ 3E 2θ), (5) 1 ∆σ NL = (r2ESθ 2 + ϕ 2E 2 − ϕ 6S 2) + ϕ 7SE + ϕ 4S θ + ϕ1E θ 2 1 + (r3ESθ 3 + χ 3E 3 + χ10S 3) + χ11SE θ  6 1 + (χ12SE 2 + χ 5S θ 2 + χ 6S 2E − χ 8S 2θ 2 + χ1E θ 2 + χ 2E 2θ). (6) That which follows is the simplification of the constitutive relations for the specific types of materials used in commercial SMR-type resonators, which usually use aluminum nitride for the piezoelectric layer, metals for the electrodes, and nonpiezoelectric dielectrics for passivation, temperature compensation, and Bragg mirror implementation. In the following sections, the thermal expansion coefficient, αE, will be used instead of the thermal pressure, τE, through the τE = −cEθαE relation, because it is a more commonly used material property. As previously mentioned, apart from the constitutive equations, Newton’s second law is essential to describe the acoustic wave propagation, because it states the relation between particle velocity, v, and stress T: ∂T ∂v = ρ . (7) ∂z ∂t



In (7), ρ is the material density, not to be confused with the electro-caloric constant ρS. The material density ρ in a film on a thick substrate depends on temperature as follows:

ρ=

ρT 0 , (8) (1 + α Si∆θSi)2(1 + α eff∆θN )

where αSi and αeff are the substrate thermal expansion coefficient and film effective vertical expansion coefficient, respectively, and ΔθSi and ΔθN are the temperature increments above room temperature in the substrate and the film. In (8), ρT0 is the material density at room tempera-

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ture. This equation has been derived by considering that the in-plane dimensions change according to the substrate thermal expansion coefficient and the vertical dimension changes according to the effective vertical expansion coefficient. In a first-order approximation, (8) can be simplified if the same temperature is assumed for the film and the substrate:

posited on a thick silicon substrate, that expand laterally in temperature according to the substrate expansion coefficient. In this situation, the vertical expansion coefficient is an effective value αeff, which depends on the substrate and the film coefficients of thermal expansion, as well as on the film Poisson coefficient [18]. A derivation of αeff can be found in Appendix II.

ρ ≅ ρT 0 + ρ NL ≅ ρT 0(1 − (2α Si + α eff )∆θN ). (9)

2) Assumptions Made After Comparison of Simulations and Measurements: Previous work has shown that the slow dynamics of temperature translate into a low-pass filter behavior of the generated third-order intermodulation distortion (IMD3) versus the envelope frequency in a classical two-tone experiment [9], and that temperature changes cannot cause third-harmonic generation at the operating frequencies of the BAW devices. For this reason, and because the thermally generated IMD3 can be modeled with first-order temperature dependence [19], we consider negligible the third-order mixed terms that are related to the thermal domain: χ3 = χ6 = χ10 = χ12 = 0.



A. Thermal Considerations Although we have described the entropy σ with (3) and (6) for completeness, we will not use these equations because we assume that the electrocaloric effect, the piezocaloric effect, the heat of polarization, and heat of deformation are negligible in the materials under consideration [13]. We will deal instead with the heat equation for the thermal domain, which takes into account the temperature rise and heat propagation through a medium [14]. Given the large area-to-thickness aspect ratio, lateral heat dissipation is negligible and therefore the one-dimensional heat equation can be used:

k  ∂ 2θ  ∂θ  , (10) = c pρ  ∂z 2  ∂t

where k and cp are thermal conductivity and specific heat, respectively. The change of in-plane dimensions in SMR resonators is dominated by the substrate and this imposes a lateralclamped condition that especially impacts how the device expands with temperature. For this reason, we will consider the change in area negligible [15], and the model will only account for dimensional changes in the vertical direction by use of Poisson’s ratio. B. Constitutive Relations for Aluminum Nitride For the specific case of aluminum nitride (AlN), several simplifications can be made by use of well-known properties of this material and comparisons between nonlinear measurements and simulations. 1) Assumptions Related to the AlN Properties: We assume that the electrocaloric and pyroelectric effects are negligible [16], which means there is no direct energy exchange, other than dissipation, between the electric and thermal domains. This assumption translates into ρS = 0. The remaining properties of AlN around room temperature change linearly with temperature [17], so we assume that the second- and third-order temperature-dependent coefficients in (4)–(6) are zero, that is, φ1 = φ4 = χ1 = χ2 = χ5 = χ8 = χ11 = 0. The expansion with temperature must be taken into account. The BAW resonator is made of thin layers, de-

3) Simplified Constitutive Equations: With the aforementioned assumptions, the stress T and electric displacement D relations for the aluminum nitride layer are

T = c E θS − e θE + ∆T (11)



D = e θS + ε S θE + ∆D, (12)

with 3 S2 E2 Eθ S + c − ϕ + ϕ 5SE 3 3 2 6 2 + ϕ 6S θ − ϕ 7E θ, (13) E ∆T = c E θα eff θ + c 2E θ

E2 E3 S2 + ε 3S θ − ϕ5 + ϕ 2E θ + ϕ 3SE + ϕ 7S θ. 2 6 2 (14)

∆D = ε 2S θ

Note that the thermal expansion effect, although a linear effect, has been included in the ΔT source. This makes the implementation of a circuit model easier, as will be shown in Section III. Eqs. (11)–(14) can also be interpreted as the result of a set of nonlinear parameters that define the nonlinear electro-thermo-mechanical constitutive equations for the aluminum nitride:  c Eθ c Eθ ϕ ϕ  c NL = c E θ ⋅  1 + 2E θ S + 3E θ S 2 + E5 θ E + E6θ θ    2c 6c 2c c (15)

ϕ ϕ ϕ   e NL = e θ ⋅  1 − 5θ S + 3θ E + θ7 θ  (16)  2e 2e e 

 ε Sθ ε Sθ ϕ ϕ  ε NL = ε S θ ⋅  1 + 2S θ E + 3S θ E 2 + S3θ S + S2θ θ  ,   2ε 6ε 2ε ε (17)

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which offers a more intuitive description of the role of each parameter. Note that the thermal expansion term is not included in the previous equations and should be kept in the linear constitutive equations. C. Constitutive Relations for Nonpiezoelectric Dielectrics and Metals Nonpiezoelectric dielectrics and metals are used in several parts of the resonator and for different purposes, especially in SMR technology. However, for a classic BAW resonator configuration, most of the electromagnetic energy is confined in the piezoelectric layer between the electrodes. This fact makes the modeling of the electric domain unnecessary for the other layers, which need only a thermo-mechanical description. Nevertheless, electric losses in the metal electrodes must be considered to properly model the linear response of the BAW resonator and the Joule-effect contribution to the temperature rise. In our model, a simple electromagnetic description of the electrodes as lumped resistors is accurate enough to properly account for the conductor losses. On the other hand, the fact that polarization in metals is negligible, and the dielectric layers, other than the piezoelectric, are not under electric field, means that piezoelectricity, pyroelectricity, and electrocaloric effects are absent in these layers. Hence, we consider only linear effects in these materials, with the exception of temperature-dependent properties, because of the lower level of the acoustic and electric fields outside the piezoelectric layer. This hypothesis will be confirmed with the demonstrated good agreement between simulations and measurements. With these considerations, we obtain the same electro-thermal constitutive relations for metals and dielectrics:

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A. Materials Stack Fig. 2 illustrates the structure of the SMR-type resonator under study, which is composed of a 1.3-µm-thick piezoelectric layer between two metal electrodes. These layers are all deposited on a Bragg mirror made of a succession of high- and low-acoustic-impedance layers. The main acoustic mode is longitudinal along the vertical z axis, and is induced by the polarization of the piezoelectric layer. Moreover, as will be discussed in Section III-B, the heat can also be considered to propagate mainly along the vertical direction until it reaches the substrate; the heat diffusion is then semi-spherical. Fig. 3 illustrates a section of the circuit implementation. Note that in Fig. 3 the piezoelectric layer implements the electric, acoustic, and thermal domains in a distributed way, whereas all the other layers model only the acoustic and heat propagation. The electric domain of the electrodes is represented by the lumped resistors R1 and R2. The current sources Q1 and Q2 couple the dissipated heat on R1 and R2, respectively, to the thermal domain. 1) Aluminum Nitride Layer: The electro-acoustic model of the piezoelectric layer is implemented in a distributed configuration based on the Mason model [12], as done in [10] and [5], [11]. Here, we have included the thermal domain to provide a complete picture of the physics of a BAW resonator. Fig. 4 shows a block diagram of the cell implementation. The governing electro-acoustic equations are described by use of an equivalent circuit, where voltage and current are

T = c E θS + ∆T , (18)

with

E ∆T = c E θα eff θ + ϕ 6S θ. (19)

III. Circuit Model The circuit model of the materials stack consists of a cascade of layers with their specific material properties, and in turn, each layer consists of a cascade of circuit cells. Each cell represents a thin section of material to reproduce the distributed nature of the acoustic propagation and thermal diffusion. The constitutive relations described in the previous section, Newton’s second law, and heat equation, are implemented at each thin section, interrelating electric, acoustic, and thermal domains. Reference [20] details the steps to derive the Mason model from the electro-acoustic equations; [9] describes the basis of the thermal model.

Fig. 2. Cross-section of a solidly mounted resonator (SMR). The figure also illustrates the thermal considerations for the modeling. The figure is not to scale, and is intended for illustrative purposes only. 

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Fig. 3. Block-diagram that illustrates the circuit implementation of three layers of the BAW resonator. Each layer is constructed as a cascade of electro-thermo-mechanical cells. The electric losses are implemented as lumped resistors and their dissipated heat is coupled to the thermal domain.

equivalent to force F and particle velocity v, respectively. Rearranging (1) and (2) in the stress T and electric-field E form provides a clear circuit-model implementation, where

T = c DθS − E =

eθ D + TC (20) ε Sθ

D − e θS − VC (21) ε Sθ

where

eθ ∆D (22) ε Sθ ∆D VC = S θ (23) ε

TC = ∆T +

c Dθ = c E θ +

(e θ)2 (24) ε Sθ

represents the stiffened elasticity. The acoustic domain results in an acoustic transmission line that obeys the telegrapher equations, with the following distributed parameters for a section of length Δz:

analogs of thermal resistivity and heat capacity respectively [19]. The thermal resistance and capacitance for a section of length Δz are related to the material thermal properties as

and

Fig. 4. Block diagram of a piezoelectric cell implementation with the governing equations for the electric, acoustic and thermal domains.

C ∆z =

∆z (25) Ac Dθ

L ∆z = ∆z ρT 0A. (26)

Eq. (26) makes use of the linear term ρT0, the temperature-dependent density, whereas the nonlinear term ρNL can be implemented as a nonlinear temperature-dependent inductance LΔz,NL = −ρT0(2αSi + αeff)ΔθN. We group the nonlinear sources in TC and VC, as done in [5] and [11], to differentiate between the linear and nonlinear parts of the circuit and provide an easier and clearer implementation. On the other hand, the thermal domain is governed by the heat equation, which also has its electrical analogy. A distributed series resistance and shunt capacitance are the



Rθ,∆z =

∆z (27) Ak

C θ,∆z = Ac p∆z , (28)

where A is the area, k is the thermal conductivity, and cp is the heat capacity. This allows us to identify the analogs of voltage and current, in the thermal domain, as temperature θ and heat rate q, respectively. Fig. 5 shows the circuit implementation of a Δz section of the aluminum nitride cell, in which the dashed box contains the electromechanical constitutive relations that are specific to the piezoelectric. We need to include losses in the model if we want to accurately capture the resonator’s behavior and account for self-heating effects. The fact that dielectric losses in aluminum nitride are minor when compared with other sources of loss in BAW resonators [21] makes us consider them to be negligible. However, in case of using a piezoelectric material with non-negligible dielectric losses, these could be easily modeled as a conductance between the electrodes. In our case, only the acoustic damping, expressed by use of a complex elasticity, in which the imaginary part is linked to the material viscosity [22], is introduced in the aluminum nitride model:

c Dθ → c Dθ + j ωη Dθ, (29)

where ηDθ is the acoustic damping coefficient, defined as ηDθ = vLτ [23]. In the previous definition, vL is the acoustic propagation velocity in the longitudinal direction and τ is the viscous relaxation time. The acoustic losses translate into a conductance as follows [22]:

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the resonator. Moreover, the resonator is in complete thermal contact with the copper chuck by use of high-thermalconductivity silver paint. The bottom thermal boundary is easily implemented in the circuit model by use of a voltage source with the reference temperature value. The atmosphere of the cryogenic chamber of the probe station is a vacuum, although measurements without vacuum have also been performed, at room temperature, to verify that the air’s acoustic impedance and thermal convection are negligible. Nevertheless, if radiation and convection effects must be taken into account, although negligible at reasonable power levels, we can use [14]

RC =

1 (32) Ah c



Rr =

1 , (33) Ah r

where hc and hr are the convection and radiation heat transfer coefficients respectively, and hr is defined as Fig. 5. Circuit-model implementation of a Δz-long section of piezoelectric.



G ∆z =

∆z . (30) Aη Dθ

Therefore, the dissipation is implemented as a heat source in the thermal domain, making the introduction of different types of dissipation more flexible:

Q ∆z =

v2 . (31) G ∆z

2) Nonpiezoelectric Dielectrics and Metals: As discussed in the previous section, materials other than the piezoelectric used in BAW resonators keep the same governing equations for the thermal and acoustic domains as the piezoelectric layer, but the electric field inside of them is negligible. Therefore, nonpiezoelectric dielectrics and metals keep the same circuit implementation of Fig. 5 but without the dashed box that contains the electro-acoustic coupling of piezoelectricity. Instead, the TC source, which is now TC = ΔT because ΔD = 0, is connected to ground. B. Thermal Boundaries Section IV presents the results of measurements on different samples that have been performed to validate the model. Therefore, the considerations and assumptions on the thermal boundaries to be included in the model must be in accordance with the specific measurement setup. To achieve a temperature-controlled environment, a cryogenic probe station has been used, and this allows for an accurate stable reference temperature at the bottom of



h r = εσ(Ts + TA)(Ts2 + TA2), (34)

with σ, ε, Ts, and TA being the Stefan–Boltzmann constant, the emissivity and temperature of the top layer and the ambient temperature, respectively. Eqs. (32) and (33) are implemented as two parallel electrical resistances connected to the top of the thermal domain model at one end and a voltage source with ambient temperature at the other end. Thermal conduction is the main heat propagation mechanism in the devices under study [8]. The resistance to heat flow, as shown in (27), is inversely proportional to the surface area. Therefore, the very high aspect ratio of the Bragg mirror segment—that is, the ratio of the inplane area to the vertical area that surrounds the Bragg mirror—translates into negligible lateral heat flow. On the other hand, the silicon substrate is very thick and the previous assumption cannot be made, so semispherical heat diffusion is considered. Fig. 5, which summarizes all these assumptions, illustrates several semispherical shells with numbers 1 to N, with the N cell being at reference temperature. Each shell represents a thermal transmission line, of the same form of that in Fig. 5, given by

R Si,∆z =

d Si/N (35) k 2πri2

C Si,∆z = 2πri2ρc pd Si/N , (36)

where dSi is the thickness of the silicon substrate and ri is the shell radius. The area of the first shell is the resonator area, so that r1 is obtained from A = 2πr12, and the other radius is calculated as

ri +1 = ri +

d Si . (37) N

rocas et al.: electro-thermo-mechanical model for bulk acoustic wave resonators

Using simulations, we have verified that use of fewer than 20 shells is sufficient to achieve convergence with accurate results. IV. Measurements and Simulations This section presents different types of measurements of BAW resonators, along with the simulations using the circuit model that have been performed to validate it. The linear performance is evaluated under different temperatures and electric-field biasing conditions, which reproduces different situations on the electro-thermo-mechanical diagram of Fig. 1 and allows us to obtain the field-derivative material properties. Moreover, intermodulation and harmonic measurements are carried out to serve as a second independent way to validate the model and access higher-order material coefficients. A. The Devices We perform measurements on two state-of-the-art oneport commercial BAW resonators, having different material stack composition that correspond to a series and a parallel acoustic resonator, respectively, in a ground-signal configuration. The structure of the different layers serves as a way to check that the nonlinearities are correctly defined as geometry-independent material properties. Although the details of the resonator structures cannot be disclosed, the devices are made of aluminum (Al), tungsten (W), aluminum nitride (AlN), silicon dioxide (SiO2), silicon (Si), and silicon nitride (Si3N4). For sake of clarity, we show only the results of the parallel resonator in the following subsections.

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and 320K and use literature values for the material temperature derivatives to obtain a first approximation. We use material properties from [6] for all materials, with the exception of the Si3N4 value [24] and the Al and AlN values, which are obtained through fine-tuning. Fig. 6 shows the results of measurements and simulations, and Table I summarizes the temperature derivatives used. C. Second-Order Nonlinearities BAW resonators generate a high-level second harmonic and their resonant frequencies change linearly with an applied electric field. Both effects are manifestations of second-order nonlinearity and no consensus on their origin exists [11], [25]. To date, there is no article predicting both effects with a single model. Therefore, we perform scattering-parameter measurements, under different bias voltages, and also second-harmonic measurements to validate, by two independent means, the model presented in Section II. In the harmonic experiments, the frequency is swept over a 200 MHz range around the mechanical resonance frequency. References [26] and [27] illustrate the usefulness of sweeping the central frequency in harmonic or intermodulation distortion (IMD) experiments because the frequency dependence gives information about the origins of the nonlinearity. This procedure is more deeply developed in [11] to illustrate how different frequency dependences of the second harmonic and IMD3 can be correlated with different constants of a set of constitutive equations. More specifically, this procedure shows that φ3 and ε 2S θ must be negligible, because, given their dependence on a specific

B. Temperature Measurements and Simulations We performed scattering-parameter measurements at different temperatures by use of a cryogenic probe station, as explained in Section III-B, with the purpose of identifying the dominant material temperature derivatives. To accomplish this, using commercial software (Advanced Design System, Agilent Technologies Inc., Santa Clara, CA), we first set up the linear circuit model with the electroacoustic material properties provided by the manufacturer and fine-tune them to perfectly agree with measurements at 293K. It is important to note that the correct linear model predicts not only the main mode resonance but also the spurious longitudinal resonances. As long as the spurious resonances are linked to specific energy distributions in the materials stack, we can use the viscosity value of each material to accurately predict the magnitude and phase of each resonance. Once the linear model at 293K is properly implemented, the next step consists of adjusting the circuit model response to fit the measurements at different temperatures and obtain the material temperature derivatives. We perform scattering-parameter measurements at 220K

Fig. 6. Measurements and simulations of the magnitude of the impedance for temperatures 220K, 293K, and 320K. The inset shows the linear dependence of the resonance and antiresonance frequencies on temperature. In the main plot, the solid lines are simulations; the dashed lines are measurements. In the inset, circles and squares represent the measurements at resonance and antiresonance, respectively, whereas the dashed line and the solid line represent the simulations at resonance and antiresonance, respectively. 

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TABLE I. Obtained Nonlinear Material Properties for Parallel and Series Resonators. φ6/cEθ (10−6/K)

φ2/εSθ (10−6/K)

φ7/eθ (10−6/K)

φ5/2eθ (10−6/K)

c3

Eθ/6cEθ

(10−6/K)

Material

Parallel

Series

Parallel

Series

Parallel

Series

Parallel

Series

Parallel

Series

Al AlN Si3N4 Si SiO2 W

−300 −60 −24.4 −75 239 −91

−300 −60 −23.8 −75 239 −91

— 150 — — — —

— 149.6 — — — —

— 10 — — — —

— 10 — — — —

— −8.5 — — — —

— −8.5 — — — —

— −18.5 — — — —

— −18.5 — — — —

field, they would produce a second harmonic with a frequency dependence that is not consistent with the measurements. Reference [11] also suggests that the second harmonic is exclusively due to φ5, which implies, as shown in Table III of Appendix I, a piezoelectric constant e that depends on the strain. Reference [11] suggests that the second harmonic is exclusively due to φ5, which from the Maxwell relations can be understood as a piezoelectric constant e that depends on the strain, as shown in Table III of Appendix I. The authors of [11] reached that conclusion after checking that the second-order nonlinearities of cE that depend on the strain (that is, c 2E θ ) must be negligible because although the frequency dependence of the second harmonic is the same as the frequency dependence caused by φ5, the third-order intermodulation distortion would be overestimated. This is true if no other causes of IMD3, which could compensate the IMD3, are considered. However, we show in this section that the hypothesis c 2E θ = 0 is consistent with two different experiments, namely second-harmonic measurements and scattering parameters

Fig. 7. Measurements and simulations of the magnitude of the impedance for an applied voltage ranging from −40 V to +40 V. The inset shows the linear dependence of the resonance and antiresonance frequencies on voltage. In the main plot, the solid lines are simulations, whereas the dashed lines are measurements. In the inset, circles and squares represent the measurements at resonance and antiresonance, respectively, whereas the dashed line and the solid line represent the simulations at resonance and antiresonance, respectively. 

with an applied dc voltage. As far as we know, this is the first time that these two different manifestations of nonlinearity are explained by use of the same nonlinear parameter. 1) Frequency Shift With DC Bias: A bias tee connected to a voltage source is introduced in the linear measurement setup to isolate the network analyzer from the dc voltage. Additionally, an open-short-load (OSL) calibration is performed for each applied voltage to determine the BAW resonator impedance. The results, for an applied voltage ranging from −40 V to +40 V, are summarized in Fig. 7. By use of simulations of the circuit model, we identify φ5 in (13) as the dominant term responsible for the frequency shift with electric field. This term in (13), as shown in Appendix I, can be seen either as a strain-dependent piezoelectric constant or an electric field dependent elasticity: φ5 = −c 2,θ E = −e 2,θ S . For this experiment, we can better understand φ5 as an electric-field-dependent elasticity because it directly depends on the applied field. We find that a value of φ5 = −25.5 N/(V∙m) agrees with the measurements, which corresponds to a relative change in cEθ of φ5/2cEθ = −3.5 ∙ 10−11 m/V. 2) Second Harmonic: We perform second-harmonic measurements to validate the model by a second independent way. The measurement setup consists of a twotone configuration, like that in [27], with a balanced output power of 20 dBm for each tone and a tone spacing of 1 kHz. The frequency of the tones is swept from below the resonance frequency to above the antiresonance frequency while keeping the tone spacing constant. The measurement system itself is characterized by use of a four-port scattering parameter measurement, so that the output of the device can be normalized with respect to its input, and the effects introduced by the measurement system can be totally de-embedded. The details on the measurement setup and the normalization procedure can be found in [27]. Fig. 8 shows the de-embedded second-harmonic output for a 20 dBm input power at each tone, along with the simulations obtained by use of the circuit model. The frequency dependence of the second harmonic clearly points to a strain-squared S2 contribution as the responsible term. The simulations confirm that the φ5 term in (14) is the dominant term for second harmonic generation. More-

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over, the same value φ5 = −25.5 N/(V∙m) that predicts the frequency shift with electric field can accurately predict the second harmonic, which corresponds to a relative change in eθ of φ5/2eθ = −8.5. Therefore, the model has been checked to be consistent with two independent manifestations of the second-order nonlinearity and reveals the dominant terms for each specific experiment. D. Third-Order Nonlinearities Third-order nonlinearity gives rise to a variety of effects such as third-harmonic and third-order intermodulation distortion generation. Moreover, different nonlinear mechanisms contribute to the IMD3 generation, including self-heating and intrinsic nonlinearity. To face the complex characterization of third-order nonlinearity, and because the tested devices do not show a measurable third harmonic, we focus on different types of measurements of the IMD3 to reveal the existent contributions. The same measurement setup and normalization process used for the second harmonic characterization is used for these measurements. 1) Self-Heating Contribution: Subsection IV-B characterizes the performance of the BAW resonator with temperature and reveals the materials’ temperature derivatives. Apart from producing a frequency shift with temperature, the dependence of materials properties on temperature also produces a parametric frequency mixing of the fundamental signal and the dynamic variations of temperature resulting from self-heating. This parametric mixing, whose details can be found in [19], gives rise to IMD3. Therefore, the modeling of the IMD3 contribution resulting from self-heating is an independent way to check that the temperature-dependent modeling is correct. In a two-tone measurement, the envelope of the signal Δf /2 = ( f2 − f1)/2 is mainly responsible for the temperature oscillations, because the dissipation process, related to the square of the signal, creates a heating component at twice that frequency. As explained in [19], the IMD3 level of the self-heating contribution-versus-envelope frequency has a low-pass filter nature because of the slow dynamics of the temperature domain. Therefore, two types of measurements must be carried out to accurately characterize the dynamic thermal behavior. The first measurement is at constant tone spacing while sweeping the central tones’ frequency, and allows us to validate the frequency dependence of the dissipation sources. The second measurement involves sweeping the tone spacing while keeping the two tones’ central (average) frequency constant and reveals the thermal impedance of the device [19]. We set up the complete circuit model presented in Section III, including the thermal domain, with the material thermal properties provided by the manufacturer and couple the dissipation in the corresponding position of the thermal domain. Fig. 9 shows measurements and simulations of the IMD3 for a tone spacing of 1 kHz. The result-

Fig. 8. Measurements and simulations of the second harmonic and second-order intermodulation response and for an input power of +20 dBm. Dashed lines with stars, rectangles, and circles represent measurements of f1 + f2, 2f1, and 2f2, respectively. Dotted line, solid line, and dashed line represent simulations of f1 + f2, 2f1, and 2f2, respectively. 

ing IMD3 is the result of dissipation in the electric and acoustic domains, both having different frequency dependences. The results of the second measurement we performed are shown in Fig. 10, along with the simulations obtained by use of the model. Measurements and simulations agree with the expected low-pass filter behavior, which is also a good way to observe the thermal time constant of the device (on the order of milliseconds). The disagreement for tone spacing above 100 kHz is because intrinsic thirdorder nonlinearity has not been introduced in the model so far, and is explained in the following subsection. An accurate modeling and understanding of the IMD3 resulting from self-heating is important because it is responsible for in-band degradation of signal integrity, resulting, for example, in an increased error vector magnitude. 2) Intrinsic Third-Order Nonlinearity: The constant IMD3 level observed above 100 kHz in Fig. 10 reveals the existence of a contribution arising from intrinsic thirdorder nonlinearity. Intrinsic nonlinearity, defined as the dependence of material properties on fields other than temperature, generates harmonics and intermodulation distortion levels that do not depend on the envelope frequency. Although one might think that the lower level of intrinsic nonlinearity, when compared with self-heating IMD3, makes this a negligible effect, the reality is that it can result in important degradation in signal quality in a BAW filter. The main undesired effect of intrinsic nonlinearity is receiver desensitization that occurs when an intermodulation product, generated by the mixing of the transmitted signal with a jammer signal, falls within the receiver band [28]. This effect is always dominated by

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Fig. 9. Measurements and simulations of the low-frequency third-order intermodulation distortion response for a tone spacing of Δf = f2 − f1 = 1 kHz. For this tone spacing, the IMD3 is dominated by self-heating effects. Dashed lines with rectangles and circles represent measurements of 2f1 − f2 and 2f2 − f1, respectively. Solid line and dashed line represent simulations of 2f1 − f2 and 2f2 − f1, respectively. 

intrinsic nonlinearities, instead of self-heating, as it occurs for high envelope frequencies. This makes the understanding and modeling of intrinsic third-order nonlinearity an important aspect of BAW technology. Third-order nonlinearities in BAW resonators have been less explored than second-order nonlinearities. The BAW resonators used in this work show no measurable third-order harmonic with the current measurement setup; therefore, only the frequency dependence of the intrinsic IMD3 can be used to unveil the dominant third-order terms [9]. The candidate terms to dominate on the IMD3 level are c 3E θ, ε 3E θ, e 3θ, χ7, and χ9. We performed two-tone measurements with tone spacing of Δf = 1 MHz to ensure that intrinsic nonlinearity dominates above self-heating effects, and swept the central frequency as was previously done for the data shown in Fig. 9. It is found by simulation that the contribution of c 3E θ/(6c E θ) = −18.5 is needed to reproduce the frequency dependence of the measurements, as shown in Fig. 11. The nonlinear term obtained is the third-order elasticity. With the nonlinear terms obtained from the frequency dependence for Δf = 1 MHz, we expect the model to be complete and able to reproduce the IMD3 level saturation at higher tone separation frequencies observed in the measurements, because this type of nonlinearity does not depend on the envelope frequency. Fig. 12 shows the agreement between measurements and simulations when intrinsic third-order nonlinearity is included in the model. V. Discussion and Conclusion We have presented a unified electro-thermo-mechanical description of the constitutive relations in a BAW resona-

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Fig. 10. Measurements and simulations of the low-frequency third-order intermodulation distortion response for a tone spacing ranging from 300 Hz to 1 MHz. The central frequency is kept constant at f0 = f1 + Δf/2 = f2 − Δf/2 = 1.845 GHz. The simulations include the self-heating effects. Dashed lines with rectangles and circles represent measurements of 2f1 − f2 and 2f2 − f1, respectively. Solid line and dashed line represent simulations of 2f1 − f2 and 2f2 − f1, respectively. 

tor. The study is based on the expansion of the Gibbs electrical energy function and provides useful relations that allow one to relate all the observables to their physical origins. To accomplish this, we translate the equations obtained into a circuit model that is used, along with measurements on BAW resonators, to characterize the linear and nonlinear terms in the model. The model obtained is able to predict the BAW resonator behavior as a function of temperature, as well as its self-heating induced nonlinearities and the harmonics and intermodulation distortion generation, in addition to the frequency shift of the acoustic resonance with dc voltage just by inclusion of the materials temperature derivatives and two intrinsic nonlinear terms. In fact, the results demonstrate that the frequency shift with dc voltage and the second-harmonic generation arise from the same nonlinear term φ5. Amplitude-frequency measurements have not been performed because this effect is not uniquely attributed to one nonlinear origin but many, including selfheating [8], and therefore is not a useful characterization method. However, this nonlinear effect is also described by the presented model. The mathematical derivation in Appendix I states that φ5 can be either understood as a strain-dependent piezoelectric coefficient or an electric-field dependent elasticity, because of the Maxwell relations for energy conservation. This can be explained from an atomistic perspective, because in zero electric-field and zero-strain situations, the atoms are in electrostatic-mechanical equilibrium. In other words, the position of each atom is given by the charge attraction and mechanical elasticity forces, given an isothermal situation. Any variation from the equilib-

rocas et al.: electro-thermo-mechanical model for bulk acoustic wave resonators

Fig. 11. Measurements and simulations of the low-frequency third-order intermodulation distortion response for a tone spacing of Δf = f2 − f1 = 1 MHz. For this tone spacing, the IMD3 is dominated by intrinsic nonlinearities. Dashed lines with rectangles and circles represent measurements of 2f1 − f2 and 2f2 − f1, respectively. Solid line and dashed line represent simulations of 2f1 − f2 and 2f2 − f1, respectively. 

rium, by an applied strain or electric field, translates into an equivalent strain-dependent piezoelectric coefficient or an electric-field-dependent elasticity. This means that, when deformation occurs, the relative positions of the atoms change, because of the redistribution of internal electrostatic-mechanical forces. In other words, the dipole moment depends on deformation in the vertical direction because the relative distances between atoms change with the deformation. This translates into a strain-dependent piezoelectric constant. This principle also gives rise to an electric field-dependent elasticity. The elasticity is the material property that relates a stress to deformation, so that as we gradually apply an electric field, we change the electrostatic forces between atoms. This fact redistributes the forces’ equilibrium, so that the effective elasticity changes. References [11] and [29] reinforce the latter argument, as both references provide the dominant nonlinear term for aluminum nitride, as e 2,θ S and c 2,θ E , respectively, both in close agreement with the values found in this work. With all these considerations, the nonlinear relations obtained are

 c Eθ ϕ ϕ  c NL = c E θ ⋅  1 + 3E θ S 2 + E5 θ E + E6θ θ  (38)   6c 2c c



ϕ ϕ   e NL = e θ ⋅  1 − 5θ S + θ7 θ  (39)  2e e 



ϕ   ε NL = ε S θ ⋅  1 + S2θ θ  . (40)   ε

The material properties have been initially obtained from measurements on a parallel resonator and then tested to exactly predict the response of different-area resonators. A slight change in φ6 and φ2, for Si3N4 and AlN respectively, is needed to adjust the series resonators. The obtained results, from devices having different area and stack configurations, prove the validity of the characterization and modeling methodology. All nonlinear material properties are summarized in Table I.

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Fig. 12. Measurements and simulations of the low-frequency third-order intermodulation distortion response for a tone spacing ranging from 300 Hz to 1 MHz. The central frequency is kept constant at f0 = f1 + Δf/2 = f2 − Δf/2 = 1.845 GHz. The simulations include the self-heating effects and the intrinsic third-order intermodulation distortion. Dashed lines with rectangles and circles represent measurements of 2f1 − f2 and 2f2 − f1, respectively. Solid line and dashed line represent simulations of 2f1 − f2 and 2f2 − f1, respectively. 

A circuit model that implements a unified physical description and accounts for the thermal, acoustic, and electrical domains and all their observable effects, is crucially needed to predict the linear and nonlinear behavior of BAW resonators in the design stage. Such a model can be used to accurately predict the linear figures of merit and nonlinear indicators, such as the error vector magnitude, and to design resonators with mitigated nonlinear effects. Appendix I Electro-Thermo-Mechanical Relations In connection with Fig. 1 and by virtue of the first law of thermodynamics, one can define the differential of internal energy of an adiabatically-insulated body as the sum of mechanical, electrical, and thermal energy [30]: dU = T dS + E dD + θ dσ, (41)



where T, S, E, D, θ, and σ are stress, strain, electric field, electric displacement, temperature, and entropy, respectively. The natural independent variables of (41) are S, D, and σ, but we are interested in using S, E, and θ as the independent variables. Therefore, we use the electric Gibbs free-energy function G(S, E, θ), obtained through the appropriate Legendre transformation: G = U − ED − σθ, whose differential form can be obtained by use of (41): dG = T dS − D dE − σ dθ. (42)



Therefore, the dependent variables T, D, and σ are related to the Gibbs function as

T =

∂G ∂S

E ,θ

, D=−

∂G ∂E

S,θ

, σ=−

∂G ∂θ

S,E

(43)

T = G S, D = −G E , σ = −G θ , (44)

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TABLE II. Material Constants. Eθ

TS c Stiffness TSS c 2E θ Second-order stiffness TSSS c 3E θ Third-order stiffness ε S θ Permittivity DE DEE ε 2S θ Second-order permittivity TE = GES = GSE = −DS Tθ = GSθ = GθS = −σS Dθ = GEθ = GθE = −σE

DEEE σθ σθθ σθθθ







∂ n +q + p f S n E q θ p f (S, E, θ) = , (45) ∂S n∂E q∂θ p n ! q ! p ! n =0 q =0 p =0

∑∑∑

which up to the third order can be written as

+ f EEEE 3 + f θθθθ 3 + f SESE + f S θS θ + f E θE θ 2

2

+ f SE θSE θ  + f SEESE + f S θθS θ + f ESSES



2

Piezoelectric constant Thermal pressure Electro-caloric constant

Table IV shows the third-order material constants, which are also related to the linear material properties of Table II. The complete nonlinear constitutive equations up to third order are

T = c E θS − e θE − τ Eθ + ∆TNL (47)



D = e θS + ε S θE + ρ Sθ + ∆D NL (48)



σ = τ ES + ρ SE + r ESθ + ∆σ NL, (49)

where the nonlinear contributions are

f = f SS + f EE + f θθ + f SSS 2 + f EEE 2 + f θθθ 2 + f SSSS 3

Third-order permittivity Heat capacity Second-order heat capacity Third-order heat capacity

−e θ −τ E −ρ S

where yx represents the partial derivative of y with respect to x. The relations in (43) can be expanded into a Taylor series, around the zero-fields point, as a three-variable function: ∞

ε 3S θ r ES r2ES r3ES

+ f E θθE θ 2 + f θSSθS 2 + f θEEθE 2, (46) where f is T, D, or σ. If we replace (44) in (46), the partial derivatives can be written in terms of G, which allows for the identification of the equivalent partial derivatives; for example, TE = GSE = −DS. Table II shows the nomenclature we will follow for the material constants. Note that the subscript denotes the order and the superscript denotes the independent variables that are kept constant. Table III shows the second-order material constants, which are also written in the first column as derivatives of the material constants of Table II. The second column is the nomenclature that indicates the order and, as an example, the first row should be read as “E-dependent heat capacity is equal to θ-dependent electro-caloric coefficient,” which allows for clarification of the equivalent second-order interactions between physical domains in Fig. 1. The third column shows the nomenclature we will use.

1 Eθ 2 (c S − ϕ 3E 2 − ϕ 4θ 2) + ϕ 5SE + ϕ 6S θ − ϕ 7E θ 2 2 1 + (c 3E θS 3 − e 3θ,EE 3 − χ 5θ 3) − χ 6SE θ  6 1 + (χ 7SE 2 + χ 8S θ 2 − χ 9S 2E − χ10S 2θ 2 − χ11E θ 2 − χ12E 2θ) (50) 1 ∆D NL = (ε 2S θE 2 − ϕ 5S 2 + ϕ1θ 2) + ϕ 3SE + ϕ 7S θ + ϕ 2E θ 2 1 + (ε 3S θE 3 + χ 9S 3 + χ1θ 3) + χ12SE θ  6 1 + (χ 4SE 2 + χ11S θ 2 − χ 7S 2E + χ 6S 2θ 2 + χ 2E θ 2 + χ 3E 2θ) (51) ∆TNL =

TABLE IV. Third-Order Material Constants. S D θθθ = σ E θθ = ρ θθ

ρ 3,S θ

χ1

Sθ ES D E θθ = σ EE θ = εθθ = rEE

ε 3S,θ = r3S,E

χ2 χ3

D θθ = σ E θ = −ρ θS = rEES

r2S,E = ρ 2S,θ

φ1

D E θ = σ EE = −ρ ES = εθS θ

−ρ 2S,E = ε 2S,θ

E Tθθθ = σ S θθ = −τ θθ

S ρ 3,E −e 3,θ E −τ 3,Eθ

φ2

S TSE θ = −D SS θ = −ρ SS

−ρ 3,S

−χ6

TEE = −DSE = −e Eθ = −εSθ

−e 2θ,E = −ε 2θ,S

φ3

Eθ Sθ TSEE = −DSSE = c EE = −εSS

c 3θ,E = −ε 3θ,S

−χ7

Tθθ = σ S θ = −τ θE = −rSES

−τ 2E,θ = −r2E,S

φ4

c 3θ,E = −r3E,S

χ8

TSE = −DSS = C EE θ = −e Sθ

E ,S Eθ TS θθ = −σ SS θ ⇒ c θθ = −rSS

−c 2θ,E = −e 2θ,S

φ5

θ TSSE = −D SSS = −e SS

−e 3,θ S

−χ9

−τ 2E,S = c 2E,θ

φ6

E TSS θ = −σ SSS = −τ SS

E −τ 3,S

−χ10

e 2,θ = τ 2,E = ρ 2,S

φ7

−e 3,θ

−χ11

r2S,E = ρ 2S,θ

φ1

θ TE θθ = −DS θθ = −σ SE θ = −e θθ E TEE θ = −DSE θ = −σ SEE = −τ EE

−τ 3,E

−χ11

D EE θ = σ EEE =

TABLE III. Second-Order Material Constants.

TS θ = −σ SS = −τ SE = c θE θ TE θ = −D S θ = D θθθ = σ E θθ =

−σ SE = e θθ −ρ θS = rEES

=

τ EE

=

ρSS

S ρ EE

θ TEEE = −DSEE = −e EE

−χ4 −χ5

rocas et al.: electro-thermo-mechanical model for bulk acoustic wave resonators

∆σ NL =

1 ES 2 (r θ + ϕ 2E 2 − ϕ 6S 2) + ϕ 7SE + ϕ 4S θ + ϕ1E θ 2 2 1 + (r3ESθ 3 + χ 3E 3 + χ10S 3) + χ11SE θ  6 1 + (χ12SE 2 + χ 5S θ 2 + χ 6S 2E − χ 8S 2θ 2 + χ1E θ 2 + χ 2E 2θ).

(52) Appendix II Effective Vertical Coefficient of Thermal Expansion

In the situation of thin films on thick substrates, the thermal expansion of the in-plane dimensions is dominated by the substrate, and its corresponding vertical dimension changes according to the material Poisson coefficient. The following is the derivation of the effective coefficient of thermal expansion for a thin film constrained by the substrate. Hooke’s law, including the coefficient of thermal expansion, for the general case reads

1 (Tx − ν p(Ty + Tz )) + α E ∆θ (53) c E,θ 1 S y = E,θ (Ty − ν p(Tx + Tz )) + α E ∆θ (54) c 1 S z = E,θ (Tz − ν p(Tx + Ty )) + α E ∆θ, (55) c Sx =

which can be rewritten as v (S + S y + S z )  c E,θ c E,θ  E  S x + p x −   1 − 2ν p α ∆θ 1 + ν p  1 − 2ν p (56) Tx =

v (S + S y + S z )  c E,θ  c E,θ E  S y + p x −   1 − 2ν p α ∆θ 1 + ν p  1 − 2ν p (57)

Ty =

v (S + S y + S z )  c E,θ c E,θ  E  S z + p x −   1 − 2ν p α ∆θ. 1 + ν p  1 − 2ν p (58)

Tz =

In the special case of lateral expansion dominated by the thermal expansion of the silicon substrate, Sx = Sy = αSiΔθ, and no vertical tension, Tz = 0, from (53)–(58), we obtain S z,N = αN ,eff∆θ, (59)



with the effective vertical coefficient of thermal expansion as

αN ,eff =

1 (α − ν N (2α Si − αN )), (60) 1 − νN N

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where αN, νN, Sz,N are the coefficient of thermal expansion, Poisson coefficient, and vertical strain for layer N, respectively. References [1] R. Aigner, J. Kaitila, J. Ella, L. Elbrecht, W. Nessler, M. Handtmann, T. R. Herzog, and S. Marksteiner, “Bulk-acoustic-wave filters: Performance optimization and volume manufacturing,” in IEEE MTT-S Int. Microw. Symp. Dig., 2003, vol. 3, pp. 2001–2004. [2] A. Reinhardt, G. Parat, E. Defay, M. Aid, and F. Domingue, “Acoustic technologies for advanced RF architectures,” in 8th IEEE Int. NEWCAS Conf., 2010, pp. 161–164. [3] R. Aigner, “SAW and BAW technologies for RF filter applications: A review of the relative strengths and weaknesses,” in IEEE Int. Ultrasonics Symp., 2008, pp. 582–589. [4] L. Mourot, P. Bar, G. Parat, P. Ancey, S. Bila, and J. F. Carpentier, “Stopband filters built in the BAW technology [Application Notes],” IEEE Microw. Mag., vol. 9, no. 5, pp. 104–116, Oct. 2008. [5] D. S. Shim and D. A. Feld, “A general nonlinear mason model of arbitrary nonlinearities in a piezoelectric film,” in IEEE Int. Ultrasonics Symp., 2010, pp. 295–300. [6] B. Ivira, P. Benech, R. Fillit, F. Ndagijimana, P. Ancey, and G. Parat, “Modeling for temperature compensation and temperature characterizations of BAW resonators at GHz frequencies,” IEEE Trans. Ultrason. Ferroelectr. Freq. Control, vol. 55, no. 2, pp. 421– 430, Feb. 2008. [7] D. Petit, N. Abele, A. Volatier, A. Lefevre, P. Ancey, and J. F. Carpentier, “Temperature compensated bulk acoustic wave resonator and its predictive 1D acoustic tool for RF filtering,” in IEEE Int. Ultrasonics Symp., 2007, pp. 1243–1246. [8] B. Ivira, R. Y. Fillit, F. Ndagijimana, P. Benech, G. Parat, and P. Ancey, “Self-heating study of bulk acoustic wave resonators under high RF power,” IEEE Trans. Ultrason. Ferroelectr. Freq. Control, vol. 55, no. 1, pp. 139–147, Jan. 2008. [9] E. Rocas, C. Collado, J. C. Booth, E. Iborra, and R. Aigner, “Unified model for bulk acoustic wave resonators’ nonlinear effects,” IEEE Int. Ultrasonics Symp., 2009, pp. 880–884. [10] Y. Cho and J. Wakita, “Nonlinear equivalent circuits of acoustic devices,” in IEEE Int. Ultrasonics Symp., 1993, vol. 2, pp. 867–872. [11] D. A. Feld and D. S. Shim, “Determination of the nonlinear physical constants in a piezoelectric AlN film,” in IEEE Int. Ultrasonics Symp., 2010, pp. 277–282. [12] W. P. Mason, Electronical Transducers and Wave Filters. New York, NY: Van Nostrand, 1948. [13] R. E. Newnham, Properties of Materials: Anisotropy, Symmetry, Structure. New York, NY: Oxford University Press, 2004. [14] F. P. Incropera, D. P. DeWitt, T. L. Berqman and A. S. Lavine, Fundamentals of Heat and Mass Transfer. New York, NY: Wiley, 2002. [15] N. B. Hassine, D. Mercier, P. Renaux, D. Bloch, G. Parat, B. Ivira, P. Waltz, C. Chappaz, R. Fillit, and S. Basrour, “Self heating under RF power in BAW SMR and its predictive 1D thermal model,” in IEEE Int. Freq. Control Symp. Joint with the 22nd European Frequency and Time Forum, 2009, pp. 237–240. [16] M. A. Dubois and P. Muralt, “Properties of aluminum nitride thin films for piezoelectric transducers and microwave filter applications,” Appl. Phys. Lett., vol. 74, no. 20, pp. 3032–3034, 1999. [17] J. D. Larson, III and Y. Oshrnyansky, “Measurement of effective kt2, Q, Rp, Rs vs. temperature for Mo/AlN FBAR resonators,” in IEEE Ultrasonics Symp., 2002, vol. 1, pp. 939–943. [18] C. Kittel, Introduction to Solid State Physics, 7th ed., New York, NY: Wiley, 1996. [19] E. Rocas, C. Collado, N. D. Orloff, J. Mateu, A. Padilla, J. M. O’Callaghan, and J. C. Booth, “Passive intermodulation due to selfheating in printed transmission lines,” IEEE Trans. Microw. Theory Tech., vol. 59, no. 2, pp. 311–322, 2011. [20] G. S. Kino, Acoustic Waves. Englewood Cliffs, NJ: Prentice-Hall, 1987. [21] R. Thalhammer and R. Aigner, “Energy loss mechanicms in SMRtype BAW devices,” in IEEE MTT-S Int. Microwave Symp. Dig., 2005, pp. 225–228.

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[22] B. A. Auld, Acoustic Fields and Waves in Solids, vol. 1, Malabar, FL: Krieger, 1990. [23] P. Muralt, J. Antifakos, M. Cantoni, R. Lanz, and F. Martin, “Is there a better material for thin film BAW applications than AlN?” in IEEE Ultrasonics Symp., 2005, vol. 1, pp. 315–320. [24] B. Smirnov, Yu. Burenkov, B. Kardashev, D. Singh, K. Goretta, and A. Arellano-López, “Elasticity and inelasticity of silicon nitride/boron nitride fibrous monoliths,” Phys. Solid State, vol. 43, no. 11, pp. 2094–2098, Nov. 2001. [25] J. D. Larson, III, S. Mishin, and S. Bader, “Characterization of reversed c-axis AlN thin films,” IEEE Int. Ultrasonics Symp., 2010, pp. 1054–1059. [26] C. Collado, E. Rocas, J. Mateu, A. Padilla, and J. M. O’Callaghan, “Nonlinear distributed model for bulk acoustic wave resonators,” IEEE Trans. Microw. Theory Tech., vol. 57, no. 12, pp. 3019–3029, Dec. 2009. [27] C. Collado, E. Rocas, A. Padilla, J. Mateu, J. M. O’Callaghan, N. D. Orloff, J. C. Booth, E. Iborra, and R. Aigner, “First-order elastic nonlinearities of bulk acoustic wave resonators,” IEEE Trans. Microw. Theory Tech., vol. 59, no. 5, pp. 1206–1213, May 2011. [28] M. Ueda, M. Iwaki, T. Nishihara, Y. Satoh, and K. Hashimoto, “Investigation on nonlinear distortion of acoustic devices for radiofrequency applications and its suppression,” in IEEE Int. Ultrasonics Symp., 2009, pp. 876–879. [29] P. Emery, A. Devos, N. Ben Hassine, and E. Defay, “Piezoelectric coefficients measured by picosecond ultrasonics,” in IEEE Int. Ultrasonics Symp., 2009, pp. 2178–2180. [30] H. B. Callen, Thermodynamics. New York, NY: Wiley, 1960.

Eduard Rocas received the Telecommunication Engineering and Ph.D. degrees from the Universitat Politècnica de Catalunya (UPC), Barcelona, in 2005 and 2011, respectively. From September 2005 to July 2006, he was involved in the simulation and modeling of advanced sonars at the Computer Vision and Robotics Group (VICOROB, University of Girona) as an FPU Grant Holder. From January 2009 to July 2011, he was a Guest Researcher at the National Institute of Standards and Technology (NIST), Boulder, CO, working on materials characterization and modeling for microelectronic devices. Since July 2011, he has been with UPC as a Research Engineer, working on the design of advanced instrumentation and electro-acoustic devices. Dr. Rocas was the finalist of the XXXII Prize for the best doctoral thesis in Defense by COIT (Colegio Oficial de Ingenieros de Telecomunicación) and the recipient of the Extraordinary Award for the best doctoral thesis in Information Technologies and Communications by UPC. Dr. Rocas is also the recipient of a Postdoctoral 2011 Beatriu de Pinós fellowship from the Catalonian government (Marie Curie co-funded Actions).

Carlos Collado received the Telecommunication Engineering and Ph.D. degrees from the Universitat Politècnica de Catalunya (UPC), Barcelona, in 1995 and 2001, respectively. He also holds a master’s degree in biomedical engineering from UPC. Since April 2005, he has been an Associate Professor with UPC, and he was Vice Dean of the Castelldefels School of Telecommunications and Aerospace Engineering (EETAC-UPC) from 2006 to 2008. In 2004, he was a visiting researcher with the University of California, Irvine (GP grant from the Catalonian government), and from 2009 to 2010, he was on a sabbatical at the National Institute of Standards and Technologies, Boulder, CO, with a BE grant from the Catalonian government. He collaborated and led several research projects for national and international public and private organizations and companies. His primary research interests include microwave devices and systems, electro-acoustic devices, and nonlinear modeling.

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Jordi Mateu (M’03–SM’10) received the Telecommunication Engineering and Ph.D. degrees from the Universitat Politècnica de Catalunya (UPC), Barcelona, in 1999 and 2003, respectively. He is an Associate professor in the Signal Theory and Communications Department, UPC, and Senior Researcher Associate at CTTC. From May to August 2001, he was a Visiting Researcher with Superconductor Technologies Inc., Santa Barbara, CA. From October 2002 to August 2005, he was research staff and coordinator of Communication Subsystems with CTTC, Spain. Since September 2004, he has held several guest researcher appointments at the National Institute of Standards and Technology (NIST), Boulder, CO, where he was a Fulbright Research Fellow from 2005 to 2006. In July 2006, he was a visiting Researcher at Lincoln Laboratory, MIT. From September 2003 to August 2005, he was a part-time Assistant Professor at Universitat Autònoma de Barcelona. In summer 1999, after graduation, he held a trainee engineer position in the Investment Technology department, Gillette, UK. He has authored or co-authored more than 45 papers in international journals, more than 60 contributions in international conferences, and three book chapters, and holds two patents. He collaborated and led several research projects for national and international public and private organizations and companies. His primary interest includes microwave devices and systems and characterization and modeling of new electronic materials, including ferroelectrics, magnetoelectric, superconductors, and acoustic devices. His recent research also includes the synthesis, design, and development of novel microwave filtering structures. Dr. Mateu was the recipient of the 2004 prize for the best doctoral thesis in Fundamental and Basic Technologies for Information and Communications, awarded by COIT (Colegio Oficial de Ingenieros de Telecomunicación) and AEIT (Asociación Española de Ingenieros de Telecomunicación). He was also the recipient of a Postdoctoral Fulbright Research Fellowship and an Occasional Lecturer Award for visiting MIT. He was second ranked in a Ramón y Cajal Contract (2005), in the area of Electrical and Communication Technologies, a National Program for promoting outstanding Young researchers. From February 2011 to June 2012, he was Vice-Dean of Academic Affairs of the Telecommunication and Aerospace Engineering School at UPC. He is a reviewer of several journals and international conferences.

Nathan D. Orloff was born in Columbia, SC, on August 10, 1981. He received the B.S. degree in physics with high honors from the University of Maryland at College Park in 2004. In 2010, he earned a Ph.D. degree in physics at the University of Maryland at College Park. His doctoral thesis concerns the study and extraction of microwave properties of strained Ruddelsden–Popper ferroelectric thin films. Dr. Orloff was the recipient of the 2004 Martin Monroe Undergraduate Research Award and the 2006 CMPS Award for Excellence as a Teaching Assistant. Dr. Orloff was also the recipient of the 2010 Michael J. Pelczar Award for Excellence in Graduate Study. He was a 2011–2013 Dean’s Postdoctoral Fellow in Biotic Gaming at Stanford University with Prof. Ingmar Reidel-Kruse. He joined the materials measurement laboratory at the National Institute of Standards and Technology in Gaithersburg, MD, in 2013 as a Rice University postdoctoral fellow under the guidance of Prof. Mateo Pasquali and Dr. Jan Obrzut.

Robert Aigner received his Ph.D. degree from Munich Technical University in 1996 for research on micromachined chemical sensors. In 1996, he was a visiting scientist at the Berkeley Sensors and Actuators Center (BSAC), where he worked on system design for inertial sensors. After returning to Germany, he joined the MEMS research group at Siemens Corporate Technology in 1997. Between 1999 and 2005, he was director of a MEMS R&D department at Infineon Technologies

rocas et al.: electro-thermo-mechanical model for bulk acoustic wave resonators and worked on a variety of MEMS devices, including automotive MEMS and RF-MEMS. The team he built and directed became pioneers in commercializing BAW technology. Since 2006, he has been director of R&D for Acoustic Technologies at TriQuint Semiconductor in Orlando, FL. His focus is to drive technology innovations in the SAW and BAW fields. He has served in several European committees for Microsystem Technology and was nominated as MEMS-expert for the European Commission. He currently is on the technical program committee of the IEEE UFFC society and serves as reviewer for JMEMS, Applied Physics Letters, and other journals. He has more than 100 patents (from 65 patent families) in the fields of BAW and MEMS granted in his name, has published more than 110 articles, and has contributed chapters to three text books.

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James C. Booth received the B.A. degree in physics from the University of Virginia in 1989 and the Ph.D. degree in physics from the University of Maryland in 1996, where the subject of his dissertation was “Novel measurements of the frequency dependent microwave surface impedance of cuprate thin film superconductors.” He has been a physicist at the National Institute of Standards and Technology since 1996, originally as an NRC postdoctoral research associate (from 1996 to 1998) and currently as the leader of the Metrology for Complex Electromagnetic Systems project. His research at NIST is focused on exploring the microwave properties of new electronic materials and devices, including ferroelectric, magneto-electric, and superconducting thin films, as well as developing experimental platforms integrating microfluidic and microelectronic components for RF and microwave frequency characterization of liquid and biological samples.

Electro-thermo-mechanical model for bulk acoustic wave resonators.

We present the electro-thermo-mechanical constitutive relations, expanded up to the third order, for a BAW resonator. The relations obtained are imple...
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