Beam resonator

Abstract

Herein is provided a resonator element (100), comprising a plurality of beam elements (101) having a length (L) and a width (W), wherein the plurality of beam elements (101) are positioned adjacent to each other and adjacent beam elements are mechanically connected to each other, and wherein the width (W) of the plurality of beam elements (101) varies within the resonator element (100) and wherein the resonator element (100) is configured to alleviate a spurious resonance mode of the resonator element (100) through having the width (W) variation

Description

[0001] BEAM RESONATOR

[0002] TECHNICAL FIELD

[0003] The present disclosure generally relates to the field of semiconductors and semiconductor apparatuses. The disclosure relates particularly, though not exclusively, to beam resonators.

[0004] BACKGROUND

[0005] This section illustrates useful background information without admission of any technique described herein representative of the state of the art.

[0006] A key performance parameter in semiconductor apparatuses, such as resonators, such as silicon MEMS resonators is the equivalent series resistance (ESR). ESR is inversely proportional to the quality factor Q of the apparatus.

[0007] Typically, semiconductor apparatuses are configured to vibrate (oscillate) in a desired main resonance mode. In certain occasions, the apparatus may adopt another, unwanted resonance mode. The performance of the semiconductor apparatus is typically adversely affected by such unwanted resonance modes.

[0008] SUMMARY

[0009] The appended claims define the scope of protection. Any examples and technical descriptions of apparatuses, products and / or methods in the description and / or drawings not covered by the claims are presented not as embodiments of the invention but as background art or examples useful for understanding the invention. It is an object of certain embodiments of the present disclosure to provide a scheme to solve at least one of the problems related to the prior art, or at least to provide an alternative to existing technology. Accordingly, certain disclosed embodiments provide for an ingenious resonator element solving at least one of the problems related to the prior art.

[0010] According to a first example aspect of the present disclosure there is provided a resonator element, comprising a plurality of beam elements having a length and a width, wherein the plurality of beam elements are positioned adjacent to each other and adjacent beam elements are mechanically connected to each other, and wherein the width of the plurality of beam elements varies within the resonator element and wherein the resonator element is configured to alleviate a spurious resonance mode of the resonator element through having the width variation.

[0011] In certain embodiments, the plurality of beam elements are fabricated into having varying widths with respect to each other. In certain embodiments, (at least) some of the plurality of beam elements are of different width with each other. In certain embodiments, (at least) some of the adjacent beam elements of the plurality of beam elements are of different width with each other. In certain embodiments, the resonator element comprises beam width gradation (within the widths of the beam elements of the resonator element).

[0012] In certain embodiments, any two adjacent beam elements of the plurality of beam elements are of different width with (respect to, respective to, in comparison to) each other. In certain embodiments, any three adjacent beam elements of the plurality of beam elements are of different width with each other. In certain embodiments, any five adjacent beam elements of the plurality of beam elements are of different width with each other.

[0013] In certain embodiments, the resonator element is an asymmetric resonator element. In certain embodiments, the resonator element comprises asymmetric beam element width variation. In certain embodiments, the resonator element is an asymmetric resonator element in its width direction. In certain embodiments, the resonator element is an asymmetric resonator element with respect to its central axis parallel to its length direction. In certain embodiments, the resonator element is an asymmetric resonator element in terms of beam element width. In certain embodiments, the plurality of beam elements having varying widths are arranged (positioned) asymmetrically in the resonator element.

[0014] In certain embodiments, outermost beam elements of the resonator element have same widths. In certain embodiments, the outermost beam elements and a centermost (central, centre) beam element have same widths. In certain embodiments, outermost beam elements of the resonator element have same widths whilst all other beam elements of the resonator element have varying widths (amongst each other). In certain embodiments, the outermost beam elements and a centermost beam element have same widths whilst all other beam elements of the resonator element have varying widths.

[0015] In certain embodiments, the width of (each of) the plurality of beam elements varies with each other less than 20% of the average width of the beam elements. In certain embodiments, the width of (each of) the plurality of beam elements varies with each other less than 15% of the average width of the beam elements. In certain embodiments, the width of the plurality of beam elements varies with each other more than 1 % of the average width of the beam elements. In certain embodiments, the width of the plurality of beam elements varies with each other more than 5% of the average width of the beam elements. In certain embodiments, the width of the plurality of beam elements varies with each other more than 10% of the average width of the beam elements.

[0016] In certain embodiments, the difference in the width of the widest beam element and the narrowest beam element is less than 20%, such as less than 15% of the width of the widest beam element. In certain embodiments, the difference in the width of the widest beam element and the narrowest beam element is more than 5%, such as more than 10% of the width of the widest beam element.

[0017] In certain embodiments, each of the plurality of beam elements has a unique (different) width in comparison to each other.

[0018] In certain embodiments, the width of the beam elements varies from another beam element by 0.1 pm to 2 pm (a width difference being in between 0.1 pm and 2 pm). In certain embodiments, the width of any two beam elements of the resonator element varies from the other beam element by 0.3 pm to 0.5 pm. In certain preferable embodiments, the width of any two beam elements of the resonator element varies from the other beam element by 0.4 pm. In certain embodiments, the width of any two adjacent beam elements of the resonator element varies from the other adjacent beam element by 0.1 pm to 2 pm, such as 0.3 pm to 0.5 pm, preferably by 0.4 pm. In certain embodiments, the width of all beam elements of the resonator element varies from the other beam elements by a 0.1 pm to 2 pm, such as by 0.3 pm to 0.5 pm, preferably by 0.4 pm.

[0019] In certain embodiments, what has been defined concerning the widths of the beam elements apply to all beam elements, except to the outermost beam elements (the rightmost beam element and left-most beam element). In other words, in further embodiments, the definitions of the beam elements do not apply to the outermost beam elements. In certain embodiments, the resonator element is a stacked beam resonator element. In certain embodiments, the stacked beam resonator element comprises a plurality of beam elements positioned side-by-side in a plane. In certain embodiments, the plurality of beam elements are positions adjacent to each other in a width direction thereof. In certain embodiments, the plurality of beam elements are positioned adjacent to each other in a width direction of the resonator element. In certain embodiments, the beam elements are separated by trenches. In certain embodiments, the beam elements are connected to each other by connection elements. In certain embodiments, each beam element is mechanically connected to another beam element by (at least) two connection elements, and adjacent beam elements are separated by trenches. In certain embodiments, the trench between the adjacent beam elements is of (has) uniform width.

[0020] In certain embodiments, the resonator element comprises a plurality of beam elements, such as seven, nine, or eleven beam elements. In certain embodiments, said adjacent beam elements are mechanically connected to each other by connection elements. In certain embodiments, the beam elements of the resonator element are arranged in a rectangular array configuration.

[0021] In certain embodiments, the resonator element is (in a shape of) a rectangle. In certain embodiments, the resonator element has an aspect ratio (ratio of length to width, when observed from above) different from 1 . In certain embodiments, the resonator element has a length-to-width aspect ratio of less than 1 . In certain embodiments, the resonator element is attached (supported, anchored) to a support structure. In certain embodiments, the resonator element is attached to a support structure from the outermost beam elements of the resonator element.

[0022] In certain embodiments, each beam element is in a shape of a (rectangular) beam. In certain embodiments, each beam element has an aspect ratio (ratio of length to width, when observed from above) different from 1. In certain embodiments, each beam element has a length-to-width aspect ratio of more than 1 .

[0023] In certain embodiments, the resonator element is fabricated on a substrate. In certain embodiments, the substrate is a wafer. In certain embodiments, the substrate is a silicon- on-insulator, SOI, wafer. In certain embodiments, the substrate comprises a silicon layer.

[0024] In certain embodiments, the resonator element comprises a material stack, the material stack comprising the silicon layer, the piezoelectric layer on top of the silicon layer, and a top electrode on top of the piezoelectric layer. In certain embodiments, the resonator element is a piezoelectric resonator element. In certain embodiments, the resonator element is (part of) a semiconductor device. In certain embodiments, the resonator element is a microelectromechanical systems, MEMS, resonator element. In certain embodiments, the resonator element is configured to operate in a megahertz frequency area. In certain embodiments, the resonator element is sized to operate in a megahertz frequency area. In certain embodiments, the resonator element is configured to operate at 32 MHz frequency. In certain embodiments, the resonator element is sized to operate at 32 MHz frequency.

[0025] In certain embodiments, the resonator element comprises a resonating element. In certain embodiments, each beam element is a resonating beam element. In certain embodiments, each beam element is a sub-element of the resonator element.

[0026] In certain embodiments, the resonator element is configured to resonate (operate, oscillate, vibrate) in an in-plane resonance mode. In certain embodiments, the resonator element is adapted to resonate in a length extensional, LE, resonance mode. In certain embodiments, the resonator element is configured to operate in an in-plane length-extensional, LE, resonance mode. In certain embodiments, the length extensional resonance mode is configured to resonate parallel to the length direction of the resonator element. In certain embodiments, the length extensional resonance mode is configured to resonate perpendicular to the width direction of the resonator element.

[0027] In certain embodiments, the resonator element is configured to resonate in a collective resonance mode. In certain embodiments, each beam element of the resonator element is configured to resonate in the (same) collective resonance mode. In certain embodiments, the resonator element is configured to resonate in a desired (main) resonance mode. In certain embodiments, each beam element of the resonator element is configured to resonate in the (same) desired resonance mode.

[0028] In certain embodiments, the width of the plurality of beam elements varies within the resonator element to (mechanically) alleviate (lessen the effect of, remove, get rid of, suppress) a spurious resonance mode of the resonator element. In certain embodiments, the spurious resonance mode is an unwanted resonance mode. In certain embodiments, the spurious resonance mode is a resonance mode other than the desired (main) resonance mode. In certain embodiments, the spurious resonance mode resonates in a resonance mode other than the desired resonance mode.

[0029] In certain embodiments, the spurious resonance mode is a flexural resonance mode. In certain alternative embodiments, the spurious mode is a width-extensional resonance mode. In certain alternative embodiments, the spurious mode is a differential resonance mode, such as a differential length-extensional resonance mode.

[0030] In certain embodiments, the spurious resonance mode is an out of plane resonance mode. In certain alternative embodiments, the spurious resonance mode is an in-plane resonance mode.

[0031] In certain embodiments, the spurious resonance mode is an out of plane flexural resonance mode. In certain embodiments, the spurious resonance mode is an in-plane widthextensional resonance mode. In certain embodiments, the spurious resonance mode is an in-plane differential length extensional resonance mode.

[0032] In certain embodiments, the resonator element comprises more than one spurious resonance mode. In certain embodiments, the width of the plurality of beam elements varies within the resonator element to alleviate spurious resonance mode(s) of the resonator element.

[0033] In certain embodiments, the resonator element is configured to alleviate a disturbance (effect, harmful effect) caused by a spurious resonance mode of the resonator element by varying the width of the beam elements (width variation) within the resonator element. In certain embodiments, the resonator element is configured to alleviate a disturbance caused by a spurious resonance mode of the resonator element by varying the width of the beam elements asymmetrically within the resonator element. In certain embodiments, the resonator element is configured to alleviate the disturbance caused by the spurious resonance mode by modifying (splitting) equivalent series resistance, ESR, of said spurious resonance mode of the resonator element.

[0034] In certain embodiments, the resonator element is configured to eliminate a collective spurious resonance mode of the beam elements of the resonator element.

[0035] According to a second example aspect of the present disclosure there is provided a resonator assembly, comprising (at least) two (more than one) resonator elements of the first aspect coupled to each other. In certain embodiments, the resonator elements are coupled to each other by a coupler. In certain embodiments, the resonator elements are coupled to each other by a coupler comprising discontinuity region(s).

[0036] In certain embodiments, the resonator assembly comprises extensional-mode resonator elements. In certain embodiments, the resonator assembly comprises a flexural mode resonator. In certain embodiments, the resonator assembly comprises a mechanical connector element which connects the flexural resonator to the extensional-mode resonator elements.

[0037] In certain embodiments, each resonator element comprises a plurality of beam elements having a length and a width, wherein the plurality of beam elements are positioned adjacent to each other and adjacent beam elements are mechanically connected to each other, and wherein the width of the plurality of beam elements varies within the resonator element and wherein the resonator element is configured to alleviate a spurious resonance mode of the resonator element through having the width variation.

[0038] In accordance with certain embodiments, embodiments of the second aspect are provided, the embodiments comprising subject matter of any single embodiment presented in connection with the first aspect, or the embodiments comprising subject matter of any of the embodiments presented in connection with the first aspect combined with subject matter presented in any other embodiment or embodiments.

[0039] Different non-binding example aspects and embodiments have been illustrated in the foregoing. The embodiments in the foregoing are used merely to explain selected aspects or steps that may be utilized in different implementations. Some embodiments may be presented only with reference to certain example aspects. It should be appreciated that corresponding embodiments may apply to other example aspects as well. In particular, the embodiments described in the context of the first aspect are applicable to each further aspect. Any appropriate combinations of the embodiments may be formed.

[0040] BRIEF DESCRIPTION OF THE FIGURES

[0041] Some example embodiments will be described with reference to the accompanying figures, in which:

[0042] Fig. 1 schematically shows a top view of a resonator element demonstrating dimensions thereof according to an example embodiment;

[0043] Fig. 2 schematically shows a top view of a resonator element comprising seven beam elements with width variation according to another example embodiment;

[0044] Fig. 3 schematically shows a top view of a resonator element comprising 1 ...n beam elements according to yet another example embodiment;

[0045] Fig. 4 schematically shows a top view of a resonator comprising two resonator elements according to an example embodiment; Fig.5 schematically shows a top view of a resonator comprising two resonator elements according to another example embodiment;

[0046] Fig. 6 schematically shows a sectional view of a piezoelectrically actuated resonator according to an example embodiment;

[0047] Fig. 7a schematically shows modifying of frequency of a spurious resonance mode according to an example embodiment;

[0048] Fig. 7b schematically shows modifying of frequency of a spurious resonance mode according to another example embodiment;

[0049] Fig. 7c schematically shows an illustration of process imperfections according to an example embodiment;

[0050] Fig. 8a schematically shows a spurious resonance mode from top view according to an example embodiment; and

[0051] Fig. 8b schematically shows another spurious resonance mode from side view according to an example embodiment.

[0052] DETAILED DESCRIPTION

[0053] In the following description, like reference signs denote like elements or steps.

[0054] Fig. 1 schematically shows a top view (from above, from up to down) resonator element

[0055] 100 according to certain embodiments. The resonator element 100 comprises a plurality of beam elements 101 having a length L and a width W. In the embodiment shown in Fig. 1 , the resonator element 100 comprises seven beam elements 101 , 10T (the number of beam elements 101 may vary depending on the embodiment). In certain embodiments, the beams elements 101 are longer L than they are wide W. In certain embodiments, the coordinate system is selected so that the x-axis resides in the width direction W of the beam elements

[0056] 101 and the y-axis in the longitudinal direction L of the beam elements 101.

[0057] The plurality of beam elements 101 are positioned adjacent to each other. In certain embodiments, the plurality of beam elements 101 are positioned adjacent to each other in a width direction thereof. The adjacent beam elements 100 are mechanically connected to each other. In certain embodiments, the resonator element 100 is formed of the plurality of beam elements 101 and a plurality of connection elements 102. In certain embodiments, said adjacent beam elements 100 are mechanically connected to each other by connection elements 102. In certain embodiments, the adjacent beam elements 101 are separated by trenches 104. In certain embodiments, the trenches 104 have a length TL (trench length). In certain embodiments, the length L of the beam element 101 comprises at least the length of the trench TL and the length of at least one connection element 102.

[0058] In certain embodiments, the beam elements 101 of the resonator element 100 are arranged in a rectangular array configuration. In certain embodiments, the resonator element has a length L (which is equal to the length of the beam element). In certain embodiments, the resonator element has a width RW (resonator element width).

[0059] As shown in Fig. 1 , the outermost beam elements are denoted by numeral 101 ’. In all other ways they correspond to the said beam elements 101 , except that they are located in the outermost positions of the resonator element 100. In certain embodiments, the resonator element 100 is attached to a support structure (not shown). In certain embodiments, the resonator element 100 is attached to a support structure from the outermost beam elements 101 ’ of the resonator element 100 (anchoring point(s) 103).

[0060] In certain embodiments, the width W of the plurality of beam elements 101 varies within the resonator element 101 to alleviate a spurious resonance mode of the resonator element. In certain embodiments, a beam element 101 differs in width from another beam element 101. In certain embodiments, each of the beam elements 101 have their respective widths W. In certain embodiments, the width W of at least one beam element 101 (or a plurality of beam elements) differs from the width W of another beam element 101 .

[0061] In certain embodiments, the resonator element 100 is of an elongated shape (having the length L smaller that their width RW). In certain embodiments, the resonator element 100 is in the shape of a rectangle (the resonator element 100 has a shape of a rectangle). In certain embodiments, the resonator element 100 has an aspect ratio (ratio of length L to width RW, when observed from above) different from 1. In certain embodiments, the resonator element 100 has a length-to-width, L-to-RW, aspect ratio of less than 1.

[0062] In certain embodiments, the beam elements 101 are of an elongated shape (having their length L larger than their width W). In certain embodiments, each beam element 101 is in the shape of a rectangular beam. In certain embodiments, each beam element 101 has an aspect ratio (ratio of length L to width W, when observed from above) different from 1. In certain embodiments, each beam element 101 has a length-to-width, L-to-W, aspect ratio of more than 1 . Fig. 2 schematically shows a top view of a resonator element 100 according to certain embodiments. In the embodiment shown in Fig. 2, the resonator element 100 comprises seven beam elements 101 , 101 ’ (the number of beam elements may vary depending on the embodiment).

[0063] In certain embodiments, the resonator element 100 comprises width variation amongst the beam elements 101 , 101 ’. In certain embodiments, any two adjacent beam elements 101 of the plurality of beam elements are of different width W1 , W2, W3, W4, W5, W6, W7 with respect to each other (in comparison with each other). In certain embodiments, the resonator element 100 is an asymmetric resonator element (with respect to beam element widths W1 , W2, W3, W4, W5, W6, W7). In certain embodiments, the plurality of beam elements 101 having varying widths W1 , W2, W3, W4, W5, W6, W7 are positioned (arranged, placed) asymmetrically in the resonator element 100. By arranging beam elements 101 having varying widths W1 , W2, W3, W4, W5, W6, W7 asymmetrically in the resonator element 100, the risk of obtaining a collective spurious resonance mode is reduced.

[0064] In certain embodiments, outermost beam elements 101 ’ of the resonator element 100 have same widths. In certain other embodiments, the outermost beam elements 101 ’ and a centermost beam element (herein shown as beam element having width W4) have same widths (herein shown as W1 = W4 = W7).

[0065] In certain embodiments, the width W1 , W2, W3, W4, W5, W6, W7 of the beam elements 101 of the resonator element 100 varies in between 8 pm to 35 pm. In certain preferable embodiments, the width W1 , W2, W3, W4, W5, W6, W7 of the beam elements 101 of the resonator element 100 varies in between 20 pm to 26 pm, preferably in between 22 pm to 24 pm, more preferably in between 22.6 pm to 23.4 pm. In certain embodiments, the resonator element 100 comprises a gradation of beam elements 101 having varying widths W1 , W2, W3, W4, W5, W6, W7, wherein the width of the beam element 101 vary from another beam element 101 by a step in between 0.1 pm and 2 pm, such as by a step of 0.4 pm. It is, depending on the embodiment, beneficial to modify the widths W1 , W2, W3, W4, W5, W6, W7 of the beam elements 101 by a suitable amount (suitable width difference), such as between 0.1 pm and 2 pm, to reduce the risk of obtaining (adapting) a collective spurious resonance mode.

[0066] In certain embodiments, the widths W1 , W2, W3, W4, W5, W6, W7 of the plurality of beam elements 101 vary with each other less than 20%, such as less than 15% of the average width of the beam elements 101 , and more than 1%, such as more than 5% or more than 10% of the average width of the beam elements 101.

[0067] In certain embodiments, the resonator element 100 comprises a first group of beam elements 101 having a first width W. In certain embodiments, the first group of beam elements 101 are asymmetrically arranged (positioned) within the resonator element 100.

[0068] In certain embodiments, the resonator element 100 comprises a second group of beam elements 101 having a second width W. In certain embodiments, the first width and the second width are different from one another (the width of a beam element 101 in the first group differs from the width W of a beam element 101 in the second group). In certain embodiments, the second group of beam elements 101 are asymmetrically arranged within the resonator element 101 .

[0069] In certain embodiments, the resonator element 101 also comprises a third group of beam elements 101 having a third width W. In certain embodiments, the third group of beam elements 101 are asymmetrically arranged within the resonator element 100. In certain embodiments, the first width, the second width and the third width are all different from one another (the width W of a beam element 101 in the third group differs from the width W of a beam element 101 in the first group and in the second group).

[0070] In certain embodiments, the above analogously applies to any fourth, fifth, sixth... group of beam elements 101 of the resonator element 100 having a fourth, fifth, sixth... width. In certain embodiments, all groups of beam elements 101 are asymmetrically positioned within the resonator element 100. In certain embodiments, all groups of beam elements 101 have varying widths in comparison to other groups of beam elements 101.

[0071] In certain embodiments, each of the plurality of beam elements 101 has a unique width W1 , W2, W3, W4, W5, W6, W7 in comparison to each other.

[0072] Accordingly, there is provided a resonator element 100, comprising a plurality of beam elements 101 having a length L and a width W, wherein the plurality of beam elements 101 are positioned adjacent to each other and adjacent beam elements are mechanically connected to each other, and wherein the width W of the plurality of beam elements 101 varies within the resonator element 100 and wherein the resonator element 100 is configured to alleviate a spurious resonance mode of the resonator element 100 through having the width W variation Fig. 3 schematically shows a top view of a resonator element 100 comprising a plurality of adjacent beam elements 101 , herein shown as 1...n beam elements 101 in accordance with certain embodiments.

[0073] In certain embodiments, the resonator element 100 is a stacked beam resonator element. A stacked beam resonator element conventionally has a basic geometry in which beam elements with equal width being connected by connection elements that are positioned at resonator edges. The geometry of the resonator element of the present disclosure differs from this geometry by varying the widths of the beam elements.

[0074] In certain embodiments, the stacked beam resonator element 100 comprises adjacent beam elements 101 (adjacent sub-elements) positioned side-by-side in a plane. In certain embodiments, the resonator element 100 comprises an anchoring point 103 to anchor beam elements to surroundings. In certain embodiments, the resonance mode of the resonator element 100 has a nodal point at anchoring point 103.

[0075] In certain embodiments, the resonator element 100 comprises n adjacent beam elements 101 where the first and nth beam element 10T have a same width W, the same width W being different from the widths W of the other beam elements 101 . In certain embodiments, the resonator element 100 comprises n adjacent beam elements 101 wherein the first, nth and n / 2 beam element 101 have a same width W, the same width W being different from the widths W of the other beam elements 101 (wherein n is an odd number). In certain embodiments, the resonator element 100 comprises n adjacent beam elements 101 wherein the first, nth and the two n / 2 beam elements 101 have a same width W, the same width W being different from the widths W of the other beam elements 101 (wherein n is an even number).

[0076] In certain embodiments, the resonator element 100 comprises a resonating element. As used herein, the term resonating element refers to the part of the resonator configured to engage in resonance mode. In certain embodiments, the resonating element comprise a plurality of resonating beams (beam elements). As used herein, the term the resonator element refers to a structure comprising both resonating part(s), as well as stationary part(s), such as a support structure.

[0077] In certain embodiments, each beam element 101 is a resonating beam element. In certain embodiments, the resonator element 100 is configured to resonate in a desired (main) resonance mode. In certain embodiments, the desired resonance mode is a collective resonance mode. In certain embodiments, each beam element 101 of the resonator element 100 is configured to resonate in the same resonance mode.

[0078] In certain embodiments, the resonator element 100 is configured to operate in an in-plane length extensional, LE, resonance mode. In certain embodiments, the in-plane length extensional resonance mode is a main resonance mode of the resonator element 100. In certain embodiments, the beam elements 101 of the resonator element 100 are configured to operate in an in-plane length extensional resonance mode. The beam elements 101 (and thus the whole resonator element 100) oscillate in the LE-mode in the direction of the y-axis (in the direction of the length L direction, parallel to the longitudinal direction).

[0079] In certain embodiments, the width W of the plurality of beam elements 101 varies within the resonator element 100 to alleviate a spurious resonance mode of the resonator element 100. Typically, resonator element(s) 100 are configured to operate (resonate, vibrate, oscillate) in a desired (collective) main resonance mode. In certain occasions, the resonator element 100 adopts another, unwanted, spurious resonance mode. This unwanted spurious resonance mode typically comprises a lower ESR than the desired main resonance mode. Therefore, the unwanted spurious resonance mode typically dissipates less energy than the desired main resonance mode, making it eventually a dominant resonance mode for selfsustained oscillation. The performance of the semiconductor apparatus comprising the resonator element 100 is adversely affected by such unwanted, spurious resonance mode(s). In certain embodiments, the spurious resonance mode is a resonance mode other than the desired main resonance mode. In certain embodiments, the resonator element 100 comprises (resonates in) a spurious resonance mode (in addition to the main resonance mode).

[0080] Figs. 4 and 5 schematically show a resonator assembly comprising a plurality of resonator elements in accordance with certain embodiments. In certain embodiments, the resonator assembly comprises (at least) two resonator elements 100 (stacked beam resonator elements). All embodiments described in context of Figs. 1 , 2 and 3 for a single resonator element apply herein as well for the resonator elements 100 of the resonator assembly.

[0081] Fig. 4 schematically shows a resonator assembly comprising two resonator elements 100 coupled to each other in accordance with certain embodiments. The resonator assembly comprises (at least) two extensional-mode resonator elements 100, and a (one or more) flexural mode resonator 310. Further, the resonator assembly comprises a (one or more) mechanical connector element 320 which connects the flexural resonator 120 to the extensional mode resonator elements. In certain embodiments, at least one of the extensional-mode resonator elements 100 of the resonator assembly comprises a piezoelectric thin-film actuator for exciting the said extensional-mode resonator to a resonance mode and thereby the whole resonator assembly to a collective resonance due to mechanical coupling of the extensional-mode resonator elements 100.

[0082] In certain embodiments, more than 50% of the mass of the resonator assembly comprise material portions of single-crystalline silicon. In certain embodiments, the resonator assembly 100 comprises an electrostatic actuator for exciting at least one of the extensional-mode resonator elements 100 to a resonance mode and thereby the whole resonator element to a collective resonance due to mechanical coupling of the extensionalmode resonators.

[0083] Fig. 5 schematically shows a resonator assembly comprising two resonator elements 100 coupled to each other in accordance with certain embodiments. In certain embodiments, the resonator assembly comprises (at least) two resonator elements 100 coupled to each other by a coupler 330. In certain embodiments, the coupler 330 is a length-extensional coupler. In certain embodiments, the resonator elements 100 are implemented in the form of stacked beam resonators having a plurality of adjacent resonating beams connected by connection element(s) and separated by trenches (thus forming a ladder-like structure).

[0084] In certain embodiments, the coupler 330 comprises discontinuity regions 340. In certain embodiments, the discontinuity regions 340 of the coupler 330 to render the coupler 330 electrically inert.

[0085] In certain embodiments, (all) the resonator elements 100 of a resonator assembly are identical with one another (each other). In certain embodiments, the resonator elements 100 of a resonator assembly are identical with one another in terms of beam width gradation. As used herein, the term beam width gradation refers to how the beam widths differ from one another within the resonator element.

[0086] In certain embodiments, (all) the resonator elements 100 of a resonator assembly are different from one another. In certain embodiments, the resonator elements 100 of a resonator assembly are different from one another in terms of beam width gradation.

[0087] In certain embodiments, the resonator assembly comprises a plurality of resonator elements. In certain embodiments, the resonator assembly comprises a plurality of resonator elements positioned adjacent to each other. In certain embodiments, every odd numbered resonator element of the resonator assembly has the same, first beam width gradation with one another. In certain embodiments, every even numbered resonator element of the resonator assembly has the same, second beam width gradation with one another. In certain embodiments, the first beam width gradation is the second beam width gradation mirrored.

[0088] Fig. 6 schematically shows an example cross section (sectional view, side view) for piezoelectric actuated resonator element residing on a substrate.

[0089] In certain embodiments, the resonator element is fabricated on a substrate. In the example embodiment of Fig. 6, a silicon on insulator (SOI) substrate (wafer) 450 is used. The reference numerals 401 and 402 denote bottom electrode and top electrode contacts, respectively.

[0090] In certain embodiments, the resonator element 100 comprises (is formed of) a material stack. In certain embodiments, the resonator element 100 comprises a material stack, the material stack comprising at least the silicon layer L4, the piezoelectric layer L2 on top of the silicon layer, and a top electrode L1 on top of the piezoelectric layer.

[0091] In the example embodiment shown in Fig. 6, the top electrode is implemented in layer L1. In this example embodiment, layer L2 is a piezoelectric layer for piezoelectric actuation of the resonator element residing in the area of denoted by 100. An opening in L2 is denoted by 420. In this example embodiments, layer L3 denotes a layer for the bottom electrode. In this example embodiment, layer L4 is a silicon layer for the resonator element (beam elements and their connecting elements). In this example embodiments, layer L5 is a buried oxide layer (SiC>2) of the SOI wafer, and layer L6 is a silicon handle layer. In certain embodiments layer L6 comprises a cavity C1 . In certain embodiments, the layer L5 follows the cavity C1 shape as shown in Fig. 6.

[0092] In certain embodiments, when a doped silicon layer is used as L4, it is possible to leave out the separate L3 bottom electrode layer. In such embodiments, the conductive doped silicon layer L4 acts as the bottom electrode. In certain embodiments, the silicon layer L4 comprises degenerately doped silicon. In certain embodiments, more than 50 % of the silicon layer L4 mass consists of degenerately doped silicon. In certain embodiments, the silicon layer L4 is doped to an average impurity concentration of at least 2*1019cm-3, such as at least 102° cm-3. In certain embodiments, the silicon layer L4 comprises single crystalline silicon. In certain embodiments, the silicon layer L4 consists essentially of single crystalline silicon. In certain embodiments, the silicon layer L4 comprises degenerately doped single crystalline silicon. In certain embodiments, the longitudinal axis L of a beam element (or all beam elements) 101 is aligned with <100> crystal direction of the beam element (not shown), such as aligned with

[0100] crystal direction of the beam element, or deviates less than 25 degrees therefrom, or less than 15 degrees therefrom in certain embodiments. In certain preferable embodiments, the longitudinal axis L of a beam element (or all beam elements) 101 is aligned with <100> crystal direction of the beam element, such as aligned with

[0100] crystal direction of the beam element, or deviates less than 5 degrees therefrom, or less than 2 degrees therefrom in certain embodiments.

[0093] Figs. 7a and 7b schematically show an impedance Z (ohm) versus frequency f (MHz) graph demonstrating the modifying of a frequency of a spurious resonance mode according to certain embodiments. The spurious resonance frequency is a function of the beam width. When all beam elements of the resonator element are of same width, there is a possibility for the beam elements to resonate at the same frequency. On the other hand, when the beam elements of the resonator element comprise width variation, the beam elements are purposefully designed to resonate at various different frequencies on their own, with a higher equivalent series resistance, ESR.

[0094] As stated above, in certain embodiments, the resonator element is configured to alleviate a disturbance caused by a spurious resonance mode of the resonator element by varying the width of the beam elements (asymmetrically) within the resonator element. In certain embodiments, the resonator element is configured to alleviate the disturbance caused by the spurious resonance mode (shown as large peak R in the reference (REF) Z vs f graph) by modifying (splitting, dividing, adjusting, separating) said spurious resonance mode of the resonator element into (several) non-harmful resonance mode(s) by varying the width of the beam elements of the resonator element. The spurious resonance mode depends on the width dimension of the beam elements of the resonator element. In Figs. 7a and 7b reference (REF) resonator element comprises beam elements of same width (with a strong low equivalent series resistance, ESR signal). In Figs. 7a and 7b resonator element 100, the width of the plurality of beam elements varies within the resonator element 100 and the resonator element 100 is configured to alleviate a spurious resonance mode of the resonator element 100 through having the width variation.

[0095] In certain embodiments, the equivalent series resistance, ESR, of said spurious mode is adjusted such that the main resonance mode remains as a dominant resonance mode. In certain embodiments, the main resonance mode is an in-plane length extensional resonance mode. In certain embodiments, the spurious resonance mode(s) has an ESR value that is at least higher than the main resonance mode ESR value. In certain embodiments, the spurious resonance mode(s) has an ESR value that is at least 3 times higher than the main resonance mode ESR value. In certain embodiments, the main resonance mode has a lower ESR value than the spurious resonance mode(s).

[0096] In other words, in certain embodiments, the spurious resonance mode R having a lower ESR than the main resonance mode is modified (split) into (several) non-harmful spurious resonance mode(s) (as shown in Figs. 7a and 7b as small peaks R’1 , R’2, R’3, R’4, R’5 in the Z vs f graphs of the resonator element 100 of the present disclosure) having high ESR by varying the width of the beam elements of the resonator element. Said modifying prevents spurious resonance mode(s) from becoming the dominant resonance mode. Thus, the main resonance mode remains the resonance mode having the lowest ESR (the dominant resonance mode). In certain embodiments, the resonator element of the present disclosure is configured to eliminate a collective spurious resonance mode of the beam elements of the resonator element.

[0097] Fig. 7c schematically shows 1 / Z (dB) versus frequency f (MHz) graph demonstrating the process imperfections according to certain embodiments.

[0098] Small variation shown in the Figs. 7a and 7b REF and 100 Z vs. f graphs (the lines of the graphs not aligning perfectly on top of each other) are demonstrated in detail in Fig. 7c. Said small variations are typically caused by imperfections on the resonator element caused by manufacturing processes. These small imperfections are not enough to reliably separate the spurious resonance mode(s) into several peaks as shown in Figs. 7a and 7b.

[0099] As shown in Fig. 7c, peaks R1 and R2 show peaks from resonator structures comprising some variation in beam element widths due to manufacturing processes. Peak R3 shows a peak from a resonator structure having all beam elements perfectly equal widths.

[0100] Without limiting the scope and the interpretation of the patent claims, certain technical effects of one or more of the example embodiments disclosed herein are listed in the following. A technical effect is providing a resonator with no harmful disturbance caused by a spurious resonance mode. A technical effect is prevention of a spurious resonance mode from becoming a collective resonance mode. A further technical effect is avoiding harmful ‘locking’ of resonance mode (oscillations are self-sustained for this resonance mode) of a resonator element to a spurious resonance mode instead of the main resonance mode. The spurious mode typically dissipates less energy than the main resonance mode and has smaller ESR in comparison to the main resonance mode. A further technical effect is avoiding the spurious resonance mode from becoming a dominant resonance mode for self-sustained oscillation. The small ESR of the spurious resonance mode is split into several high ESR spurious resonance modes by varying the width of the beams of the resonator element, which renders the main resonance mode of the resonator element the smallest ESR resonance mode thus preventing the spurious resonance mode from becoming the dominant resonance mode.

[0101] The increase in the ESR of the spurious resonance modes is dependent on the resonance of the beam elements of the resonator element along their width dimension. In other words, the ESR of the spurious mode is inversely proportional to the number of beams resonating collectively (since the spurious resonance modes depend on the beam width). Thus, varying the width of the beam elements reduces the number of beam elements contributing to any resonance peak, thereby increasing the ESR. The ESR of the split spuriouses are designed to be higher than the main resonance mode. Thus, varying the width of the beam elements of the resonator element, the spurious resonance mode(s) can be alleviated.

[0102] It should be noted that minor marginal imperfections in width(s) of the beam elements caused by for instance errors during the resonator element manufacturing processes are not sufficient to reliably alleviate the spurious resonance mode(s) of the resonator element. In the present disclosure, the beam elements are intentionally designed and fabricated into varying widths.

[0103] A further technical effect is leaving the main resonance mode (in certain embodiments, the in-plane length extensional resonance mode) intact in terms of performance, as well as providing a resonator element with stable high performance.

[0104] The spurious resonance mode is an unwanted resonance mode (other than the desired main resonance mode). Figs. 8a and 8b schematically show examples of spurious resonance modes according to an example embodiment, in which the main resonance mode is an in-plane length extensional resonance mode. Fig. 8a schematically shows a spurious resonance mode from a top view. In the example embodiment shown in Fig. 8a, the spurious resonance mode is a width-extensional resonance mode, in particular an in- plane width-extensional resonance mode. In this example spurious resonance mode, the displacement occurs in-plane, but in width direction instead of the desired length direction.

[0105] Fig. 8b schematically shows a spurious resonance mode from a side view. In the example embodiments shown in Fig. 8b, the spurious resonance mode is a flexural resonance mode, in particular an out of plane flexural resonance mode. In this example spurious resonance mode, the displacement occurs out of plane, instead of the desired length direction. In certain alternative embodiments (not shown), the spurious mode is a differential resonance mode, such as an in-plane differential length-extensional resonance mode.

[0106] Various embodiments have been presented. It should be appreciated that in this document, words comprise, include, and contain are each used as open-ended expressions with no intended exclusivity.

[0107] The foregoing description has provided by way of non-limiting examples of particular implementations and embodiments a full and informative description of the best mode presently contemplated by the inventors for carrying out the invention. It is however clear to a person skilled in the art that the invention is not restricted to details of the embodiments presented in the foregoing, but that it can be implemented in other embodiments using equivalent means or in different combinations of embodiments without deviating from the characteristics of the invention.

[0108] Furthermore, some of the features of the afore-disclosed example embodiments may be used to advantage without the corresponding use of other features. As such, the foregoing description shall be considered as merely illustrative of the principles of the present invention, and not in limitation thereof. Hence, the scope of the invention is only restricted by the appended patent claims.

Claims

CLAIMS1 . A resonator element (100), comprising a plurality of beam elements (101 ) having a length (L) and a width (W), wherein the plurality of beam elements (101 ) are positioned adjacent to each other and adjacent beam elements are mechanically connected to each other, and wherein the width (W) of the plurality of beam elements (101 ) varies within the resonator element (100) and wherein the resonator element (100) is configured to alleviate a spurious resonance mode of the resonator element (100) through having the width (W) variation.

2. The resonator element of claim 1 , wherein any two adjacent beam elements (101 ) of the plurality of beam elements (101 ) are of different width (W) with each other.

3. The resonator element of claim 1 or 2, wherein the resonator element (100) is an asymmetric resonator element.

4. The resonator element of any preceding claim, wherein outermost beam elements (10T) of the resonator element have same widths (W).

5. The resonator element of claim 4, wherein the outermost beam elements (101 ’) and a centermost beam element (101) have same widths (W).

6. The resonator element of any preceding claim, wherein the width (W) of the plurality of beam elements (101 ) varies with each other less than 20%of the average width of the beam elements (101 ).

7. The resonator element of any preceding claim, wherein each of the plurality of beam elements (101) has a unique width (W) in comparison to each other.

8. The resonator element of any preceding claim, wherein the width (W) of the beam elements (101) varies from another beam element (101 ) by 0.1 pm to 2 pm.

9. The resonator element of any preceding claim, wherein the resonator element (100) is a stacked beam resonator element.

10. The resonator element of any preceding claim, wherein the resonator element (100) is configured to operate in an in-plane length-extensional, LE, resonance mode.11 . The resonator element of any preceding claim, wherein the resonator element (100) is configured to alleviate the disturbance caused by the spurious resonance mode by modifying equivalent series resistance, ESR, of said spurious resonance mode of the resonator element (100).

12. The resonator element of any preceding claim, wherein the resonator element (100) is configured to eliminate a collective spurious resonance mode of the beam elements (101 ) of the resonator element (100).

13. A resonator assembly comprising at least two resonator elements (100) of any of claims 1-12 coupled to each other.

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