PIEZOELECTRIC CROSS-SECTIONAL LAME MODE TRANSFORMER

CROSS REFERENCE TO RELATED APPLICATIONS

This application claims the priority of U.S. Provisional Application No. 62/240,934 filed October 13, 2015 and entitled "Piezoelectric Cross-Sectional Lame Mode Transformer", which is hereby incorporated by reference in its entirety.

STATEMENT REGARDING FEDERALLY SPONSORED RESEARCH OR DEVELOPMENT

This invention was developed with financial support from Grant Number DARPA N- ZERO-HR001 1-15-C-0138 awarded by the DARPA. The U.S. Government has certain rights in the invention

BACKGROUND

Micro Electro Mechanical (MEM) resonators are key enablers for the development of miniaturized and low-power multi-band radio-frequency (RF) systems capable of operating in the crowded modern commercial and military spectral environment. The use of micro- and nano-electromechanical systems to form low-loss passive filters presents challenges as a result of their still low Figure of Merit. Because of this reason, pre-existing resonator technologies, such as Surface Acoustic Wave (SAW), are still preferred despite the impossibility of monolithically integrating them on the same chip than the rest of the electronic components.

SUMMARY

In view of the foregoing, the present inventors have recognized and appreciated the advantages of a transformer with the features described herein to be employed in future RF components including micro- and nano- electromechanical systems.

In one aspect, the instant invention provides a piezoelectric transformer comprising: a piezoelectric layer having first and second opposed surfaces extending in a length direction and a width direction, and having a thickness, T, between the opposed surfaces;

an input portion comprising: an input interdigital electrode comprising a plurality of input conductive strips disposed on the surfaces of the piezoelectric layer, each of the plurality of input conductive strips extending in the length direction and having a width, W _in , extending in the width direction, and

an input port configured to receive an input signal and in electrical communication with the input interdigital electrode; and

an output portion comprising:

an output interdigital electrode comprising a plurality of output conductive strips disposed on the surfaces of the piezoelectric layer and interdigitated with the plurality of input conductive strips, each of the plurality of output conductive strips extending in the length direction and having a width, W _ou t, extending in the width direction, and

an output port configured to transmit an output signal and in electrical communication with the output interdigital electrode;

wherein the input portion and the output portion are configured to excite a two- dimensional Lame mode of vibration in the piezoelectric layer in response to an input voltage applied to the input port to produce a voltage gain at the output port.

In another aspect, the instant invention provides a method of making the above- described piezoelectric transformer, the method comprising lithographically patterning the input interdigital electrode and the output interdigital electrode in contact with the surfaces of the piezoelectric layer to provide a desired resonant frequency range.

Yet another aspect of the invention is a method of producing a voltage gain comprising: (a) providing the transformer of claim 1 ; and (b) operating the transformer to provide the voltage gain.

Another aspect of the invention is a method of using a cross-sectional Lame mode resonator as a voltage-transformer.

Yet another aspect of the invention is a micro-electromechanical device including the above-described transformer.

The invention can also be summarized with the following list of embodiments.

1. A micro-electromechanical piezoelectric transformer comprising:

a piezoelectric layer having first and second opposed surfaces extending in a length direction and a width direction, and having a thickness, T, between the opposed surfaces;

an input portion comprising:

an input interdigital electrode comprising a plurality of input conductive strips disposed on the surfaces of the piezoelectric layer, each of the plurality of input conductive strips extending in the length direction and having a width, W _in , extending in the width direction, and

an input port configured to receive an input signal and in electrical communication with the input interdigital electrode; and

an output portion comprising:

an output interdigital electrode comprising a plurality of output conductive strips disposed on the surfaces of the piezoelectric layer and interdigitated with the plurality of input conductive strips, each of the plurality of output conductive strips extending in the length direction and having a width, W _ou t, extending in the width direction, and

an output port configured to transmit an output signal and in electrical communication with the output interdigital electrode;

wherein the input portion and the output portion are configured to excite a two- dimensional Lame mode of vibration in the piezoelectric layer in response to an input voltage applied to the input port to produce a voltage gain at the output port.

2. The transformer of embodiment 1 , wherein the piezoelectric layer comprises at least one of aluminum nitride, lithium niobate, lithium tantalate, zinc oxide, gallium nitride, scandium nitride, and quartz.

3. The transformer of embodiment 1 , wherein the piezoelectric layer comprises aluminum nitride.

4. The transformer of embodiment 1 , wherein the input portion and the output portion are asymmetrical, the asymmetry configured to produce the voltage gain.

5. The transformer of embodiment 1 , wherein the plurality of input conductive strips are interdigitated with the plurality of output conductive strips with a pitch, P, in the width direction, and the pitch is so configured to maximize an electromechanical coupling coefficient for the resonator

6. The transformer of embodiment 5, wherein the pitch is in a range of about 50 nm to about 100 μηι.

7. The transformer of embodiment 5, wherein the thickness of the piezoelectric layer is about equal to the pitch.

8. The transformer of embodiment 5, wherein the thickness of the piezoelectric layer is equal to the pitch within about 0.5%.

9. The transformer of embodiment 1 , wherein the resonant frequency of the transformer is lithographically defined. 10. The transformer of embodiment 1 , wherein input and output portions are electrically asymmetrical.

1 1. The transformer of embodiment 10, wherein an effective capacitance at the input port differs from an effective capacitance at the output port.

12. The transformer of embodiment 1 , wherein the input port and the output port are mechanically asymmetrical.

13. The transformer of embodiment 12, wherein the input conductive strips and the output conductive strips differ in one or more of strip size, width, length, and area.

14. The transformer of embodiment 1 , wherein an acoustic wavelength in the width direction is about equal to the thickness of the peizoelectric layer.

15. The transformer of embodiment 1 , wherein the voltage gain is equal to or larger than about 100 for quality factors greater than about 2000.

16. The transformer of embodiment 1 , wherein the thickness of the piezoelectric layer is from about 50 nanometers to about 100 micrometers.

17. The transformer of embodiment 1 , wherein e31 and e33 piezoelectric coefficients of the piezoelectric layer are coherently combined to excite the two-dimension al Lame mode of vibration.

18. The transformer of embodiment 1 , wherein a frequency of the Lame mode of vibration ranges from about 1 MHz to about 100 GHz.

19. A method of making the piezoelectric transformer of embodiment 1 , the method comprising, lithographically patterning the input interdigital electrode and the output interdigital electrode in contact with the surfaces of the piezoelectric layer to provide a desired resonant frequency range.

20. A method of producing a voltage gain comprising:

(a) providing the transformer of embodiment 1 ;

(b) operating the transformer to provide the voltage gain.

21. Use of a cross-sectional Lame mode resonator as a voltage-transformer.

22. A device comprising the transformer of embodiment 1.

23. The device of embodiment 22, wherein the device is an AC-DC converter or a DC-DC converter.

BRIEF DESCRIPTION OF THE DRAWINGS FIG. 1A is a schematic representation of thickness field excitation (TFE) for cross- sectional Lame mode resonators (CLMRs) and degenerate cross-sectional Lame mode resonators (dCLMRs), according to one embodiment.

FIG. 1 B is a schematic representation of lateral field excitation (LFE) for CLMRs and dCLMRs, according to one embodiment.

FIG. 2 depicts the distribution of the resonance frequency (f _r ) of a CLMR as a function of the acoustic wavelength along the lateral direction (λ _χ ), according to one embodiment.

FIG. 3 depicts distributions of displacements ½ and ¾ simulated through finite element analysis (FEA) along cut lines (BB' and BB") in the cross-section of a CLMR.

FIG. 4 depicts piezoelectric coupling coefficient k simulated through two- dimensional FEA, according to one embodiment.

FIG. 5 depicts for TFE CLMRs of different T _A | _N /A _X and mode number n values, according to one embodiment.

FIG. 6 depicts k for LFE CLMRs of different T _A | _N /A _X and mode number n values, according to one embodiment.

FIG. 7 compares resonance frequencies f _r simulated through FEA with resonance frequencies obtained analytically for TFE dCLMRs, as the acoustic wavelength varies, according to one embodiment.

FIG. 8 compares resonance frequencies f _r simulated through FEA with resonance frequencies obtained analytically for LFE dCLMRs, as the acoustic wavelength varies, according to one embodiment.

FIG. 9 depicts the resonance frequency f _r simulated through FEA as the coverage of the interdigitated electrode(s) a varies, for TFE CLMRs and LFE CLMRs, according to one embodiment.

FIG. 10 depicts the he distribution of the shown as the coverage of the interdigitated

FIG. 1 1 depicts the steps of fabricating a TFE CLMR by using a four-mask microfabrication process, according to one embodiment.

FIG. 12 depicts an SEM image of a fabricated AIN TFE CLMR, according to one embodiment.

FIG. 13 depicts the distribution of admittance amplitude as a function of frequency for the AIN TFE CLMR of FIG. 12. FIG. 14 depicts the simulated k _t ² , the simulated admittance Y, and the simulated mode- shape of total displacement for a TFE CLMR, according to one example.

FIG. 15 depicts the simulated k _t ² , the simulated admittance Y, and the simulated mode- shape of total displacement for a LFE CLMR, according to one embodiment.

FIG. 16 depicts the steps of fabricating a LFE CLMR by using two-mask microfabri cation process, according to one embodiment.

FIG. 17 depicts a scanning electron microscope (SEM) image of a fabricated AIN LFE CLMR, according to an embodiment.

FIG. 18 depicts the distribution of admittance amplitude as a function of frequency for the AIN LFE CLMR of FIG. 17.

FIG. 19 depicts a schematic representation of a 2-port cross-sectional Lame mode transformer (CLMT).

FIG. 20 depicts the simulated open-circuit voltage gain of CLMTs assuming a quality factor of 5000, a pitch of the IDTs equal to 5 μηι and when varying the thickness of the AIN plate between 1 μηι to 7 μηι.

FIG. 21A depicts an equivalent circuit model of a CLMT.

FIG. 21 B depicts the Thevenin's equivalent used to compute the voltage V _out a the input of the electrical transformer in FIG. 22A.

FIG. 21 C depicts a 2-port Y-matrix representation adopted to compute V ₀ Vm and G _v from the circuit shown in FIG. 22B.

FIG. 22A depicts simulated G _v ^(max) as a function of W and assuming a Q| _0a d-value equal to 1000.

FIG. 22B depicts simulated frequency of operation (f _res ) as a function of W and assuming a Qioad-value equal to 1000.

FIG. 23 shows 2D-FEA simulated Y _in - and G _v - as functions of the frequency (f) of the input signal, V _in and assuming a Q| _0a d-value equal to 1000.

FIG. 24 shows a scanned-Electron-Microscope picture of a fabricated CLMT.

FIG. 25 shows measured Y _in relative of the fabricated CLMT (red); analytically predicted Y _in using the Q| _0a d, C _in , C _out , k _t ² and η values measured experimentally (black). Measured G _v of the fabricated CLMT (orange); and analytically predicted G _v (blue).

DETAILED DESCRIPTION OF THE INVENTION Following below are more detailed descriptions of various concepts related to, and embodiments of, micro-electromechanical resonators and methods of producing the same. It should be appreciated that various concepts introduced above and discussed in greater detail below may be implemented in any of numerous ways, as the disclosed concepts are not limited to any particular manner of implementation. Examples of specific implementations and applications are provided primarily for illustrative purposes.

MEM resonators have been researched for their ability to attain high quality factors (Q) and large electromechanical coupling coefficients (k _t ² ) in small volumes, thereby achieving desirable Figure of Merit. As used herein, k ² is a measure of the conversion efficiency between electrical and acoustic energy in piezoelectric materials; Figure of merit is the product of the quality factor (Q) and k ² . The Figure of Merit may directly determine the motional resistance in a resonator, impact oscillator design by setting the required gain (i.e., power consumption) and phase noise of oscillator, and impact filter design by setting insertion loss in properly terminated filters and device bandwidth.

Aluminum Nitride (AIN) Film-Bulk-Acoustic resonators (FBAR) can replace off-chip surface acoustic wave (SAW) devices in commercial products, hence significantly reducing the form-factor of the RF front-ends. The AIN FBAR technology relies on the e ₃₃ piezoelectric coefficient of AIN to transduce resonant vibration along the thickness of an AIN plate. Since the device resonance frequency (f _r ) is set by the thickness of the AIN plate (T _A IN), the resonance frequency cannot be tuned lithographically. Thus, integration of multi-frequency FBAR based filters on a same chip may increase the fabrication complexity (i.e., by mass-loading or trimming).

AIN contour-mode resonator (CMR) technology allows lithographical tuning of the resonance frequency. The AIN CMR relies on the e ₃₁ piezoelectric coefficient of AIN to transduce resonant vibration along an in-plane direction of an AIN plate (e.g., width extensional or length extensional motion). The lithographically set lateral dimension of the device determines the resonance frequency of the device. Therefore, CMRs operating in the Ultra- High- (UHF) and Very-High- (VHF) frequency ranges can be fabricated on a same chip. However, the electromechanical coupling coefficient (k _t ² ) of AIN CMRs is lower than that of FBARs due to the intrinsically lower amplitude of the e ₃₁ piezoelectric coefficient compared to the e ₃₃ . Thus, FBAR-based filters are still preferred to the CMR-based ones for the implementation of low insertion loss and wideband passive filtering networks. Cross-Sectional Lame mode resonators

Resonators disclosed herein are a new class of AIN MEM resonators based on the piezoelectric transduction of a Lame mode or a degenerate Lame mode in the cross-section of an AIN plate, called cross-sectional Lame mode resonators (CLMRs) or degenerate cross- sectional Lame mode resonators (dCLMRs). The CLMRs and dCLMRs rely on a coherent combination of the e ₃₁ and e ₃₃ piezoelectric coefficients of AIN to transduce a two dimensional (2D) mechanical mode of vibration, which is characterized by longitudinal vibrations along both the width and the thickness of the AIN plate. For the CLMRs, the peak-to-peak displacement along the width direction is the same as the peak-to-peak displacement along the thickness directions. For the dCLMRs, the peak-to-peak displacements along the width direction and the thickness direction are different. CLMRs and dCLMRs can achieve high values of electromechanical coupling coefficient, for example, as high as 7%. In addition, since such a 2D mode of vibration depends on the lateral dimensions of the plate, CLMRs/dCLMRs operating at different resonance frequencies can be lithographically defined on a same substrate without requiring additional fabrication steps. Thus, both high electromechanical coupling coefficient and integration of multi-frequency and low insertion-loss filters on a same chip can be achieved without additional costs and fabrication complexity.

Referring to FIG. 1A, a schematic representation of thickness field excitation (TFE) for CLMR/dCLMR is shown, according to one embodiment. The CLMR/dCLMR may be part of nano-electromechanical or micro-electromechanical based RF systems to be used for applications such as sensing and communication. The TFE CLMR/dCLMR in FIG. 1A includes a piezoelectric plate 1 11 , a top interdigitated electrode 1 12 disposed on a top surface of the plate 1 11 , and a bottom interdigitated electrode 1 16 disposed on a bottom surface of the plate 1 11. The plate 1 11 may have any suitable dimensions. Depending on the geometry of the resonator, the term "dimension" may refer to any dimension of interest in the resonator. For example, the dimension may refer to thickness, width, height, length, diameter, radius, etc. According to one embodiment, the plate may have a thickness of less than or equal to about 10 microns - e.g., less than about 9 microns, about 8 microns, about 7 microns, about 6 microns, about 5 microns, about 4 microns, about 3 microns, about 2 microns, about 1 micron, about 900 nm, about 800 nm, about 700 nm, about 600 nm, about 500 nm, about 400 nm, about 300 nm, about 200 nm, about 100 nm, or less. According to another embodiment, the plate is a nano-plate, referring to a plate whose largest dimension is in the nanometer range, such as less than or equal to about 1 micron, or any of the aforedescribed values with respect to dimension. According to another embodiment, nano-plate may refer to a plate with at least one dimension in the nanometer range, such as less than or equal to about 1 micron, or any of the aforedescribed values with respect to dimension. According to another embodiment, the plate is a micro-plate, referring to a plate whose largest dimension is in the micrometer range, such as less than or equal to about 1 mm. According to another embodiment, micro-plate may refer to a plate with at least one dimension in the micrometer range, such as less than or equal to about 1 mm.

The plate 1 11 may be made of any suitable piezoelectric material. According to one embodiment, the plate 11 1 may include a compound, such as a nitride, such as an aluminum nitride (AIN). According to another embodiment, the plate may include at least one of aluminum nitride, lithium niobate, lithium tantalate, zinc oxide, gallium nitride, scandium nitride, and quartz.

The top electrode 1 12 and the bottom electrode 1 16 of the resonator may be made of any suitable material. According to one embodiment, the electrodes 1 12 and 116 may include a metal, such as a noble metal, such as platinum or gold. In some embodiments, the top electrode 1 12 and the bottom electrode 1 16 are in direct contact with the piezoelectric plate 1 11. In some embodiments, at least one of the top electrode 1 12 and the bottom electrode 116 is not in directed contact with the piezoelectric plate 11 1. The top electrode 1 12 may include multiple conductive strips, each strip having a width \N _e and arranged with a pitch p from adjacent strips (i.e., p is the combination of the width of a strip and the width of the space between two adjacent strips). The bottom electrode 1 16 has an identical pattern as the top electrode 1 12. As shown, two electric field components can be excited simultaneously, Ei along the lateral direction and E ₂ along the thickness direction. Ei has field lines connecting adjacent conductive strips forming the top interdigitated electrode 1 12. E ₂ has field lines connecting two conductive strips facing each other, one conductive strip from the top interdigitated electrode 112 and the other from the bottom interdigitated electrode 1 16.

Referring to FIG. 1 B, a schematic representation of lateral field excitation (LFE) for CLMR/dCLMR is shown, according to one embodiment. The LFE CLMR/dCLMR in FIG. 1 B includes a piezoelectric plate 121 and a top interdigitated electrode 124 disposed on a top surface of the plate 121. The plate 121 is similar to the pate 1 11 of FIG. 1A. The top electrode 124 is similar to the top electrode 112 of FIG. 1A. Different than the TFE CLMR/dCLMR, the LFE CLMR/dCLMR does not include a bottom electrode. Therefore, electrical field component Ε _Ϊ along the lateral direction can be excited but not electrical field component along the thickness direction. As will be discussed in greater detail below, LFE CLMR/dCLMR may simplify the fabrication of the device at the expense of a lower k _t ² , comparing to TFE CLMR/dCLMR. Experiments demonstrated a k _t ² of about 2.5% and a quality factor Q of about 1900 for a LFE CLMR/dCLMR operating at 2.8 GHz, resulting in a k _t ² Q product (i.e. Figure of Merit (FoM)) close to 46.

Principle of operation of cross-sectional Lame mode resonators (CLMRs)

When an alternating voltage is applied to the interdigitated electrode(s), a cross- sectional Lame mode of vibration may be excited in the resonators shown in FIGS. 1A-B. Starting from the equations of motion in solids, the Lamb wave theory can identify all modes of vibration that can be described by a one dimensional (1 D) displacement vector in a plate. Thus, the modes of vibration can be characterized by a predominant displacement along one geometrical dimension of the plate, whose size determines the frequency of operation. Different from 1 D modes of vibration, CLMRs/dCLMR shows comparable displacements along two dimensions, for example, displacement along the lateral direction (JjQ and displacement along the thickness direction (JQ of the plate 1 11 or 121 of FIG. 1. For the CLMRs, the peak-to-peak displacement along the lateral direction is the same as the peak-to-peak displacement along the thickness directions. For the dCLMRs, the peak-to-peak displacements along the lateral direction and the thickness direction are different. Equation (1) is the general expression of displacements along the lateral (¾) and thickness (JQ directions for the CLMR:

cos(^ _x x) sin(fi _z z)

- sin(j¾.x) cos _z z). (1), wherein β _χ and β _ζ are the wave-vectors relative to the motion along the x- and z-directions.

The resonance frequency (f _r ) of the CLMR can be obtained by solving the equations of motion (Equations (2) and (3)), which describe the distribution of the x- (μ _χ ) and z- displacements (μ _ζ ) in an AIN plate, with proper boundary conditions.

^C " ^^ ^{+ C} 55 ^^ + 2C ₁₅ ^ _r + (C ₅₅ + C ₁₃ ) ^ _r = p - (2 _r ) - μ _χ (2),

^C i5 + ^C *5 + C ₃₃ -^-Γ + (C ₅₅ + C ₁₃ ) = ρ ^■ (2nf _r ) ^■ μ _ζ (3), wherein are components of the stiffness matrix of the piezoelectric material of which the plate is made (e.g., AIN) and p is the mass density of the piezoelectric material.

By setting μ _χ and μ _ζ in Equations (2) and (3) to _j t½ and JT _Z in Equation (1), respectively, Equations (2) and (3) are simplified as:

Cii/?* ² - (C ₁₃ + C ₅₅ ) j¾ + C ₅₅ /? _z ² = p ^■ (2nf _r y (4) C^z ² - (C ₁₃ + ^ β _χ β _ζ + ϋ ₅₅ β _χ ² = p ^■ (2nf _r ) ² (5). Equations (4) and (5) have four sets of (β _χ , β _ζ ) solution, but only one set corresponds to positive wave-vectors along both vibrational directions, hence enabling the excitation of the cross-sectional Lame mode (CLM) in the resonator. The set of β _χ , _ζ ) is:

B p (2π/ _Γ ) ²

(6),

C11C33-C55 ^ HAB = £¾ ⁽⁷⁾ '

wherein A = (C _lt - C ₅₅ )(C ₁₃ + C ₅₅ ) ² and B = (C ₃₃ - c ₅₅ ).

All sides of AIN CLMRs behave as stress-free boundaries. Thus, _/ ζ and satisfy the following boundary conditions:

Wx ^X =— WAIN (8) ^J .'

K = ¹ ™ AIN (9),

wherein n is the mode number along the x-direction, m is the mode number along the z- directions, W _MN is the width of the AIN plate, which is equivalent to n ^■ p (See FIGS. 1A-B), and T _AlN is the thickness of the AIN plate. Analysis below discusses the vibration mode with the minimum vibrational order allowed by both TFE and LFE CLMRs (i.e. n equal to 2 and m equal to 1). In this scenario, W _MN coincides with the acoustic wavelength along the x-direction (λ _χ ), and λ _χ =2ρ.

By comparing Equation (8) and Equation (6), the resonance frequency, f _r , of the CLMR

By combining Equations (7) through (9), the following A _X /T _A | _N ratio should be satisfied in order to enable excitation of the CLM in the AIN plate:

(11).

^T AIN

FIG. 2 illustrates the distribution of the resonance frequency f _r as a function of the acoustic wavelength λ _χ . The curve in FIG. 2 represents the analytically found distribution; the dots represent the simulated f _r value through finite element analysis (FEA) as A _x varies between 1 and 10 μηι. T _ALN has been adjusted, for each A _x value, to satisfy Equation (1 1). FIG. 2 also shows the geometry used to compute f _r and the two A _x values experimentally explored (i.e., λ _χ = 3 μηι for the "LFE experiment" and A _x = 8 μηι for the "TFE experiment"). The analytically found distribution of f _r with respect to λ _χ shows excellent agreement with the simulated trend through FEA.

FIG. 3 depicts distributions of ¾ and simulated through FEA along cut lines (BB' and BB") in the cross-section of a CLMR. In the simulation, a W _AlN of 1.5 μηι (mode number n=1) is used and a resonator quality factor Q of 2500 is assumed. The displacement reaches the maximum at (x,z) = (0.75 μηι, 0.75 μηι). As expected, for CLMR, the simulated distributions of½ and JT _Z show the same peak-to-peak value (See Equation (1)) along cut lines BB' and BB".

The resonator may be configured to resonate at any appropriate frequency. According to one embodiment, the resonator resonates at a frequency of at least about 10 MHz, e.g. , at least about 50 MHz, about 100 MHz, about 200 MHz, about 300 MHz, about 400 MHz, about 500 MHz, about 600 MHz, about 700 MHz, about 800 MHz, about 900 MHz, or higher. According to another embodiment, the resonator resonates at a frequency of about 10 MHz to about 100 GHz, e.g. , about 50 MHz to about 90 GHz, about 100 MHz to about 80 GHz, about 200 MHz to about 70 GHz, about 300 MHz to about 60 GHz, about 400 MHz to about 50 GHz, etc.

For MEM piezoelectric resonators, piezoelectric coupling constant (K ² ) identifies the maximum electromechanical coupling coefficient (k _t ² ) for a specific mode of vibration. k ² is a measure of the conversion efficiency between electrical and acoustic energy in piezoelectric resonators. In particular, k ² represents the portion of the electrical energy, stored in the resonator's static capacitance (C ₀ ), which is converted into mechanical motion (i.e. , proportional to the ratio between motional and static capacitances, C _m /C ₀ ). k ² of a resonator can be directly extracted from its electrical response as:

2 _ Ιΐ ² fp ^~ fs _ . m (12

1 ~ 8 fs ² ~ 8 C ₀ ^

wherein f _p is the parallel resonance frequency, and f _s is the series resonance frequency of the resonator.

The K ² of a first order CLMR (i.e. , fundamental Cross-Sectional Lame mode of vibration, n and m both equal to 1) can be analytically derived as discussed below. Since CLMRs displace along both lateral and thickness directions, their piezoelectric coupling coefficient is the sum of two components. One component (¾?) is produced by the lateral motion (i.e., ¾) whereas the second component (K ) is originated from the thickness vibration (i.e., jQ. Since the electric field is exclusively distributed across the thickness of the AIN plate (z-oriented), the K ² associated with each of the two displacement components of the CLM can be computed by solving its corresponding Christoffel equation. The so found K ² value (Kc _hr ) is an effective measure of the actual piezoelectric coupling coefficient if the mode-shape is uniform along the directions orthogonal to the vibrational one. In this scenario K ² _ff can be approximated as:

If the resonator were to displace along the x-direction, with displacement distribution, μ _χ , not uniform along the z-direction (i.e. μ _χ = cos(^ _x x) ^■ S(z)), its effective K ² (K ² _rr ) would be lower than ¾ _r - l ^{n tne} CLM case where ¾ and JT _Z are not uniform along thickness and width of the AIN plate, K _x and K _z are computed as:

^i / _c [ sin^ _z z)dz = ^K ² _hr[x] (14),

_K 2 ₌ ^ ^ _{5ίη(βχΧ άχ =} - ² K _Chr[z] (15), wherein ^anc ' ^K cnr[z] ^{are tne} piezoelectric coupling coefficients derived from

Christoffel equation, respectively, for the lateral and vertical longitudinal vibration components of the CLM.

^K chr[x] ^{and K} chr[z] ^are computed as:

wherein C _lat is the effective stiffness coefficients of the CLMR along the x-direction, C _thic is the effective stiffness coefficient of CLMRs along the z-directions, e ₃₁ is an AIN piezoelectric coefficient equal to -0.58 C/m ² , e ₃₃ is another AIN piezoelectric coefficient equal to 1.55 C/m ² , and ε is the AIN dielectric permittivity.

Applying Equations (6) and (7), C _lat and C _thic are:

Ctnic = Ciat TT^{ = 300 GPa (19). Therefore, for CLMs excited in AIN plates, K ² (K ² _CLM ) is:

K ² CLM = ^ . ^K Chr[x] + ^K Chr[z\ ) = ⁷ - ^{1 %} ( ²⁰ ) -

The K ² CLM value obtained from Equation (20) can be verified by the piezoelectric coupling coefficient k _t ² , which is extracted from its simulated admittance (See Equation (12)), through 2D-Finite Element Analysis (FEA), as shown in FIG. 4. A schematic representation of the specific CLMR geometry used in the simulation is also shown in FIG. 4. In particular, a CLMR characterized by a W _A IN of 1.5 μηι and a T _A LN of 1.45 μηι (See Equation (1 1)) is used. A resonator length (L) equal to 50 μηι and no damping is assumed. In addition, the presence of the IDTs is neglected by exciting the devices through infinitesimal excitation boundaries. As evident from A schematic representation of the specific CLMR geometry used in the simulation is also shown in FIG. 4., the FEM computed electromechanical coupling coefficient closely matches the value derived through the simplified analytical model (See Equation (20)). The validity of Equation (20) is hence confirmed.

FEM

A comparison between and the previously reported K values for AIN FBARs and AIN CMRs is shown in Table 1.

Table 1. Comparison of piezoelectric coupling coefficient of different types.

A. Higher-order CLMs in AIN thin plates

The performance of a first order CLMR (i.e., fundamental Cross-Sectional Lame mode of vibration, n and m both equal to 1) is discussed above. In order to reach small values of input impedance, piezoelectric resonators often recur to higher-order vibrational modes (i.e. n > 1) excited through the use of conductive interdigitated structures (IDTs). For a conventional S ₀ mode Lamb-wave, the pitch of the IDTs used to excite the S ₀ mode Lamb-wave resonators may define the acoustic wavelength and, consequently, the frequency of operation. However, when higher order CLMs are excited in AIN plates, the resonance frequency and electromechanical performance depend on both λ _χ and T _A IN (See Equation (1 1)). In particular, when the resonance frequency of the Lamb-wave S ₀ mode propagating along the lateral direction (i.e. x) matches the resonance frequency of the longitudinal mode propagating along the thickness direction (i.e. z), a large electromechanical coupling coefficient can be expected through the excitation of a CLM in the AIN plate. This phenomenon can be studied, through Finite Element Analysis, by extracting of the S ₀ mode for different Τ _Α | _Ν /λ _χ values and for a representative set of n- values. This can be done through the identification, for each investigated case, of the metallized (f _m ) and the non-metallized (f ₀ ) frequencies. Once f _m and f ₀ are found, k can be obtained through Equation (12) by replacing f _s with f _m and f _p with f ₀ . Referring to FIG. 5, k£ according to Equation (12) for TFE CLMRs of different Τ _Α | _Ν /λ _χ and n values is shown. In calculating k£ , the presence of the metallic IDTs is neglected.

T FEM

Curve 501 corresponds to k£ changing with Τ _Α | _Ν /λ _χ for mode number n=2; curve 502 for n=5; curve 503 for n=15, curve 504 for n=40; and curve 505 for n→∞. In simulating the case related to n→ oo, periodic boundary conditions are applied to both displacement and electric fields, at the lateral edges of a second order CLMR (i.e., W _A | _N = λ _χ ).

Referring to FIG. 6, according to Equation (12) for LFE CLMRs of different Τ _Α | _Ν /λ _χ

T FEM

and n values is shown. In calculating k£ , the presence of the metallic IDTs is neglected.

T FEM

Curve 601 corresponds to k£ changing with Τ _Α | _Ν /λ _χ for mode number n=2; curve 602 for n=5; curve 603 for n=15, curve 604 for n=40; and curve 605 for n→∞. In simulating the case related to n→ oo, periodic boundary conditions are applied to both displacement and electric fields, at the lateral edges of a second order CLMR (i.e., W _A | _N = λ _χ ).

As shown, a maximum k£ value (k£ ) is found at a Τ _Α | _Ν /λ _χ that approaches the optimum value (-0.5) enabling the excitation of a CLM in the AIN plate (See Equation (11)), for both the TFE and the LFE.

For the case of TFE shown in FIG. 5, k^ ^max reduces as n increases. This reduction originates from the increase of the parasitic capacitance produced by the lateral component of the modal distribution of the electric field. In particular, k _t ² for a higher-order vibrational mode

(N), k? ^n~N , can be related to k _t ² for n equal to 1 ,/^ ^η_1 . The relationship between k^ ^n~N and k\ ^n_1 is as follows.

, ₂ n=JV N-Cfif ¹ _ N-Cfif ¹ _ j _r 2 ^{n= 1} 1

wherein C^ ⁼¹ is the resonator motional capacitance when n is equal to 1 , C ₀ ^n=N is the resonator static capacitance when n is set to N, C ₀ ^V and Co ^at are the capacitances associated with the electric field modal distributions along the thickness and lateral directions, respectively. According to Equation (21), the negative impact of Co ^at on the device kt2 is minimum when N is equal to 1 and reaches maximum when N approaches infinity. In particular, when N→ oo, k ^n~N can be rewritten as:

It is worth noting that the value of Co ^at depends on T _A | _N /A _X . In particular, Co ^at is minimum for T _A | _N /A _X → 0 while increases for larger T _A | _N /A _X values. Nevertheless, when Τ _Α | _Ν /λ _χ → 0.5, Co ⁱ is still smaller than C ₀ ^V . Therefore, boosting of the k associated with the excitation of the CLM in the structure is not significantly compromised. In addition, the Τ _Α | _Ν /λ _χ value at which the k _t ² value reaches the maximum slightly lowers when n is increased. This may be due to the different sensitivity of Ci _a t and C _t hic to Co ^at , which originates, in piezoelectric plates, from the different piezoelectric coefficients involved in the motion along the lateral and thickness directions.

For the case of LFE shown in FIG. 6, a much lower sensitivity of with respect to n is found. This is mostly due to the fact that Co ^at remains significantly larger than C ₀ ^V for all T _A | _N /A _X values and is independent of n.

Degenerate cross-sectional Lame mode resonators (dCLMRs)

As discussed above, CLMRs/dCLMRs have displacements along two dimensions, for example, displacement along the lateral direction (JjQ and displacement along the thickness direction (JT _Z ) of the piezoelectric plate. For CLMRs, the peak-to-peak displacement along the lateral direction is the same as the peak-to-peak displacement along the thickness directions. For dCLMRs, the peak-to-peak displacements along the lateral direction and the thickness direction are different.

CLM excitation can be enabled when the ratio of acoustic wavelength to the thickness of the piezoelectric plate (A _¾ T _A IN) satisfies Equation (11). For the minimum vibrational order, the acoustic wavelength is two times the pitch of the interdigitated electrode(s) (i.e., λ _χ = 2p). For dCLMRs, the ratio of the acoustic wavelength (i.e., the pitch of the interdigitated electrode(s)) to the thickness of the piezoelectric plate no longer satisfies Equation (1 1). dCLMRs can be formed by stationary motion along the thickness and lateral directions of the plate and are possible for a much wider range of T _A | _N /A _x values. The mode degeneracy is enabled by the special dispersive properties of AIN that allow the lateral coupling of each conductive strip of the IDTs that vibrates along the thickness direction. When the coupling happens, stationary waves occur along both directions of the plate and consequently, the electromechanical performance and resonance frequency depend on the motion along both directions.

Equation (10) shows that as the value of acoustic wavelength (i.e., the pitch) changes, the resonance frequency would change accordingly. Thus, the dCLMRs may achieve various resonance frequencies by varying the pitch of the interdigitated electrode(s) in lithographic processes without the need to change the thickness of the piezoelectric plate. As discussed in greater detail below, it is possible for dCLMRs to achieve k _t ² values that are comparable to the best k _t ² values attained in the non-degenerate CLMRs in a broad range of lithographically defined operating frequencies (i.e., different A _x -values for a given T _A IN)- Thus, it is possible to use dCLMRs to achieve, simultaneously, high k _t ² and a lithographic tunability of f _r .

As discussed above, The mode degeneracy is enabled by the special dispersive properties of AIN that allow the lateral coupling of each conductive strip of the IDTs that vibrates along the thickness direction. The dispersion equation for two-dimensional modes of vibration relates the wave-vectors relative to both directions (See Equations (8) and (9)) to the resonance frequency as follows:

wherein v _eq is an equivalent sound velocity for the two-dimensional motion. Equation (23) can be used to describe the resonance frequency, f _r , of degenerate CLMRs as a function of λ _χ . To do so, v _eq is calculated. v _eq can be estimated as if the small difference of sound

velocities relative to the motion along the two directions is neglected. f _r in Equation (23) may be set equal to the resonance frequency of non-degenerate CLMRs in Equation (10). Although Equation (10) is obtained for CLMRs in which Equation (11) is strictly satisfied. The validity of such approximation is valid for dCLMRs because the difference between C _lat and C _thic is small

(See Equations (18) and (19)). Assuming that v _eq is almost independent of for a limited range of - ^{τ 1} - values around the optimum (-0.5), f _r can be calculated as: f _r ^ ^—- |i + f¾) ² . (24).

^{J r} 2Ύ _ΑίΝ } V λ _χ ) ^ 4p

Referring to FIG. 7, resonance frequencies f _r simulated through FEA is compared with resonance frequencies obtained analytically for TFE dCLMRs, as the acoustic wavelength λ _χ varies. Curve 701 represents the simulated f _r , and curve 702 represents the analytical ^ with the presence of interdigitated electrode(s) neglected in both approaches. The thickness of the AIN plate (T _A | _N ) is 4 μηι, which is the same value as used in the TFE experiment in FIG. 2. λ _χ varies between about 7 μηι and about 11 μηι. It can be seen from FIG. 7 that the analytical f _r closely matches the simulated f _r . FIG. 7 also shows that degenerate TFE CLMRs achieve a lithographic frequency shift of -350 MHz around 1.2 GHz by varying λ _χ from about 7 μηι to about 11 μηι. Such variation does not lead to substantial decrease of k _t ² .

Referring to FIG. 8, resonance frequencies f _r simulated through FEA is compared with resonance frequencies obtained analytically for LFE dCLMRs, as the acoustic wavelength λ _χ varies. Curve 801 represents the simulated f _r , and curve 802 represents the analytical ^ with the presence of interdigitated electrode(s) neglected in both approaches. The thickness of the AIN plate (TAIN) is 1 .5 μηι, which is the same value as used in the LFE experiment in FIG. 2. λ _χ varies between about 2.4 μηι and about 3.6 μηι. It can be seen from FIG. 8 that the analytical f _r closely matches the simulated f _r . FIG. 8 also shows that degenerate LFE CLMRs achieve a lithographic frequency shift of -670 MHz around 3.2 GHz by varying λ _χ from about 2.4 μηι to about 3.6 μηι. Such variation does not lead to substantial decrease of k _t ² .

Frequency tuning through metal coverage variation

The resonance frequency of the CLMRs may be tuned by changing the coverage (i.e., a =We/p) of the interdigitated electrode(s). As used herein, the coverage of the interdigital electrode(s) refers to a ratio of the width of each conductive strip (l/l _e ) to the pitch of adjacent conductive strips (p). As both ½ and JT _Z change along the x-direction, the equivalent mass density (p ^(e<7) ) of CLMRs is a function of the coverage a. As a varies, the effective sound velocity of the CLMRs would change and, consequently, the operating frequency of the device would change. In other words, the effective sound velocity and resonance frequency of the resonators can be lithographically changed by varying l/l _e for a given p.

Referring to FIG. 9, resonance frequency f _r simulated through FEA is shown for two- finger (i.e., n=2) TFE CLMRs and LFE CLMRs, as the coverage of the interdigitated electrode(s) a varies. Curve 901 is for the TFE CLMRs, and curve 902 is for the LFE CLMRS. CLMRs having two fingers may not achieve a device input impedance of 50 Ω. However, the input impedance of 50 Ω may be achieved through the adoption of resonator arrays. In computing f _r , it is assumed that the top and bottom IDTs of the TFE CLMRs are made of platinum and each have a thickness of 0.07 _A IN- The top IDTs of the LFE CLMRs are made of aluminum and have a thickness of 0.07 _A | _N . The simulation results show that a lithographic frequency variation of ~ 30% can be attained on the TFE CLMRs, while the resonance frequency of the LFE is less sensitive to a. Larger frequency tuning can be attained by using thicker platinum IDTs on the TFE CLMRs.

Referring to FIG. 10, the distribution of the k£ is shown as the coverage of the interdigitated electrode(s) a varies for TFE CLMRs and LFE CLMRs of FIG. 9. Curve 1001 is for the TFE CLMRs, and curve 1002 is for the LFE CLMRs. As discussed above, a lithographic frequency variation of -30% can be attained on TFE CLMRs using platinum IDTs (as in the experimental demonstration), and larger frequency tuning can be achieved if thicker platinum

IDTs are used. At the same time, a larger than 5% can be achieved on the TFE CLMRs for such high frequency tuning. In contrast, for the LFE CLMRs using aluminum IDTs, both the resonance frequency and k£ are less sensitive to a (as in our experimental demonstration). This may be due to the larger impact that platinum electrodes have on the effective sound velocity (v _eq ) of the device comparing to aluminum electrodes.

Fabrication of CLMRs/dCLMRs and experimental results

The CLMR/dCLMRs described herein may be fabricated by appropriate microfabrication process. According to another embodiment, the resonators may be fabricated by a four-mask microfabrication process. The four-mask fabrication process may include: disposing a first electrode layer over a substrate, patterning the first electrode layer to form a bottom interdigitated electrode, disposing a piezoelectric layer over the substrate, disposing a second electrode layer over the piezoelectric layer, patterning the second electrode layer to form a top interdigitated electrode, etching the piezoelectric layer to form a piezoelectric micro-plate, and releasing the micro-plate from the substrate. The substrate may comprise, or be, any suitable material, such as silicon. According to one embodiment, the disposing of the electrode layers may include any suitable process, such as a sputter deposition process. According to one embodiment, the patterning of the electrode layers to form interdigitated electrodes may include any suitable process, such as a lift-off process. According to one embodiment, the disposing of the piezoelectric layer may include any suitable process, such as a sputter deposition process. The piezoelectric layer may include any suitable material, such as the aforedescribed piezoelectric materials. According to one embodiment, the etching of the piezoelectric layer to form a piezoelectric micro-plate may include any suitable process, such as an ion conductive plasma (ICP) process. The forming of the micro-plate may include forming a perimeter of the nano-plate. According to one embodiment, the releasing the piezoelectric layer from the substrate may include any suitable process, such as an isotropic etching process.

According to one embodiment, the resonators may be fabricated by a two-mask microfabrication process. The two-mask fabrication process may include: disposing a piezoelectric layer over a substrate, disposing an electrode layer over the piezoelectric layer, patterning the electrode layer to form an interdigitated electrode, etching the piezoelectric layer to form a piezoelectric micro-plate, and releasing the micro-plate from the substrate. The substrate may comprise, or be, any suitable material, such as silicon. According to one embodiment, the disposing of the piezoelectric layer may include any suitable process, such as a sputter deposition process. The piezoelectric layer may include any suitable material, such as the aforedescribed piezoelectric materials. According to one embodiment, the disposing of the electrode layer may include any suitable process, such as a sputter deposition process. According to one embodiment, the patterning of the electrode layer to form an interdigitated electrode may include any suitable process, such as a lift-off process. According to one embodiment, the etching of the piezoelectric layer to form a piezoelectric micro-plate may include any suitable process, such as an ion conductive plasma (ICP) process. The forming of the micro-plate may include forming a perimeter of the nano-plate. According to one embodiment, the releasing the piezoelectric layer from the substrate may include any suitable process, such as an isotropic etching process.

TFE CLMRs were fabricated and their performance was experimentally characterized. A device was formed by a 4 μηι thick AIN layer and two 0.1 μηι thick platinum IDTs, one IDT disposed on the top surface of the AIN layer and the other IDT disposed on the bottom surface of the AIN layer. Platinum was chosen as the bottom IDT material due to the need of growing a high quality AIN film, and as the top IDT material in order to preserve high acoustic symmetry in the cross-section of the device. The coverage of the platinum IDTs (a) was 0.5. The pitch of the IDTs setting (i.e., λ _χ /2) was chosen so as to match the AIN thickness.

Referring to FIGS. 11 (a)-(e), steps of fabricating TFE CLMRs are shown according to one embodiment. The LFE CLMRs were fabricated using a four-mask microfabrication process. At step (a), a 100 nm thick Pt film was deposited on top of a high resistivity Silicon wafer 1 120 through a 10 nm thick Ti layer used as adhesion layer, and was patterned through a lift-off process to form the bottom IDT 1 110. At step (b), a 4 μηι thick AIN film 1 130 was sputter- deposited on the wafer. The AIN film 1 130 may be wet etched to form vias. At step (c), a 100 nm thick Pt film was deposited on the AIN film 1130 through a 10 nm thick Ti layer, and was patterned through a lift-off process to form the top IDT 1 140. At step (d), the AIN film 1 130 was etched by ICP in Cl ₂ based chemistry to define the width of the AIN plate. This was done through the use of a hard mask made out of 2 μηι of Si0 ₂ so as to attain steep AIN sidewall angles (>75°). At step (e), the Silicon substrate underneath the resonator was released through XeF ₂ isotropic etching. The SEM image of the fabricated device is shown in FIG. 12.

The electrical performance of the fabricated TFE CLMRs were measured, in air, by connecting the device, through Ground-Signal-Probes (GSG), to a commercial Network Analyzer (Agilent E5071C). Referring to FIG. 13, the measured admittance of the device and its fitting to the Butterworth-van Dyke (BVD) model are shown - they match well. The TFE CLMR used for the measurement has a T _A IN of 4 μηι and a λ _χ of 8 μηι. The measured admittance was de-embedded from the directly extracted parasitic capacitance due to the probing pads (11 fF). The TFE CLMR shows a main resonant peak at about 1.02 GHz, a quality factor, Q, about 600, and a k _t ² larger than 4.6 %. This k _t ² value is higher than the k _t ² of other piezoelectric resonators using I DTs to actuate and sense mechanical vibrations in AIN plates. The experimental demonstration of such a high k _t ² confirms the advantages of the disclosed CLMR technology, which is enabled by the combined use of the AIN e ₃₁ and e ₃₃ piezoelectric coefficients.

The fabricated device showed a relatively lower quality factor than the quality factor of other AIN resonators operating in the same frequency range. It may mostly be caused by a large amount of energy lost through anchor dissipations. It is believed that a refinement of the anchor design would help maximize the quality factor attained by TFE CLMRs. It is worth noting that the selected pitch size (4 μηι) did not yield the highest possible k _t ² value (-5.7% for the given material stack and mode order number). This might be due to the presence of the metallic I DTs which causes a slight variation of the optimum T _A | _N /A _X value for the optimum transduction of the CLM and, consequently, the achievement of the highest electromechanical coupling coefficient. This was confirmed in FIG. 14.

FIG. 14 shows the simulated k _t ² as a function of the acoustic wavelength λ _χ for the TFE CLMRs. In the simulation, AIN TFE CLMR with a thickness of 4 μηι is used. The I DTs are 0.1 μηι thick platinum and the coverage a is set to 0.5. From Fig. 14, the highest F EM

kj; (kt -5.7%) was attained at a pitch size of 5 μηι (i.e. λ _χ =10 μηι). It is also worth noting that a k^ higher than 4% can be achieved for a relatively large range of λ _χ values, thus not only when Equation (1 1) is strictly satisfied. As discussed above, this fact is enabled by the capability of exciting degenerate Lame modes in AIN plates. The ]^ ^max js larger than what is predicted when the presence of the metallic I DT is neglected. This may be due to the fact that the I DTs lower both the effective stiffnesses in the lateral (C _lat ) and thickness (C _thic ) directions, thus effectively increasing ¾ir[x] and ¾ _r[z] . FIG. 14 also shows the simulated admittance and the simulated mode-shape of the displacement relative to two adjacent fingers for λ _χ = 8 μηι.

LFE CLMRs were fabricated and their performance was experimentally characterized. For example, a device was formed by a 1 .5 μηι thick AIN layer and a 0.2 μηι thick aluminum I DT disposed on the top surface of the AIN layer. Different than the fabricated TFE CLMR discussed above, aluminum instead of platinum, which has a lower resistivity and a sound velocity closer to that of AIN, was chosen to be the I DT material. The coverage of the aluminum interdigitated electrode (a) was 0.5.

FIG. 15 shows the simulated k _t ² as a function of the acoustic wavelength λ _χ for the LFE CLMRs. FIG. 15 also shows the simulated admittance and the simulated mode-shape of the displacement relative to two adjacent fingers for λ _χ = 3.2 μηι. Similar to the TFE CLMRs discussed above, due to the presence of the metallic I DT used to excite the electric field in the AIN layer, the highest k ² value is expected at a T _A IN _x value that is slightly different from what was found when the presence of the metal layers is neglected. Consequently, in order to maximize k ² , λ _χ is adjusted through FEA. FIG. 15 shows that maximum k ² is attained at λ _χ close to 3.2 μι ι.

Referring to FIG. 16(a)-(d), steps of fabricating an AIN LFE CLMR are shown according to one embodiment. The LFE CLMRs were fabricated using a two-mask microfabrication process. At step (a), a 1.5 μηι thick AIN film 1650 was sputter-deposited on top of a high resistivity silicon wafer substrate 1630. At step (b), a 200 nm thick Al film was deposited on top of the AIN film and patterned through etching to define the top interdigitated electrode 1610. At step (c), the AIN film was etched by ICP in Cl ₂ based chemistry to define the width of the AIN plate. At step (d), the silicon substrate underneath the AIN resonator was released through XeF ₂ isotropic etching. An SEM image of the fabricated device is shown in FIG. 17.

The electrical performance of the fabricated LFE CLMR was measured, in air, by connecting the device, through Ground-Signal-Ground probes (GSG), to a commercial Network Analyzer (Agilent E5071C). Referring to FIG. 18, the measured admittance of the device and its fitting to the Butterworth-van Dyke (BVD) model are shown - they match well. The LFE CLMR used for the measurement has a T _A | _N of 1.5 μηι and a λ _χ of 3.2 μηι. The LFE CLMR shows a main resonant peak at about 2.82 GHz, a quality factor, Q, about 1855, and a k _t ² about 2.48% which closely matches the FEM predicted value in FIG. 15. This k ² value is slightly larger than what is predicted when the presence of the metallic IDT is neglected. It may be due to the fact that the aluminum lowers both the effective stiffnesses in the lateral (C _lat ) and thickness (C _thic ) direction, thus leading to higher and ¾ _r[z] . The measured k ² and Q results in a k ² -Q product (i.e. Figure of Merit (FoM)) close to 46, which is higher than the FoM of any LFE AIN MEM resonators disclosed before that employed a single top IDT fabricated through a 2-mask fabrication process.

Cross-Sectional Lame mode transformers and experimental results

To date, transformers are generally large, lossy and hardly implementable at high frequency; hence, their use in low-power integrated applications has not be taken into consideration so far. In another aspect, the instant invention provides a novel class of MicroElectroMechanical (MEM) piezoelectric transformers capable of achieving high open- circuit voltage-gain in excess of 100 (for quality factors greater than 2000) between input and output of 2-port AIN CLMRs (FIG. 19). CLMRs with two physical ports can also be made by actuating and sensing the motion through different portions of top and bottom IDTs. In particular, when such portions are formed by the same number of identical strips and are electrically terminated on equal load (Z| _0ad ), 2-port CLMRs are reciprocal and symmetric. However, when these conditions are not verified, they behave as transformers, hence showing different voltage and current levels at their input and output ports.

Fig. 19 illustrates an embodiment of a transformer 2010 with an input portion 2012 and an output portion 2014 disposed on a piezoelectric layer 2013 having first and second opposed surfaces extending in a length direction and a width direction. The piezoelectric layer (AIN in Fig. 19) has a thickness T (TAIN in Fig. 19) between the opposed surfaces. The input portion includes an input interdigital electrode 2016 with of a plurality of input conductive strips 2018 disposed on the surfaces of the piezoelectric layer. Each of the plurality of input conductive strips extends in the length direction and has a width, W _in , extending in the width direction. The input portion 2012 also includes an input port 2020 configured to receive an input signal and in electrical communication with the input interdigital electrode. The output portion 2014 includes an output interdigital electrode 2022 with a plurality of output conductive strips 2024 disposed on the surfaces of the piezoelectric layer and interdigitated with the plurality of input conductive strips 2018. Each of the plurality of output conductive strips 2024 extends in the length direction and has a width, W _ou t, extending in the width direction. The output portion 2014 also includes an output port 2026 configured to transmit an output signal and in electrical communication with the output interdigital electrode. The input portion and the output portion are configured to excite a two-dimensional Lame mode of vibration in the piezoelectric layer in response to an input voltage applied to the input port to produce a voltage gain at the output port. The input conductive strips can be interdigitated with the output conductive stripe with a pitch P in the width direction.

The transformation mechanism is originated from mechanical or electrical asymmetries altering the sound-velocity at the output-port with respect to its value at the input port. For instance, the use of different electrical loads connected at input and output ports produces a variation in the effective capacitance that, because of the piezoelectric effect, stiffs the device at each port. As a result, the magnitude of the displacement field at the two ports is significantly different. In particular, CLMTs show a larger displacement at the input port than that at the output port. As their insertion-loss is negligible, a lower displacement value at their output port must result into a larger voltage level (V _ou t) than that applied at the input port (V _in ). Similarly, the introduction of mechanical asymmetries, such as the use of different strip size for the input and output portions of the IDTs, would also produce a discrepancy in the effective sound velocity relative to input and output ports, hence producing a voltage-gain. Differently from any other Lamb-wave piezoelectric transformer demonstrated to date, CLMTs use the special characteristics of two-dimensional modes of vibration to achieve much larger Gv than those based on contour-modes of vibration in piezoelectric plates, which was verified through Finite Element Analysis (FEA).

FIG. 20 shows that when the pitch of the I DTs is chosen to be close to the AIN thickness, Gv increases significantly with respect to what achieved by contour-mode resonators, which operate at much lower Τ _Α | _Ν /λ _χ values. G _v is directly proportional to the device figure of merit (FoM), which is defined as the product between quality factor (Q) and k _t ² , and it is maximum when the output port is left floating. A simulation was conducted of the voltage level at the output terminal of a 920 MHz CLMT formed by a 4 μηι thick AIN film sandwiched between two 100 nm thick platinum I DTs. The pitch of the I DTs was fine-tuned (\ΛΑ-5μηι) so as to take into consideration the change of the T _A IN A _x value that enables the excitation of a non- degenerate Lame mode when metallic electrodes are also used to actuate the motion in the device.

An experiment was conducted with simulated G _v value when assuming different values of quality factor, Q. As expected, G _v increases proportionally with respect to the assumed quality factor. In particular, a G _v value as high as 140 is expected by assuming a quality factor close to 4400, hence approaching the maximum demonstrated in AIN resonators operating in the same frequency range.

The electrical behavior of a CLMT can be modelled using an equivalent circuit (FIG. 21). This circuit has been widely used to analyze the performance of previously developed piezoelectric transformers. The equivalent circuit is composed by a series combination of motional resistance (R _m ), motional inductance (L _m ) and motional capacitance (C _m ), two shunt capacitances representing the device input and output capacitances (C _in and C _ou t, respectively) and an electrical transformer. The turn-ratio (η) of the electrical transformer maps the ratio between the transduction coefficients relative to the output- and the input-ports. The open-circuit voltage gain (G _v ) can be defined as the ratio between V _ou t and V _in , when the output port is kept open.

Vout is related to the voltage (V _ou t) across the input-port of the electrical transformer (FIGS. 21A-B) as follows:

V ₀ ut ^— V _out ^■ η (25) To compute an analytical expression for G _v , we first expressed V _ou t (FIGS. 21A-B) as a function of the magnitude of the input voltage, V _in , and of the frequency, f . To do so, we replaced the electrical transformer in FIG. 21 with the Thevenin impedance seen at its input-port (FIG. 21 B). The resulting circuit in FIG. 21 B was then represented through a 2-port admittance matrix (Y-matrix). The use of such representation permits to easily compute the expression of out/ _in and, by using Eq. (25), the desired expression of G _v as follows: ^out

« —

ψΥ ₂ :

8k (2nf _res )

(26)

(_ ₍ 2 _{+ (} 2 t) ²

Qload

From Eq. (26), it is straightforward to find the maximum G _v -value (G _v ^(max) ), which is attained when f is equal to the resonance frequency (i.e. f _res ) relative to the device input- admittance (Yj _n ). The expression of G _v ^(max) is reported in Eq. (27). (max) _ kt QloadCjn ²

v — o _r

^{O L} out

Y _in was instead computed as: where Yn , Y ₁₂ , Y21 , and Y22 are the coefficients of the equivalent Y-matrix (FIG. 21 C) relative to the circuit shown in FIG. 21 B. It is important to point out that G _v ^(max) (Eq. (27)) is proportional to the value of the resonator figure-of-merit (FoM oc Qi _oad ^■ k*) , which further highlights the advantage of CLMTs over transformers based on other laterally vibrating piezoelectric resonator technologies.

A. Simulated Performance of 920 MHz 3-fingers CLMTs through Finite Element Simulations

The design of a CLMT operating around 920 MHz was conducted through 2D-FEA. In particular, the dependence of both f _res (FIG. 22A) and G _v ^(max) (FIG. 22B) from the pitch of the I DT (i.e. W) was investigated. Maximum G _v ^(max) -value was attained when W was set to 5.5 μηι. Such W-value (W ^(tx) ) permits to maximize the input-/( _f ² . In addition, sensitivity of G _v ^(max) to the width of the metal-strip (W _e i ^(out) ) forming the output portion of the IDT was detected. Therefore, in order to attain the highest G _v ^(max) -value, W _e i ^(out) was chosen to be about half μπι) of the width of the metal strips forming the input portion of the IDT μηι). As verified by FEA, this choice enables a -10% improvement in the G _v ^(max) -value compared to the case. This improvement is originated from a slight increase of the device k ² . In addition f _re s was found to decrease proportionally to W (FIGS. 22A-B). Furthermore, a k ² in excess of 4.1 % was estimated by 2D-FEA (FIG. 23). Such a high /Rvalue enables a G _v ^(max) -value in excess of 40 when assuming a Q| _0a d-value equal to 1000.

B. Measured Performance of a 920 MHz 3-finger CLMT

A SEM-picture of a fabricated CLMT is shown in FIG. 24. The performance of a 3-finger 920 MHz CLMT was characterized (FIG. 25). The device was tested, in air, through a RF probe-station using GSG-probes. Values of k _t ² , Q _\03d and G _v ^(max) in excess of 4%, 1000 and 39, respectively, were directly extracted from the measured Y-matrix. A value of η 0.23 was instead experimental variation, Af _res , of f _res

and the analytical model (Figure 6 and Eqs. 2-3), based on the experimentally extracted values of k _t ² , Qioad, /?, Cm and C _out , fit well the measured data. However, a small variation of f _re s was measured with respect to its predicted value through FEA. This discrepancy is due to fabrication uncertainties associated with the actual thickness of the IDTs with respect to the value considered in the FEA.