Analytical and Finite-Element Modeling of a Localized-mass Sensor

Humberto Campanella, Jaume Esteve Centro Nacional de Microelectrónica CNM-IMB

Consejo Superior de Investigaciones Científicas CSIC Bellaterra (Barcelona), Spain E-mail: hcampane@cnm.es

Emile Martincic Institut d’Electronique Fondamentale IEF

Paris, France

Pascal Nouet Université de Montpellier II UM2

LIRMM (UMR CNRS-UM2) Montpellier, France

Arantxa Uranga, Nuria Barniol Dept. Electronics Engineering

Universitat Autònoma de Barcelona UAB Bellaterra (Barcelona), Spain

Abstract—Analytical and finite-element models of a localized-mass sensor fabricated with thin-film bulk acoustic wave resonator (FBAR) are reported. A Mason’s model-based analytical approach to understand the non-linear behavior of the sensor is introduced. On the other hand, careful finite-element analysis (FEA) of the sensor is carried out for different mass-loading configurations. Based on both modeling and experimental results, the non-linear behavior of the localized-mass sensor’s responsivity is confirmed.

I. INTRODUCTION Thin-film bulk acoustic wave resonators (FBARs) exhibit

high frequency and mass sensitivities, thus making this technology a suitable candidate for bio-molecular or chemical-detection applications [1]-[2]. Our group reported for the first time the concept of FBAR-based localized-mass sensor [3] and the effects of location and size of the localized-mass in the sensor's responsivity [4]. This work complements previous experimental achievements by proposing two modeling approaches of the localized-mass sensor.

Analytical and numerical models are implemented to studying the wave propagation and the field-coupling through the bulk of the resonator’s acoustic layer, respectively. The analytical approach is based on the Mason’s model [5]. This model has inspired 1D and 2D-like descriptions of FBAR-based uniform-film mass sensors [6], and other piezoelectric resonators [7], among others. Finite-element-modeling (FEM), on the other hand, predicts the electro-mechanical response of the resonator by means of coupled-field numerical analysis. Both 2D and 3D FEM analyses of FBARs implemented through commercial or custom modeling tools can be found in the available literature [8]-[9]. However, no 3D-like analytical model for

FBAR-based localized-mass sensors has been reported to our extent.

The contribution of this paper is to proposing 3D-like models, to study the sensor’s responsivity and mode-shaping. This paper is organized as follows: in section II, the sensor’s concept and the fabrication technology of the FBARs are introduced. Next, in section III, the FEM implementation is described. In section IV the analytical sensor’s model is presented. Both analytical and FEM results are contrasted with experimental measurements. Finally, we conclude and propose future paths of research in section V.

II. TECHNOLOGY AND CONCEPT OF THE SENSOR The localized-mass sensor has been implemented using

FBARs as the sensing device. By means of the in-house CNM25 process, FBARs were implemented as a sandwiched aluminum nitride (AlN) membrane (1μm-thick), sputtered on top of a platinum/titanium (Pt/Ti) layer, fabricated on a silicon (Si) substrate, and released by means of front-side reactive-ion-etching (RIE) of the Si substrate. Additionally, a silicon oxide (SiO2) layer is deposited on top of the silicon substrate, prior to the bottom metal's deposition. With this configuration, the resonance frequency of the devices is found in the band of 2.4 GHz.

For the purpose of sensor's concept demonstration, several experiments comprising focused-ion-beam-assisted deposition of a localized metalorganic precursor containing Pt on the top electrode of FBARs were carried out, according to the procedures we previously presented in [10]. Mass-loading of the FBAR was performed inside a FEI DB Strata235 FIB, a dual beam instrument. In Fig. 1(a), a cross-sectional schematic illustration of the FBAR with localized-mass deposited on the top electrode is observed. An

This work was sponsored by SEIKO EPSON Corporation

1-4244-2581-5/08/$20.00 ©2008 IEEE 367 IEEE SENSORS 2008 Conference

exemplary FBAR device with a square-shaped, localized-mass on its top electrode is shown in Fig. 1(b).

The size and location of the mass have been proven to affect the localized-mass sensor’s performance, as we have demonstrated in [5]. For example, the responsivity [g/Hz] is better when the mass is deposited on the center of the electrode. Although thorough experimental study of these aspects has been carried out, finite-element and analytical models have also been developed in order to understand the mechanisms behind the non-linear performance of the mass sensor.

III. FINITE-ELEMENT MODELING Finite-element-model (FEM) analysis of the mass-to-

resonator system was implemented using ANSYS ® (ANSYS Inc., Canonsburg, PA, USA). Modal and harmonic responses for various mass-loading configurations were analyzed. Thus, different locations and geometries of the localized-mass on the top electrode of an FBAR were considered. The behavior of the FBAR with the deposited-mass is described by the piezoelectric constitutive equations coupling the mechanical and electrical domains in the piezoelectric material [11].

The FEM geometry of the FBAR model is the same of the fabricated devices: An electrode of area 50μm×70μm was chosen, the AlN and Pt layers having thicknesses of 1000nm and 100nm, respectively. The ANSYS model of Fig. 2 depicts the meshing of an FBAR with localized-mass on top of its electrode. The boundary conditions are defined

as a voltage applied across the AlN layer, which is mechanically clamped, as long as the electrodes are, by two of their four parallel walls (clamping conditions: mechanical displacements ux, uy, and uz in the x, y, and z directions, respectively, are equal to 0).

First, the possible influence of meshing noise was studied: Different models, each one with meshing variations according to the mass’ location and size, were implemented. The results of the harmonic analysis showed an important influence of the meshing. For that reason, the meshing was fixed to make the element size equivalent to the smallest mass deposited on the FBAR. In this way, we extracted harmonic responses with no influence of the meshing. This is critical to subsequent mass-loading analyses. Once the meshing-dependent noise of the model is controlled, various configurations of the localized-mass sensor were modeled: Masses with the same geometry and located in different positions of the electrode were included in the model, the center and the lateral regions of the electrode being the mass-deposition locations. The curves of Fig. 3 show that the resonance frequency value is sensitive to the location in which the mass is deposited, the highest frequency shift obtained for the case of the center-located mass deposition. These results were contrasted to experimental characterization data, as observed in the plot. This comparison evidences that the location of the mass is a key factor affecting the resonance frequency, the center of the electrode being the most sensitive area to frequency changes.

The sensor’s mode-shaping dependence on the mass location is another topic that can be studied through the use of FEM analysis. In this way, the mechanical displacements of the electrode for each frequency and mass location were calculated. The topographic model images of Fig. 4 show how the mode-shaping is modified when the mass is deposited on different positions of the electrode. These results coincide with the analytic modeling and with the conclusions of previous works on the amplitude distribution at resonance.

Figure 1. FBAR-based localized-mass sensor: (a) Schematic illustration ofthe concept; and (b) SEM image of a square-shaped mass deposited on thetop electrode of an FBAR.

Figure 2. Finite-element model of FBAR with localized-mass on its top

electrode with meshing: Electric field E is defined applying a 0V potentialonto the lower electrode and a 1V potential onto the upper electrode.Mechanical displacements ux, uy and uz are shown equal to 0.

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IV. THE SPREAD-DISPLACED-EQUIVALENT-MASS (SDEM) ANALYTICAL MODEL

Based on the FBAR Mason’s model [5], a three-dimensional (3D) representation of the equivalent transmission-line model of the mass sensor was implemented in MATLAB ®. Compared to previous work on FBAR analytical modeling, the localized-mass sensor’s model describes the 3D-like geometry of FBARs when a localized-mass is deposited on the top electrode of the resonator.

We propose a mass-modified equivalent-circuit model of the longitudinal-mode FBAR. Thus, information of the size and (x, y) location of the mass is included to fit the non-linear behavior of the sensor. For this purpose, we introduce the Spread-Displaced-Equivalent-Mass (SDEM) concept. Its main contribution is that a 3D system can be described by 1D-models, with no need of 3D-native acoustic models. According to this, and after spatial-transform manipulation, the localized-mass of thickness t, lateral area A, and position (xi,yi) is modeled as a spread thin-film of thickness tSDEM covering the whole electrode’s surface. Since the mass-responsivity of the sensor is non-linear and depends on both the size and location of the mass, the main considerations behind the SDEM concept are:

I) The center of the FBAR’s electrode is the most sensitive region of the sensor, then

II) For a given mass located at the center, the SDEM is thicker than that of another identical mass located far away from the electrode’s center

III) For a given location of a mass m1 with thickness t1, and area Amass1, the SDEM is thicker than that of another mass m2 with identical mass, lower thickness t2, and higher

area Amass2.

IV) Localized-mass deposition within a wide range of mass-values lead to non-linear responsivity behavior

In consequence of the foregoing considerations, the spatial-transform functions involved in the SDEM dimensioning include mass location (x-y coordinates) and geometry (thickness, width, length) variables. In this way, the 3D-equivalent-circuit model of the localized-mass sensor is built by modifying the classical 1D Mason’s model. Assuming perfect isolation between the FBAR and the substrate, said model is depicted in the circuit representation of Fig. 5, where the SDEM is the additional uniform-film layer deposited on the FBAR’s top electrode. Referring to this circuit, the analysis of the localized-mass sensor starts from the 1D-equation defining the electrical impedance ZIN of the loaded FBAR [12].

The value of tSDEM is a non-linear function of the thickness t, the width w, the length l, the location coordinates of the deposited mass (xm, ym), and the FBAR’s electrode area A. Final calculation of the electrical impedance ZIN of the loaded-FBAR requires recursive calculation of ZT and ZB. Regarding the spatial location of the mass and its effects on the value of tSDEM, the findings of previous work on the amplitude distribution of longitudinal-mode crystal resonators were used to model the 3D-topography of the sensor [13]. In Fig. 6, both experimental and modeling results are compared. According to consideration IV on SDEM modeling, the curves of Fig. 6(a) show the analysis results after localized-mass deposition on the center of the top electrode, the analyzed bodies having different sizes and identical mass. In Fig. 6(b), and according to consideration III, all the samples have the same size (Amass= w×l) but different thickness t, thus changing the mass. Good agreement between experimental and modeling data is observed.

V. CONCLUSIONS Two modeling approaches of FBAR-based localized-

mass detectors have confirmed the location and mass-layout dependence of the responsivity, as we experimentally found out. Various FE models were implemented, the same modifying the size and location of the mass. Analytical modeling of the sensor implemented a 3D-like version of the Mason’s model, thus introducing the concept of SDEM. In both models, we succeeded to achieving good agreement between simulation and experimental data. Future evaluation of the active-sensor area is also an interesting subject of study. Our analytical and FEM models will aid on this task.

Figure 3. Frequency response sensitivity to different-located mass

deposition (distance measured with respect to the electrode’s center).

Figure 4. Mode shaping of the FBAR at resonance after localized-mass deposition of a 2.0μm×2.0μm-sized mass (ANSYS) located at: (a) no mass; (b) the

center; (c) 20 μm away from the center of the electrode.

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Figure 5. Modified Mason’s equivalent-circuit mode of the localized-mass sensor, including the SDEM circuit element.

Figure 6. Experimental and SDEM modeling results: (a) Frequency change against size of the localized-mass (constant mass of 3.6×10-12g, different thickness and area to preserve the mass); (b) Frequency change against the amount of mass (same area, different thickness, in order to change the mass).

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