Lateral Spurious Mode Suppression in Lithium Niobate A1 Resonators

This work presents an improved design that exploits dispersion matching to suppress the spurious modes in the lithium niobate first-order antisymmetric (A1) Lamb wave mode resonators. The dispersion matching in this work is achieved by micro-machining the lithium niobate thin film to balance the electrical and mechanical loadings of electrodes. In this article, the dispersion matchings of the A1 mode in lithium niobate based on different metals are analytically modeled and validated with finite-element analysis. The fabricated devices exhibit spurious-free responses with a quality factor of 692 and an electromechanical coupling coefficient of 28%. The demonstrated method herein could overcome a significant hurdle that is currently impeding the commercialization of A1 devices.


I. INTRODUCTION
A S5G promises to open new horizons for paradigmshifting applications, miniature wideband filters in sub-6 GHz are one of the outstanding challenges in the front-end. Currently, the commercial solutions are surface acoustic wave (SAW) resonators and thin-film bulk acoustic wave (BAW) resonators [1], [2]. However, their moderate electromechanical coupling (k 2 t < 10%) is insufficient to meet several allocated 5G new bands [3]- [7]. Although the bandwidth can be increased by integrating passive electromagnetic components with acoustic resonators, the enhancement comes at the cost of complex manufacturing processes and large sizes [8].
Alternatively, the first-order antisymmetric (A1) Lamb wave mode resonators based on lithium niobate (LiNbO 3 ) thin films have recently been studied as a compelling solution for sub-6-GHz wideband filters due to their high k 2 t (>20%) and record-break FoM [9]- [12]. Despite their prospect of enabling wideband and low loss filters, the demonstrated A1 devices so far are all laden with the lateral spurious modes [13]- [17]. The presence of lateral spurious modes remains a major bottleneck for further advancing A1 devices into real applications as it creates unwanted ripples in comprising filters [18]- [20].
To overcome this challenge, this work focuses on the suppression of the lateral spurious modes in LiNbO 3 A1 resonators. We first identify the origins of the lateral spurious modes in the conventional LiNbO 3 A1 design that consists of elusively top interdigital electrodes on a suspended LiNbO 3 thin film. It is concluded that the dispersion mismatch between metalized and unmetalized sections of the LiNbO 3 thin film causes the most significant lateral spurious modes.
An improved design that exploits dispersion matching across the resonator is then proposed and analyzed. The dispersion matching is achieved by micromachining the LiNbO 3 thin film to form a recessed structure for top electrodes. The recessed electrodes have been used in SAW resonators for better energy confinement (higher dispersion mismatch) [21], [22]. In a similar fashion but for a contrasting purpose, this work utilizes recessed structures to minimize the trapping of the acoustic energy. The dispersion matching of A1 in LiNbO 3 based on different metal electrodes is analytically modeled and validated with finite element analysis. The relationship between the recessed depth and electrode thickness is discussed. To validate our analysis and modeling, different designs have been fabricated on a 650nm-thick Z-cut LiNbO 3 thin film with all of them showing near spurious-free measured responses. These devices have shown strong potential for enabling high-performance A1 devices for future 5G front-ends.

II. THEORETICAL ANALYSIS AND MODELING A. A1 and Its Lateral High-Order Spurious Modes
To efficiently excite the A1 in a LiNbO 3 thin film, the top-only interdigital transducers (IDTs) are typically used for the least fabrication complication [ Fig. 1(a)]. To achieve high performance, the acoustic energy is confined in the main body of the devices by etching through LiNbO 3 thin film to form free boundaries, and the energy confinement may introduce unwanted higher order A1 modes, which are treated as spurious modes. To simplify the relationship between the fundamental A1 and other higher order A1 spurious modes, the resonator cross section can be viewed as a 2-D cavity. The resonant frequency ( f A1 0 ) of the A1 mode in a 2-D cavity with  a thickness of t and length of l is given by [23], [24] where v t and v L are the acoustic velocities in the vertical and lateral directions. Based on our previous work, the thickness of LiNbO 3 (t) is chosen to be 650 nm in this work for 5G new radio (NR) [10]. To achieve sufficient static capacitance for system impedance matching, the LiNbO 3 A1 devices typically consist of multiple pair electrodes or unit cells as shown in Fig. 1(a). Each unit is treated as a quasi-isolated resonant body from the rest. The length (l) of the resonant body, which influences the resonant frequency, depends on the structure of the A1 devices. In the ideal case where A1 is excited in a LiNbO 3 thin film without electrical and mechanical loading, l is decided by the distribution of the applied electric fields. More specifically, l is the distance between the respective zero-E-field planes under the signal and ground electrodes [ Fig. 1(b)]. At such planes, the mechanical deformations of A1 are decoupled with the electrical field. These zero-E-field planes are referred to onward as the electrical boundaries. Without considering the mechanical loading, l is equal to the sum of the electrode separation (G) and electrode width (W e ). However, in the actual LiNbO 3 A1 devices, which use top electrodes for the generation of the electric field, both electrical boundaries and mechanical interfaces affect the value of l. The mechanical interfaces refer to the positions of the top electrodes as they are created by the acoustic impedance mismatch between the LiNbO 3 sections with and without top electrodes. Specifically, as seen in Fig. 1(c), the whole resonator body can be modeled as alternatingly cascading the high impedance and low impedance sections [24]. Similar to EM, acoustic waves reflect at the interface or boundary between two media with different acoustic impedance. The mechanical boundaries divide the A1 devices into two sets of sections. The first set of sections are the LiNbO 3 sections without top electrodes, for which the value of l is equal to the value of G. The other set of sections are the LiNbO 3 sections with top electrodes, for which the value of l is equal to the value of W e .
As l can have multiple values in (1), A1 modes with different resonant frequencies can be excited in the device with top electrodes. Among these A1 modes, the fundamental mode features the largest k 2 t . From the point of energy, k 2 t of the excited A1 depends on the mutual energy (U m ) between the electrical and mechanical domains. U m is the integration between the electrical field and stress. As the A1 mode confined by the electrical boundaries (l = W e + G) features the largest mutual energy, it can be treated as the fundamental mode. In contrast, the A1 modes confined by the mechanical interfaces are treated as the higher order spurious modes.
As the mechanical interfaces lead to the internal reflections of the acoustic waves, multiple orders of the lateral A1 spurious modes can be presented. The resonant frequencies of the higher order A1 spurious modes ( f mn 0 ) in the same 2-D cavity are given by [25] f mn where m and n are the mode orders in the vertical and lateral directions, respectively. α is the ratio between the velocities in vertical and lateral directions. According to Hook's law of elasticity, the specific spurious modes only can be generated in the case where U m is nonzero. Based on our previous work, only the higher order spurious modes with odd orders in the vertical and lateral directions can be excited from the nonzero integral of U m [25]. As the performance of the comprised filters is mainly affected by the spurious modes near the passband, the resonant frequencies of the spurious modes close to f A1 0 will be identified and analyzed. In a LiNbO 3 thin film of several hundred nanometers in thickness, α is much larger than t/l in (2). In this case, for higher order spurious modes (m > 1) in the vertical direction, the f mn 0 would be around m times higher than f A1 0 , which are far away from the passband. For the lateral higher order A1 modes with m = 1, especially the third-order (n = 3), the resonant frequencies ( f 1n 0 ) are close to f A1 0 . In addition to the resonant frequencies, the lateral higher order A1 modes also feature a high k 2 t . From the point of energy, k 2 t of the higher order A1 also depends on the U m between the electrical and mechanical domains. Assuming the stress field of the higher order A1 modes follows the sine distribution in the lateral direction, k 2 t of the mth order A1 mode is 1/m 2 of the fundamental A1 mode [25]. Considering the large k 2 t of the fundamental A1, third-order A1 would feature k 2 t over 3%, leading to ripples over a wide frequency range.

B. Mismatch at Mechanical Interfaces
To further understand the mechanical interfaces induced by the electrical and mechanical loadings, the electrical loading is first studied. Due to piezoelectricity, the electrical loading leads to nonzero mechanical stress in the LiNbO 3 slab covered by electrodes. As a result, the acoustic impedances are different for the LiNbO 3 sections with or without electrodes, inducing the acoustic reflections of the waves at the electrode edges [26]- [29]. To validate the effect of the electrical loading, the massless top electrodes are defined in the COMSOL-based finite-element analysis (FEA). Fig. 2 presents the simulated response of the lateral third-order A1 mode excited due to the electrical loading. Additionally, the mechanical loading from the top electrodes leads to the change of the equivalent density and Young's modules at the electrode edges. While the reflection caused by electrical loading does not vary with electrode thickness, the reflection from the mechanical loading increases for thicker or heavier electrodes. The greater reflections subsequently induce more significant spurious modes [30].
To validate our analysis, the A1 mode devices are simulated with the FEA. As shown in Fig. 3, the simulated response based on the structure shown in Fig. 1 presents three main resonances. Consistent with our analysis, one of these three resonances is the fundamental A1 featuring the largest k 2 t , while others at higher frequencies are the lateral third-order and fifth-order A1. It is worth noting that the induced spurious modes are more significant after applying mechanical loading, which is consistent with our analyses (Figs. 2 and 3). As a comparison shown in Figs. 2 and 3, the ideal design is simulated by applying ideal lateral electric fields across the LiNbO 3 slab without placing metallic electrodes on the top surface. Additionally, the lateral boundary conditions of the LiNbO 3 slab are set to be periodic to avoid the reflections of acoustic waves at the two ends. The spurious-free response of the ideal design validates our analysis that the mechanical interfaces caused by the electrical and mechanical loadings are the primary sources of the lateral higher order A1 spurious modes.
To study the reflections at the mechanical interfaces quantitatively, we use the dispersion mismatch to scale as the dispersion in specified film stacks takes the electrical and mechanical loadings into consideration simultaneously. As the electrical and mechanical loadings determine the reflection coefficient, the larger dispersion mismatch indicates a higher reflection coefficient. The dispersion curves of A1 in the LiNbO 3 sections with and without electrodes are calculated and plotted in Fig. 4. Aluminum electrodes of 70 nm in thickness are first used. At the same eigenfrequency, A1 has different wavelengths in the LiNbO 3 sections with and without electrodes. This outcome is consistent with the displacement  mode shapes in the COMOSL-based FEA. Although aluminum is a comparatively light material and preferred for reducing the reflections, the mismatch in dispersion caused by electrodes is still significant.
In addition to the lateral higher order A1 modes, the dispersion mismatch at the mechanical interfaces also can generate the higher order fundamental symmetric (S0) and antisymmetric (A0) modes near the targeted frequency range [31]. In the previous work, the reflections are partially suppressed by reducing the feature size of the electrodes to make the ratio between G/(G + W e ) close to 1 to partially suppress the spurious modes [12], [13], [24], [32]. However, this method cannot entirely suppress the higher order A1 modes and requires a small feature size of the transducers (W e ), which limits the freedom of design and leads to reduced power handling capability.
To sum up, we have identified the origins of the spurious modes and the dispersion mismatch between metalized and  unmetalized sections is the main reason. A new design is needed to achieve dispersion matching.

C. Dispersion Match by Recessed Electrodes
From the analysis of the spurious mode origins, an intuitive method to mitigate spurious modes is to tune the dispersion in metalized sections to match the dispersion characteristics in the unmetalized sections. As shown in Fig. 5, we conceive a recessed structure to adjust the thickness of LiNbO 3 in the metalized sections to shift their dispersion characteristics. In practice, the thickness of the top electrodes (t e ) should be close to the recessed depth (t r ) to minimize the surface discontinuities. Similar to the analysis before, we first use aluminum as the top electrodes to validate our proposal. To balance the electrical and mechanical loading from 70-nm-thick Al, the 650-nm-thick LiNbO 3 thin film needs to be thinned down to 560 nm. The dispersion of A1 in different film stacks is compared in Fig. 6(a).
The proposed LiNbO 3 A1 devices are simulated with different recessed depths (t r ) and 70-nm-thick aluminum as electrodes. All structures have the same G of 4 μm, W e of 3 μm, and cell number of 10 (Table I). In the recessed design with a 30 nm depth, the dispersion mismatch still exists,  I  DESIGN PARAMETERS OF A1 RESONATORS   TABLE II  PARAMETERS OF TYPICALLY USED METALS   TABLE III DESIGNS BASED ON DIFFERENT METALS and the lateral third-order A1 spurious mode is pronounced [ Fig. 6(b)]. After increasing the recessed depth to 90 nm, which is the optimized value for dispersion matching in 650-nm-thick LiNbO 3 thin film, the FEM-simulated response presents a spurious-free result [ Fig. 6(c)]. Further increasing the recessed depth breaks the balance and regenerates spurious modes. However, the spurious modes excited in the recessed structure with a depth of 120 nm are subdued [ Fig. 6(d)]. This is because the electric field is optimized in the recessed structures, causing the electrical boundaries, at where the electric field strength is zero, to approach the mechanical interfaces at the edges of the top electrodes. As shown in Fig. 7, the electric fields are focused in the LiNbO 3 sections without electrodes and closer to be the ideal lateral electric field (no vertical components) in the more deeply recessed structure.
In addition to Al, the recessed structure also can be applied to other metals. Table II lists the parameters of typically used metals (Ti, Cu, W, Au, and Pt) for acoustic devices. Table III presents the designs of recessed devices based on different metals. To show feasibility, the dispersion of film stacks involving these metals is calculated to find the optimal combination. As shown in Fig. 8(a), (c), (e), (g), and (i), which are ordered by densities, a heavier metal leads to a more substantial mismatch due to its greater mechanical loading effect. As a comparison, the FEM-simulated results  based on the conventional and recessed structures (with the corresponded optimized designs) are presented in Fig. 8(b), (d), (f), (h), and (j). The spurious modes caused by the electrical and mechanical loadings are all suppressed in the recessed designs.

D. Implementation of the Recessed Structure
In addition to the suppression of the spurious modes, the recessed designs also can help to increase the static capacitance per unit area (referred to as distributed C 0 onward),  resulting in smaller device footprints for matching to 50 . Fig. 9 shows the distributed C 0 in the recessed designs normalized to its counterpart in a conventional configuration (t r = 0). The results suggest that a larger t r leads to a greater distributed C 0 , and the trend is more significant in a device with smaller G. To maximize distributed C 0 , G should be as small as possible while t r should be as large as possible. However, as described in the previous section, G also affects the wavelength of A1. As shown in Fig. 10, a smaller wavelength leads to a smaller k 2 t . Therefore, a tradeoff should be made between the footprint (or distributed C 0 ) and k 2 t . Considering the large FBW of 5G NR, the minimum value of G is set to be 4 μm to achieve sufficient k 2 t . Due to the limitations of our in-house fabrication, the width of the top electrodes is set to be 3 μm, and the recessed depth (t r ) is limited to be less than 100 nm.
Although all investigated metals can, in theory, work in the recessed design, some of them are not practicable due to the required small thickness for dispersion matching and its potential high electric loss. In practice, the material for top electrodes should feature high conductivity and allow for sufficient thickness. Therefore, thick electrodes with a density of less than LiNbO 3 's are preferred. Based on the parameters listed in Tables II and III, aluminum is the best option. In this article, we focus on implementing aluminum as top electrodes to demonstrate the proposed method.

III. EXPERIMENTAL RESULTS AND DISCUSSION A. Fabrication of A1 Resonators with Recessed Electrodes
The designed A1 resonators were fabricated on a 650-nmthick Z-cut LiNbO 3 thin film following the process described in [24]. Additional steps, as shown in Fig. 11, are incorporated to micro-machine the recessed electrodes. The photoresist is  first patterned for defining the recessed structure and top electrodes. The LiNbO 3 sections, which will be covered by electrodes, are thinned in an inductively coupled plasma (ICP)reactive ion etching (RIE) system. The photoresist remaining after the step of LiNbO 3 thinning further serves as the photoresist for electrodes lift-off, thus achieving the self-alignment of electrodes and recessed sections. 70-nm-thick aluminum is subsequently evaporated and lifted-off as top electrodes in the recessed sections. In the last step of the process, the Si under LiNbO 3 is removed with XeF 2 -based dry etching to suspend the devices. To reveal the difference between the conventional and recessed structures, the devices based on these two structures with the same layout are fabricated. The layout of the fabricated A1 mode devices is shown in the microscope image [ Fig. 12(a)]. It is worth noting here that the release windows are added in the resonator's transverse directions for minimizing releasing radius, improving energy confinement, and suppressing transverse spurious modes [24], [33]. The SEM images of the fabricated conventional and recessed devices are shown in Fig. 12(b) and (c), respectively. The zoomed-in views clearly show the difference between the electrodes protruding off the LiNbO 3 surface [ Fig. 12(c)] and the electrodes situating in the recessed grooves [ Fig. 12(e)]. In the recessed designs, the thickness of LiNbO 3 under the electrodes is thinned down to be 560 nm, and the thickness of aluminum in the recessed grooves is 70 nm.

B. Admittance Responses
Three different groups of devices with different lateral dimensions are designed, and their parameters are listed in Table IV. In each group, both conventional design (t r = 0) and recessed design (t r = 90 nm) were fabricated to demonstrate the feasibility of our proposed method.
The fabricated devices were characterized at room temperature in the air with a Keysight N5249A PNA network analyzer. The comparisons of the measured responses based on three groups with different lateral dimensions are shown in Fig. 13. Consistent with our theoretical analyses, the devices based on the conventional design show several spurious responses with significant k 2 t , while the fundamental A1 features a low k 2 t (<20%). The resonant frequencies of the excited higher order A1 in the conventional designs are also consistent with (2) that third-order and fifth-order A1 feature higher resonant frequencies in the group with larger G.
On the other hand, all of the devices employing the recessed electrodes exhibit near spurious-free responses with a maximum Q 3 dB (quality factor at the resonance frequency) of 692 and k 2 t of 28% (Table IV). Good agreement is obtained between the measurement and the analysis. Consistent with the calculated dispersion curves [ Fig. 6(a)], A1 in the recessed design features higher resonant frequency than the conventional design. In addition to k 2 t and resonant frequencies, the recessed devices achieve high Qs, suggesting that the surface micromachining of LiNbO 3 does not pose a lower Q limit than the existing loss-inducing factors. Comparing these three recessed devices with the same W e , the Q 3 dB is higher for a greater G (Table IV). This is likely caused by the lower metal coverage and subsequently smaller mechanical loss from the metal.

IV. CONCLUSION
In this work, we have demonstrated a new method based on the dispersion matching to suppress the lateral spurious modes in LiNbO 3 A1 mode resonators. Based on the analysis of the lateral spurious modes, dispersion matching is identified as the key enabler for their suppression. It can be achieved by micromachining the LiNbO 3 thin film to form the recessed electrodes. The fabrication process for the recessed electrodes has been demonstrated. All fabricated devices based on the proposed method exhibit spurious-free responses with high Q (maximum of 692) and enhanced k 2 t (28%). The design variations show the broad applicability of our proposed structures. Upon further optimization, this method would advance LiNbO 3 A1 mode devices to become a promising signal processing solution in the next-generation 5G front-ends. Her research interests include design and microfabrication techniques of MEMS resonators, filters, and wireless communication systems.
Ms. Gao has won the Best Student Paper