MEMS resonator

Abstract

A microelectromechanical (MEMS) resonator comprising a resonator structure having a plurality of beam elements (1-6) with a certain geometry and connecting elements (C1-C5), wherein the plurality of beam elements (1-6) are positioned adjacent to each other and adjacent beam elements (1-6) are mechanically connected to each other by connecting elements (C1-C5), wherein the geometry of the beam elements (1-6) or the connecting elements (C1-C5) varies within the resonator structure.

Description

TECHNICAL FIELD

[0001] The present invention relates generally to micro-electro-mechanical (MEMS) resonators. BACKGROUND

[0002] This section presents information which is useful background information for the understanding of the present invention but does not admit any prior art.

[0003] A key performance parameter for micro-electro-mechanical (MEMS) resonators, such as silicon MEMS resonators, used for frequency reference applications is the equivalent series resistance (ESR). The ESR is inversely proportional to the quality factor Q of the resonator and it is therefore generally desirable to maximize this parameter. It is generally very critical for maximizing the Q factor of a MEMS resonator to maximize the anchoring, i.e. the connection of the resonator to its surrounding substrate. SUMMARY

[0004] It has been observed that it is desirable to anchor the resonator in such a way that the nodes of the resonant mode shape coincide with the anchoring points. A desirable property of the resonator and its mode shape is that the nodes exist at the edges of the resonator, since the anchoring is typically done by silicon tethers connected to the edges of the resonator.

[0005] It has furthermore been observed that another factor that can lead to energy loss and thereby reduce the Q factor of the resonator is damping due to the presence of a gas atmosphere surrounding the resonator.

[0006] A stacked beam resonator comprises a plurality of resonator beams positioned side by side in a plane, separated by trenches and connected by connection elements.

[0007] It is an object of certain embodiments of the present invention to optimize the geometry of a stacked beam resonator.

[0008] According to a first example aspect of the present invention, there is provided a micro-electro-mechanical resonator, comprising: a resonator structure having a plurality of beam elements and connection elements with a certain geometry, wherein the plurality of beam elements are positioned adjacent to each other and adjacent beam elements are mechanically connected to each other by connection elements, wherein

[0009] The geometry of the beam elements or the connection elements varies within the resonator structure.

[0010] In certain embodiments, the geometry is varied by changing at least one size parameter of the resonator. In certain embodiments, the variation of the geometry of the beam elements and / or the connection elements is to provide a node(s) of the resonant mode shape in relation to the perimeter of the resonator. In certain embodiments, the node(s) coincide with one or more anchoring points anchoring the beam elements to the surrounding environment, or are relatively close to the anchoring point(s).

[0011] In the present context, at least one dimensional parameter is for example a parameter or a set of parameters (comprising at least two parameters) defining a shape, orientation or dimension of e.g. the beam elements and / or their mechanical connection to each other.

[0012] Stacked beam resonators typically have a basic geometry in which beam elements of equal width are connected by rectangular connecting elements located at the edges of the resonator. Embodiments of the present invention deviate from this basic geometry.

[0013] In certain embodiments, a beam element differs in width from another beam element. The beam elements have their respective widths, and in certain embodiments, the width of at least one beam element (or of multiple beam elements) differs from the width of another beam element.

[0014] In certain embodiments, a beam element differs in shape from another beam element. The beam elements have their respective shapes, and in certain embodiments, the shape of at least one beam element (or of multiple beam elements) differs from the shape of another beam element.

[0015] In certain embodiments, a beam element differs in tilt angle (direction) from another beam element. In such embodiments, at least two adjacent beam elements are positioned at a tilt angle with respect to each other. In certain embodiments, the tilt of adjacent beam elements occurs in the plane of the resonator (“in-plane tilt”).

[0016] In certain embodiments, a beam element differs in its longitudinal position within the resonator structure from another beam element. In such embodiments, the longitudinal positions of at least two adjacent beam elements differ from each other.

[0017] In certain embodiments, a connecting element differs in its shape from another connecting element.

[0018] In certain embodiments, a connecting element (or at least one connecting element, or multiple connecting elements) differs in its displacement from a resonator edge from another connecting element.

[0019] In certain embodiments, the width of at least one beam element differs from the width of another beam element, and at least one of these connecting elements is spaced apart from the respective end of the relevant adjacent beam element (or at least from the end of one of said adjacent beam elements, i.e. the end closer to the connecting element in question) by a distance.

[0020] In certain embodiments, the width of an outermost beam element is smaller than the width of its (directly) adjacent beam element.

[0021] In certain embodiments, the resonator comprises a trench extending from a resonator edge to the interior of the resonator structure until it reaches the edge of one of the connecting elements. In certain embodiments, the trench is straight in shape.

[0022] In certain embodiments, the resonator comprises n adjacent beam elements, wherein the first and the n-th beam elements have a width different from the width of the other beam elements, and the connecting elements between the first and the second beam element, and between the (n-1)-th and the n-th beam element are each spaced apart from the respective end of the beam elements by a distance.

[0023] In certain embodiments, the resonator structure is formed by a plurality of beam elements and connecting elements.

[0024] In certain embodiments, the resonator is configured to operate in a planar length extension (LE) mode. In certain embodiments, the beam elements of the resonator are configured to operate in the planar length extension mode.

[0025] In certain embodiments, the width of a beam element is an average width or an effective width. In certain embodiments, the plurality of beam elements have their respective lengths. In certain embodiments, the length is an average length or an effective length. In certain embodiments, the length of a beam element is greater than its width.

[0026] In certain embodiments, the coordinate system is chosen such that the x-axis lies in the width direction of the beam elements, and the y-axis lies in the longitudinal direction of the beam elements. The beam elements (and thus the entire resonator) oscillate in the LE mode along the direction of the y-axis.

[0027] In certain embodiments, the beam elements have an elongated shape (their length is greater than their width). They can be, for example, rectangular, rounded rectangular, rounded elliptical, hourglass-shaped, tapered, rotated hourglass-shaped, or asymmetrically elongated. In certain embodiments, the shape (or form) of one or more beam elements is different from the shape of another one of these beam elements.

[0028] In certain embodiments, at least one connecting element is different from another connecting element. In certain embodiments, the difference is in shape, size, position, and / or orientation.

[0029] In certain embodiments, the resonator structure with the plurality of beam elements and connecting elements is a symmetric structure. In certain embodiments, the resonator structure has reflectional symmetry. In certain embodiments, the resonator structure has mirror symmetry. In certain embodiments, the mirror symmetry is about the x-axis and / or the y-axis (the directions of the axes are defined later in the specification). In certain embodiments, the intersection of the x-axis and the y-axis is at the center of mass of the resonator structure. In certain embodiments, the resonator structure has rotational symmetry. In certain embodiments, the rotational symmetry is two-fold rotational symmetry.

[0030] In certain embodiments, the resonator structure comprises an asymmetry in the form of at least one of the connecting elements that is positioned asymmetrically.

[0031] In certain embodiments, the general structure of the resonator is symmetric, but the structure includes specific details that break the symmetry, such as the mentioned asymmetrically placed connecting elements.

[0032] In certain embodiments, the resonator comprises a semiconductor material. In certain embodiments, the resonator comprises silicon. In certain embodiments, the resonator comprises degenerately doped silicon. In certain embodiments, more than 50% of the resonator mass consists of degenerately doped silicon. In certain embodiments, the resonator comprises a silicon body that is doped to an average impurity concentration of at least 2*1018cm-3, such as at least 1019cm-3, such as at least 1020cm-3. 19 cm -3 , such as at least 1019cm-3, such as at least 1020cm-3. 20 cm -3 .

[0033] In certain embodiments, the resonator is configured to be actuated in a piezoelectric manner or in an electrostatic manner.

[0034] In certain embodiments, the longitudinal axis of one (or all) of the beam elements is aligned with the

[100] crystal direction of the beam element, or deviates therefrom by less than 25 degrees. In certain preferred embodiments, the longitudinal axis of one (or all) of the beam elements is aligned with the

[100] crystal direction of the beam element, or deviates therefrom by less than 15 degrees. In certain preferred embodiments, the longitudinal axis of one (or all) of the beam elements is aligned with the

[100] crystal direction of the beam element, or deviates therefrom by less than 5 degrees. In certain preferred embodiments, the longitudinal axis of one (or all) of the beam elements is aligned with the

[100] crystal direction of the beam element, or deviates therefrom by less than 2 degrees.

[0035] In certain embodiments, the microelectromechanical resonator is a stacked beam resonator. In certain embodiments, the beam elements are positioned in a rectangular array configuration. In certain embodiments, the main resonant mode shape of the resonator has node(s) at the perimeter of the resonator. In certain embodiments, the resonator comprises anchor points to anchor the beam elements to the surrounding environment, wherein the main resonant mode shape of the resonator has a node at the anchor points. In certain embodiments, the resonant mode refers to a fundamental mode. In certain embodiments, the resonant mode refers to an overtone.

[0036] In certain embodiments, the width, longitudinal arrangement or orientation of at least one of the beam elements is different from the width, longitudinal arrangement or orientation, respectively, of another beam element; or

[0037] At least one of the connecting elements connecting adjacent beam elements is spaced apart from at least one respective end portion of the relevant adjacent beam element by a distance.

[0038] In certain embodiments, displacement from a resonator edge refers to displacement in the y-direction and from the resonator edge closer to the connecting element in question. Similarly, in certain embodiments, the distance by which the connecting element(s) are spaced apart from the respective end(s) of the beam element(s) refers to the spacing in the y-direction from the end(s) of the beam element(s) closer to the connecting element(s) in question.

[0039] In certain embodiments, the longitudinal positions of the centers of mass of adjacent beam elements differ in the y-direction. In certain embodiments, the longitudinal position of a beam element refers to the position in the y-direction.

[0040] In certain embodiments, the resonator is a stacked beam resonator and the plurality of beam elements are positioned in a rectangular array configuration, and wherein the resonator structure comprises a top electrode implemented in a layer on top of the piezoelectric layer.

[0041] In certain embodiments, the resonator comprises a width of the trench between the edges of the beam elements forming the resonator and the silicon frame firmly (solidly) mounted around it, which width is selected to minimize or avoid acoustic resonance at the trench area.

[0042] In certain embodiments, the width of the trench between the edges of the beam elements forming the resonator and the silicon frame firmly mounted around it is given by the equation Trench width = 4 pm x (40 MHz / Freq) x (n + ½)

[0043] where "Freq" is the frequency of the resonator, "n" is a positive integer 0, 1, 2, 3... or the width of the trench deviates at most 20% from the width given by the equation.

[0044] The above embodiments and examples are described later in this specification for the purpose of explaining selected aspects or steps that can be used in embodiments of the application. It should be understood that the respective embodiments are applicable to other example aspects as well. Any suitable combination of embodiments can be formed. BRIEF DESCRIPTION OF DRAWINGS

[0045] The application will now be described by way of example only, with reference to the accompanying drawings, in which:

[0046] Figure 1 A microelectromechanical (MEMS) resonator according to certain embodiments is shown;

[0047] Figure 2 A microelectromechanical (MEMS) resonator according to certain embodiments is shown; Figure 1 Experimental results for a MEMS resonator of the type shown;

[0048] Figure 3 Simulation results for MEMS resonators of the type shown are shown. Figure 1 Simulation results for MEMS resonators of the type shown are shown.

[0049] Figures 4a-4i Alternative resonator geometries according to certain embodiments are shown.

[0050] Figure 5a And 5b Connection elements in certain alternative embodiments are shown.

[0051] Figure 6 Alternative beam geometries according to certain embodiments are shown.

[0052] Figure 7 Alternative resonator geometries according to certain embodiments are shown.

[0053] Figure 8 And 9 Further other resonator geometries according to certain embodiments are shown in detail.

[0054] Figure 10 A cross-sectional view of a piezoelectrically actuated resonator according to certain embodiments is shown.

[0055] Figure 11 A cross-sectional view of an electrostatically actuated resonator according to certain embodiments is shown; and

[0056] Figure 12 Embodiments are further shown in which the resonator geometry varies within the resonator structure. DETAILED DESCRIPTION

[0057] In the following description, like numbers refer to like elements throughout.

[0058] Figure 1 A microelectromechanical (MEMS) resonator according to certain embodiments is shown. In more detail, Figure 1 A top view of a stacked beam resonator is shown, which comprises six adjacently positioned beam elements 1...6 having widths W1...W6 and a length L. The beam elements 1...6 are coupled to each other by connection elements. In the embodiment shown, every two adjacent beam elements are coupled by two connection elements. One of the connection elements between the first and second beam elements 1, 2 has been drawn in the figure and is denoted by C1 (the other connection elements can be denoted similarly, e.g. the connection element connecting beams 2 and 3 is denoted by C2, the connection element connecting beams 3 and 4 is denoted by C2, and so on). The connection elements have widths WC1...WC5 and they are located at distances LC1...LC5 from the edges of the resonator 100 (with respect to the fifth connection element, the width WC5 and the distance LC5 from the edge have been drawn in the figure).

[0059] Beam elements 1...6 are separated from each other by trenches (rectangular shaped in this embodiment). The trench between beam elements 1 and 2 is denoted TW1, the trench between beam elements 2 and 3 is denoted TW2 (not indicated in the figure), and so on. The trench separating the entire resonator 100 from its surroundings is denoted T in the figure. AL and AR denote the left and right anchor positions, respectively, which are located in the center of the left and right sides of the resonator 100.

[0060] The geometrical parameters in the figure are LI...L6 = 91.75 pm, W2...W5 = 28 pm, TW1...TW5 = 5 pm, LC2...LC4 = 0 pm, LC1 = LC5 = 5 pm, WC1...WC5 = 5 pm.

[0061] Figure 2 Experimental results related to the MEMS resonator of the type shown in Figure 1 are shown. As an example, the Q-factor of a 40 MHz parametrically varied resonator fabricated on a silicon wafer was experimentally measured. The normal to the silicon wafer was in the

[100] direction, and the side of the resonator beam elements was aligned with the <100> direction. The parameter D was varied between 8...16.5 pm, and approximately 5 resonators were measured for each variation at atmospheric pressure. The resonator design geometry was similar to the one described in Figure 1 , except that the beam element widths W1 and W6 were varied according to the formula W1 = W6 = 28 pm - D. The best performance was reached at D = 14 pm, i.e. at W1 = W6 = 14 pm.

[0062] Figure 3 Simulation results related to the MEMS resonator of the type shown in Figure 1 are shown. More specifically, Figure 3 the figure shows the figure of merit FOM Figure 2 of the geometrical structures discussed in anchor COMSOL Multiphysics, a FEM analysis software, was used for the calculations. The FOM anchor is defined as

[0063] FOM anchor = u_x_anchor / |max(u_y_long_side)|

[0064] where u_x_anchor is the x-displacement at the right anchor point AR (see Figure 1 ) of the relevant resonant mode shape, and max(u_y_long_side) is the maximum y-displacement of the resonant mode on the x-top edge of the resonator. The best value, i.e. the position at which FOM anchor = 0, is close to W1 = W6 = 14 pm, which is in very good agreement with the best value found experimentally in Figure 2 .

[0065] In the disclosed embodiments, the coordinate system is chosen such that the x-axis lies in the width direction of the beam elements, while the y-axis lies in the longitudinal direction of the beam elements. The beam elements (and thus the entire resonator) oscillate in the y-axis direction in the LE mode.

[0066] As Figures 1-3 is shown, it is practical to implement a MEMS resonator in which the width of at least one beam element differs from the width of another beam element. It is furthermore practical to implement a MEMS resonator in which at least one of the connecting elements connecting adjacent beam elements is spaced apart from the end of the adjacent beam element (or from the resonator edge) by a distance. The main resonant mode shape of the resonator has a node occurring at the resonator periphery. It is optimal in the example shown in Figures 1-3 that the rightmost and leftmost beam elements have a width that is half the width of the other beam elements having the shown geometry, and that the connecting elements between the first and second beam elements and between the fifth and sixth beam elements are spaced apart from the resonator edge by the same distance due to the wide grooves between the beam elements. In this way, the node of the relevant main resonant mode shape coincides with the anchor point. It should be noted, however, that the presented geometry is not the only preferred geometry, but that the geometry can be varied and designed according to the relevant embodiment.

[0067] Figures 4b-4i An alternative geometry is shown in which a node can be obtained at the resonator periphery, for example at the anchor point in the side edge of the outermost beam elements 1 and 6. Figure 4a A basic geometry is shown in which each beam element 1-6 has an equal width, and each connecting element is at a distance zero from the relevant resonator edge E.

[0068] Figure 4b A resonator geometry is shown in which the width of the beam elements 1 and 6 is modified (in this embodiment, narrowed), and the connecting elements C2 and C4 between the beam elements 2 and 3 and 4 and 5 are modified (displaced by a distance from the resonator edge E). These modifications are with respect to the basic geometry shown in Figure 4a

[0069] Figure 4c Another alternative geometry is shown in which the width of the beam elements 1 and 6 is modified (in this embodiment, narrowed), and the connecting element C3 between the (centralmost) beam elements 3 and 4 is modified.

[0070] Figure 4d Another alternative geometry is shown in which the width of the (centralmost) beam elements 3 and 4 is modified (in this embodiment, narrowed), and the connecting elements C2 and C4 between the beam elements 2 and 3 and 4 and 5 are modified.​

[0071] Figure 4e Another alternative geometry is shown in which the width of the beam elements 2 and 5 is modified.

[0072] Figure 4f Another alternative geometry is shown which generally corresponds to the basic geometry shown in Figure 4e However, in this embodiment one of the connecting elements Cl between the beam elements 1 and 2 and one of the connecting elements C5 between the beam elements 5 and 6 is modified (moved away from the respective edge E by a distance), but the other is not. The modified connecting element Cl is the one closer to the edge E, while the modified connecting element C5 is the one closer to the other or opposite edge E. Thus, Figure 4f The geometry shown in

[0073] Figure 4g Another alternative geometry is shown in which the symmetry is further reduced compared to the geometry shown in Figure 4f In the geometry shown in Figure 4f one of the connecting elements Cl between the beam elements 1 and 2 and one of the connecting elements C5 between the beam elements 5 and 6 is modified, while Figure 4g A geometry is shown in which only one connecting element is modified (here: one of the connecting elements C5).

[0074] Figure 4h Another alternative geometry is shown which otherwise corresponds to the geometry of Figure 4g In the alternative shown in Figure 4h the width of each beam unit is equal. Accordingly, an equal beam element is shown in which only one connecting element is modified (moved away from the respective edge E).

[0075] Figure 4i A further alternative geometry with equal width of the beam elements is shown. In this embodiment the connecting elements Cl between the beam elements 1 and 2 and the connecting elements C5 between the beam elements 5 and 6 are modified (moved away from the respective edge E by a distance). The resonator structure formed by the beam elements and the connecting elements is thus a symmetric structure (equal beam elements and the outermost connecting elements are symmetrically shifted).

[0076] Figures 4b-4i The exact dimensions in

[0077] The connecting elements between the beam elements can be realized as shown in Figure 1 - 4 and Figure 5aThe shown x-directional rectangles. However, the shape of one or more of the connecting elements can be different, like Figure 5b The connecting elements C1, C2, C4 and C5 therein have been arranged as tilted elements.

[0078] Furthermore, instead of rectangles, the beam elements can be another elongated shape. The elongated geometry supports length extension (LE) resonance modes, and thus the alternative beam geometries shown as examples in Figure 6 The different beam types are: rounded rectangle, rounded ellipse, hourglass, tapered, rotated hourglass, and asymmetric elongated. In these embodiments, it can become suitable to consider the average width (or effective width) and average length (or effective length) when referring to the width and length of a beam element.

[0079] Figure 7 Yet other alternative resonator geometries are shown according to certain embodiments. Herein, the center of mass of different beam elements varies in the longitudinal direction of the beam elements. Thus, the center of mass position of the beam units is not equal in the y-direction.

[0080] Figure 7 The uppermost figure demonstrates a geometry where the center of mass of each subsequent beam element is displaced further in the longitudinal direction (y-direction) compared to the center of mass of the previous beam element. The middle figure shows a geometry where the centermost beam unit (or its center of mass) is displaced the most, while the outermost beam units are displaced the least. The lowermost figure in turn shows a geometry where the beam units (or their center of mass) are displaced in an alternating fashion, e.g. by every second beam unit. Figure 7 The connecting elements shown in are connecting adjacent beam elements in such a way that they are located at the end of one of the beam elements they connect, but at a distance from the end of the other beam element.

[0081] Figure 8 and 9 Details of other resonator geometries according to certain embodiments are shown. In Figure 8 In the shown embodiment, the beam elements differ in their tilt angle (orientation) from another beam element. The beam element n is positioned with a tilt angle with respect to the adjacent beam element n+1. The beam units n and n+1 are connected by a connecting element Cn located at the end of the beam units. Due to the tilt, the connecting element Cn is slightly different in size at the end (the other connecting element is longer). Furthermore, the trench between the beam elements n and n+1 is not a perfect rectangle, but is expanded towards one edge of the resonator. The tilt of the adjacent beam elements occurs within the plane of the resonator (“in-plane tilt”). In Figure 9 In the shown embodiment, the shape and tilt angle of the adjacent beam elements varies within the resonator structure. The connecting elements are not shown Figure 9Figures 1A and IB show a basic resonator structure.

[0082] Depending on the embodiment, the presented resonator structure is adapted for piezoelectric or electrostatic actuation.

[0083] Figure 10 An example cross-section showing piezoelectric actuation is shown. A silicon-on- insulator (SOI) wafer 150 is applied. Reference numerals 201 and 202 represent bottom and top electrode contacts, respectively. The top electrode is implemented in layer LI. Layer L2 is a piezoelectric layer for the piezoelectrically actuated resonator (located in the area denoted by 100). The opening in L2 is denoted by 120. Layer L3 represents a layer for the bottom electrode. Layer L4 is a doped silicon layer for the resonator body (beam elements and their connecting elements). L5 layer is the buried oxide layer of the SOI wafer, and L6 layer is the silicon handling layer. In certain embodiments, when a doped silicon layer is used as L4, a separate L3 bottom electrode layer can be omitted. In such embodiments, the conductive doped silicon layer acts as the bottom electrode.

[0084] Figure 11 An example cross-section showing electrostatic actuation is shown. A silicon-on- insulator (SOI) wafer (denoted here as 150') is applied. Reference numerals 203 and 204 represent contacts of layer L4 with two different electrode potentials, respectively. Layer L4 is a doped silicon layer for the resonator body (beam elements and their connecting elements). The area of the resonator is denoted by 100. There is a gap for electrostatic actuation between the resonator edge and the remaining layer L4. L5 layer is the buried oxide layer of the SOI wafer, and L6 layer is the silicon handling layer.

[0085] The number of beam elements, as well as the number of connecting elements between the beam elements, can vary according to the embodiment. Thus, while the presented examples have shown 6 beam elements, in more general cases the number of beam elements is different from 6. The basic resonator structure or resonator unit (stacked beam resonator) disclosed here can be replicated. For example, in certain embodiments, multiple resonator units can be placed in parallel, or in certain embodiments, in series. The basic resonator structure is not anchored to its surrounding environment by its outermost beam elements, but can be anchored by the non-outermost side beam elements (e.g., one or more inner beam elements). Anchoring to the "surrounding environment" can be replaced by attachment to another resonator unit.

[0086] Figure 12 Further embodiments are illustrated in which the resonator geometry is varied within a resonator structure similar to that shown in, e.g., Figures 1A and IB. Figures 1-3 Figure 12 ​The MEMS resonator 100 shown includes a resonator structure having a plurality of elongated beam elements 1-n, where, for example, n = 12. Each of the beam elements 1-n is a stacked beam element positioned parallel to each other to form a rectangular array configuration. Therefore, the longitudinal axes of the beam elements 1-n are parallel. The shape of the beam elements 1-n is preferably rectangular. The centroid of each beam element 1-n is aligned along a single line (i.e., a single line passes through the centroid of each beam element in the beam elements 1-n). Each beam element 1-n preferably has an equal length.

[0087] The resonator 100 is separated from its surroundings by a groove T surrounding the resonant structure. Similarly, each beam element 1-n is separated from its adjacent beam element by a groove T. Each beam element 1-n is also connected to its adjacent beam element by a connecting element. Preferably, the number of beam elements between each pair of adjacent beam elements is two (i.e., adjacent beam elements are connected to each other by exactly two connecting elements). The resonator 100 is mechanically connected to its surroundings at the edge of the outermost beam element. The location of the anchor point is determined by... Figure 12 Reference numerals AL (left anchor, at the left edge of the resonator) and AR (right anchor, at the right edge of the resonator) are used in the designation. Therefore, preferably, the resonator 100 is anchored to its surrounding environment (e.g., substrate) in the lateral direction of the outermost beam elements 1 and n, at the midpoint of the outer long sides of the outermost beam elements 1 and n.

[0088] The position where beam element 1-n intersects the groove T at its first end (in the y direction) is denoted as edge E1 (front edge of the resonator), and the corresponding edge n at the opposite end of beam element 1- is denoted as edge E2 (rear edge of the resonator).

[0089] exist Figure 12 In the example shown, the widths of the beam elements and the positions of the connecting elements differ within the structure. More specifically, the outermost beam elements 1 and n have widths smaller than the widths of the other beam elements. Preferably, the width of the first beam element 1 is equal to the width n of the last beam element, and the widths of the remaining beam elements 2 to n-1 are equal to each other but greater than the width of beam element 1 (or n).

[0090] Furthermore, each connecting element is zero distance from its nearest resonator outer edge E1 or E2, except that the connecting element connecting the outermost beam element and its adjacent beam element is spaced apart from its corresponding edge E1 or E2 (or from the corresponding end of the associated adjacent beam element 1-n). Therefore, the straight boundary line extending along edges E1 and E2 is discontinuous at the point between the outermost beam element and its adjacent beam element. At that point, the trench T penetrates horizontally from the boundary between the outermost beam element and its adjacent beam element. Thus, the trench T extends from the resonator edge E1 or E2 into the interior of the resonator structure until it reaches the edge of the associated spaced-apart connecting element.

[0091] Figure 12 The resonator structure shown represents a symmetrical resonator structure. It has mirror symmetry with respect to the central x-axis and central y-axis.

[0092] The beam elements of resonator 100 are configured to vibrate in a resonant mode extending (LE) along their longitudinal (y-direction). The resonator structure includes a piezoelectric layer for piezoelectric actuation of the resonator.

[0093] In a preferred embodiment of the invention, the resonator 100 is enclosed in a cavity under low pressure to reduce energy loss due to gas damping. However, in practical implementations, the pressure around the resonator may be insufficient to reduce gas damping to a negligible level. In practical implementations of the invention, it has been observed that at the edges of the beam (or resonator)... Figure 12 The trench between E1 or E2 and the surrounding firmly mounted silicon frame ( Figure 12 Acoustic resonance may occur in the groove T). Several beam resonators according to embodiments of the present invention were fabricated, and their resonant characteristics were characterized by impedance measurements. The beam resonators include resonators with frequencies of 40 MHz, 32 MHz, and 24 MHz. The width of the groove T between edges E1 and E2 (reference) Figure 12 The groove widths (defined as T) are 2μm, 4μm, 5.5μm, 6μm, 8μm, 9μm, 10μm, 11μm, 12μm, and 13μm. These widths produce a high Q-factor and therefore low gas damping. 低气体阻尼 And the groove width (defined as T) that produces a low Q factor and thus high gas damping. 高气体阻尼 The measurements taken under environmental pressure were analyzed. The results for the trench width were able to fit the empirical model equations.

[0094] T 低气体阻尼 = 4μm(40MHz / Freq)(n+1 / 2), and

[0095] T 高气体阻尼= 4pm (40MHz / Freq)(n+1),

[0096] where "Freq" is the frequency of the beam resonator according to embodiments of the application, and "n" is an integer number of acoustic wavelengths (n = 0, 1, 2,...).

[0097] Without limiting the scope and spirit of the patent claims, certain technical effects of one or more exemplary embodiments disclosed herein are listed below. One technical effect is to provide an optimized geometry for stacked beam resonators. Another technical effect is to improve the flexibility of MEMS resonator design.

[0098] The foregoing description has provided a full and informative description of the best mode presently contemplated by the inventors for carrying out the present application. However, it is to be understood that the application is not to be limited to the precise details of the embodiments shown, since obvious modifications and variations can be made therein by those skilled in the art. For example, while the embodiments described herein have been described in the context of a beam resonator, the principles described herein can be applied to other resonator designs, such as a bulk acoustic wave resonator.

[0099] Furthermore, some of the features of the disclosed embodiments of this application can 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 application, and not in limitation thereof. Hence, the scope of the application is only restricted by the appended patent claims.

Claims

1. A microelectromechanical resonator, comprising: a resonator structure having a plurality of beam elements and connecting elements with a certain geometry, wherein the plurality of beam elements are positioned adjacent to each other and adjacent beam elements are mechanically connected to each other by the connecting elements at both ends, wherein the geometry of the beam elements or the connecting elements varies within the resonator structure.

2. The resonator according to claim 1, wherein the resonator is a stacked beam resonator and the plurality of beam elements are positioned in a rectangular array configuration, and wherein the resonator structure comprises a top electrode implemented in a layer on top of a piezoelectric layer.

3. The resonator according to claim 1 or 2, wherein a beam element differs from another beam element in width.

4. The resonator according to claim 1 or 2, wherein a beam element differs from another beam element in shape.

5. The resonator according to claim 1 or 2, wherein a beam element differs from another beam element in tilt angle.

6. The resonator according to claim 1 or 2, wherein a beam element differs from another beam element in longitudinal position of the beam element within the resonator structure.

7. The resonator according to claim 1 or 2, wherein a connecting element differs from another connecting element in shape of the connecting element.

8. The resonator according to claim 1 or 2, wherein a connecting element differs from another connecting element in displacement of the connecting element from a resonator edge.

9. The resonator according to claim 1 or 2, wherein at least one beam element differs from another beam element in width and at least one of the connecting elements spacing the adjacent beam elements apart by a distance from the respective end of the adjacent beam elements.

10. The resonator according to claim 1 or 2, wherein a width of an outermost beam element is smaller than a width of an adjacent beam element of the beam element.

11. The resonator according to claim 1 or 2, comprising a trench extending from a resonator edge to an interior of the resonator structure until the trench reaches an edge of one of the connecting elements.

12. The resonator according to claim 1 or 2, comprising n adjacent beam elements, wherein a first beam element and an n-th beam element have a width different from a width of the other beam elements and the connecting elements between the first and second beam elements and between the n-1-th beam element and the n-th beam element are spaced apart by a distance from the respective end of the beam elements.

13. The resonator according to claim 1 or 2, comprising an anchor point anchoring a beam element to the surrounding environment, wherein a main resonant mode shape of the resonator has a node at the anchor point.

14. The resonator according to claim 1 or 2, wherein the beam elements have an elongated shape.

15. The resonator according to claim 14, wherein the beam elements have a rectangular, rounded rectangular, rounded elliptical, hourglass, tapered, rotated hourglass, or asymmetric elongated shape.

16. The resonator according to claim 1 or 2, wherein at least one connecting element differs from another connecting element.

17. The resonator of claim 1 or 2, wherein the beam elements of the resonator are configured to operate in in-plane length extensional modes.

18. The resonator of claim 1 or 2, wherein the resonator structure of the resonator having the plurality of beam elements and the connecting elements is a symmetric structure.

19. The resonator of claim 1 or 2, wherein the resonator structure includes an asymmetry in the form of at least one of the connecting elements positioned asymmetrically.

20. The resonator of claim 1 or 2, wherein the resonator comprises a semiconductor material.

21. The resonator of claim 20, wherein the resonator comprises silicon.

22. The resonator of claim 21, wherein the resonator comprises degenerately doped silicon.

23. The resonator of claim 21, wherein more than 50% of the resonator mass is comprised of degenerately doped silicon.

24. The resonator of claim 21, wherein the resonator comprises a silicon body doped to an average impurity concentration of at least 1 x 1012 atoms / cm3.

51. The method of claim 50, wherein the average impurity concentration is at least 1 x 1012 atoms / cm3.

25. The resonator of claim 1 or 2, wherein a longitudinal axis of a beam element is aligned with a [100] crystal orientation of the beam element.

26. The resonator of claim 1 or 2, wherein a longitudinal axis of a beam element is misaligned with a [100] crystal orientation of the beam element by less than 25 degrees, or less than 15 degrees, or less than 5 degrees, or less than 2 degrees.

27. The resonator of claim 1 or 2, comprising a width of a trench between edges of the beam elements forming the resonator and a silicon frame firmly mounted around the edges, the width selected to minimize or avoid acoustic resonance at the area of the trench.

28. The resonator of claim 27, wherein the width of the trench between the edges of the beam elements forming the resonator and a silicon frame firmly mounted around the edges is given by the equation: Trench Width = 4 pm x (40 MHz / Freq) x (n + ½) with an error range of + / - 20%, and wherein "Freq" is the frequency of the resonator and "n" is a positive integer 0, 1, 2, 3,....

Images