Temperature compensated plate resonator

Abstract

The invention relates to a microelectromechanical resonator device comprising a support structure and a semiconductor resonator plate doped to a doping concentration with an n-type doping agent and being capable of resonating in a width-extensional resonance mode. In addition, there is at least one anchor suspending the resonator plate to the support structure and an actuator for exciting the width-extensional resonance mode into the resonator plate. According to the invention, the resonator plate is doped to a doping concentration of 1.2*10.sup.20 cm.sup.−3 or more and has a shape which, in combination with said doping concentration and in said width-extensional resonance mode, provides the second order temperature coefficient of frequency (TCF.sub.2) to be 12 ppb/C.sup.2 or less at least at one temperature. Several practical implementations are presented.

Claims

1. A microelectromechanical resonator device comprising; a support structure, a semiconductor resonator plate doped to a doping concentration with an n-type doping agent and being capable of resonating at least partly in a width-extensional resonance mode, at least one anchor suspending the resonator plate to the support structure, and an actuator for exciting said width-extensional resonance mode into the resonator plate, wherein the resonator plate is doped to a doping concentration of at least 1.2*10.sup.20 cm.sup.−3 and has a shape which, in combination with said doping concentration and in said width-extensional resonance mode, provides a second order temperature coefficient of frequency (TCF.sub.2) of 12 ppb/C.sup.2 or less at at least one temperature.

2. The resonator device according to claim 1, wherein the shape of the resonator plate has an aspect ratio of 1.1-1.6.

3. The resonator device according to claim 1, wherein the shape of the resonator plate has an aspect ratio larger than 1.3, and the resonator plate is provided with a piezoelectric thin film forming part of said actuator.

4. The resonator device according to claim 1, wherein the shape of the resonator plate is rectangular.

5. The resonator device according to claim 1, wherein the shape of the resonator plate is elliptical.

6. The resonator device according to claim 1, wherein the shape and doping concentration of the resonator plate are such that the first order temperature coefficient of frequency (TCF.sub.1) of the resonator device is 1 ppm/C or less at said at least one temperature.

7. The resonator device according to claim 1, wherein the shape of the resonator plate is non-square and non-circular and has an aspect ratio of 2 or less.

8. The resonator device according to claim 1, wherein the resonator plate has an axis of symmetry which is aligned with, a direction of the semiconductor crystals with 5 degrees accuracy.

9. The resonator device according to claim 8, wherein said axis of symmetry coincides with a longitudinal axis of the resonator plate.

10. The resonator device according to claim 1, wherein the resonator plate further comprises a base plate having an aspect ratio of 1.2-1.6 and at least one protrusion extending laterally from the base plate.

11. The resonator device according to claim 10, wherein the base plate is adapted to resonate in said width-extensional mode and the protrusions in a flexural, torsional or length-extensional mode.

12. The resonator device according to claim 1, wherein the plate resonator is doped to a doping concentration of 1.25*10.sup.20 cm.sup.−3 or more.

13. The resonator device according to claim 1, wherein the shape and doping concentration of the resonator plate are such that the second order temperature coefficient of frequency (TCF.sub.2) is 6 ppb/C.sup.2 or less at said at least one temperature.

14. The resonator device according to claim 1, wherein the shape and doping concentration of the resonator plate are such that the first order temperature coefficient of frequency (TCF.sub.1) of the resonator device is 0.5 ppm/C or less and the second order temperature coefficient of frequency (TCF.sub.2) is 3 ppb/C.sup.2 or less at said at least one temperature.

15. The resonator device according to claim 1, wherein its total temperature drift of frequency is less than 50 ppm over a temperature range spanning at least 125° C.

16. The resonator device according to claim 1, wherein said actuator further comprises a piezoelectric actuator acoustically coupled with the resonator plate.

Description

BRIEF DESCRIPTION OF THE DRAWINGS

(1) FIGS. 1A-1E show resonator geometries according to selected embodiments of the invention.

(2) FIG. 2A illustrates a graph of total frequency drift vs. temperature for a WE mode resonator with optimized lateral aspect ratio at different doping concentrations.

(3) FIG. 2B shows a graph of second order TCF vs. doping concentration for a WE resonator whose linear TCF is made zero by optimal design.

(4) FIG. 2C shows TCF.sub.1=0 and TCF.sub.2=0 curves of a WE mode resonator as a function of doping concentration and in-plane aspect ratio of the base portion of the resonator.

(5) FIGS. 2D shows graphs of temperature coefficients of the elastic parameters Cy of silicon as a function of doping concentration n.

(6) FIG. 2E shows measured first and second order TCF of a Lame mode with the plate sides in [100]-orientation as a function of carrier concentration. The two data points with the highest carrier concentration (n>=10e19cm-3) were first discovered in connection with the present invention.

DETAILED DESCRIPTION OF EMBODIMENTS

(7) FIG. 1 shows a rectangular resonator plate 10A having a length L and width W. The longer sides of the plate 10A are oriented along the [100] direction of the silicon crystal. Main deformation direction of the resonator plate 10A in a WE resonance mode is along the width axis of the plate and is illustrated with dashed arrows. Preferably, the deformation is symmetrical with respect to the longitudinal axis of the plate 10A.

(8) FIG. 1B shows resonator plate 10B also having a generally rectangular shape but having rounded corners. Apart from this, the resonator is similar to and also operates similarly to that of FIG. 1A.

(9) FIG. 1C shows an elliptical resonator plate 10C as another embodiment. The longer main axis of the plate has a length of L and the shorter a length of W. Optimal values and ratio of L and W may differ from those of FIGS. 1A and 1B. The elliptical plate can also resonate in a width-extensional mode, that is, along the direction of the shorter main axis.

(10) FIG. 1D shows still another embodiment of the resonator, having an octagonal resonator plate 10D.

(11) FIG. 1E shows a modified rectangular plate resonator 10E. It is provided with four protrusions 12E, which are adapted to resonate flexurally as the base plate resonates width-extensionally. The protrusions are preferably beams with an aspect ratio of 5 or more. The protrusions can be tilted with respect to the [100] crystal direction or parallel to it. In practice, the resonance mode of such compound plate is a compound mode where the WE mode dominates in the base plate and a flexural mode in the protrusions. The protrusions can be used to tune the TCF characteristics of the resonator and/or to allow for nodal anchoring of the resonator.

(12) FIG. 1E exemplifies only one variation of basic geometries shown in FIGS. 1A-1D. Any one of the illustrated basic plate geometries, or any other base plate geometry, can be provided with one or more protrusions or other extensions in order to adjust its properties. Typically, the protrusions or extensions are dominated by another resonance mode than the base plate. The term aspect ratio herein refers to the dimensions of the base plate being dominated by the WE mode.

(13) All the embodiments described above have a symmetry axis of the plate parallel to the [100] crystal direction. This is a preferred case, because the plate remains symmetric with respect to the silicon crystal and no parasitic resonances are created to the plate. Thus, the quality factor of the resonator remains high. However, the plate can also be rotated in the plane of the wafer by 1-45 degrees, for example, with respect to the [100] direction in order to adjust its resonance characteristics.

(14) Anchoring of the plates 10A-10D to the supporting structure (not shown) is preferably carried using anchoring elements (not shows) spanned between the supporting structure and one or more nodal or quasinodal points of the resonance mode at the perimeter of the plate 10A-10D. In the illustrated symmetric cases, the nodal points are at the points where the axis of symmetry along the longitudinal dimension of the plates 10A-10D intersect the perimeter of the plate, i.e., for example in the middle of shorter sides of the plate IDA, 10B or 10D or at the longitudinal distal points of plate 1C.

(15) FIG. 2A illustrates total frequency drift Δf vs, temperature T measured for an optimized SE/WE mode resonator (rectangular shape with aspect ratio ranging from 1 to 1.2 and sides in the [100] direction) at three different n-doping concentrations. As can be seen the opening curvature of the frequency-vs-temperature curve (i.e. TCF.sub.2) decreases with increased doping level. In more detail, the evolution of TCF.sub.2 is shown in FIG. 2B. Using a linear approximation in this optimal case, it can be estimated that TCF.sub.2 is zeroed at approximately 1.2*10.sup.20 cm.sup.−3. Assuming that the second-order TCF grows monotonously when doping gets larger than 11*10.sup.19cm.sup.−3 and that the linear TCF gets smaller simultaneously only relatively slowly, there exist a doping level and a single point (=a definite aspect ratio) on the WE-SE continuous branch, where both TCF.sub.1 and TCF.sub.2 are zero.

(16) FIG. 2A shows that a total frequency drift of less than 50 ppm over the industrial range of −40 . . . +85C can be achieved with optimized geometry when the doping concentration is in the range according to the invention. Further extrapolation using the data shown in FIG. 2B on the other hand supports that a drift of less than 25 ppm is feasible.

(17) FIG. 2C shows in detail zero TCF.sub.1 and TCF.sub.2 curves of a WE mode resonator (according to FIG. 1A) , when n-doping concentration and aspect ratio L/W are varied. As can be seen, both TCF.sub.1 and TCF.sub.2 are zeroed simultaneously at about 1.3*10.sup.20 cm.sup.−3 concentration and aspect ratio of 1.3. This qualitative result demonstrates the feasibility of the invention. FIG. 2C represents an optimal situation for a rectangular doped silicon resonator element. The simulation has been performed for a resonator with a thickness of 10 μm, width (W) of 320 μm, and whose length (L) has been varied. Simulations with thicker geometries indicate that the optimal aspect ratio is shifted to slightly lower values: when the resonator thickness is increased to 50 μm, the optimal aspect ratio is approximately 1.1. In practice, additional materials which may be present in order to provide an operational actuator, change the position of the curves to some extent. For example, a metal electrode and a piezoelectric material layer required for piezoactuation move the curves towards the right hand side and down. Consequently, the TCF.sub.1=TCF.sub.2=0 point, where the curves intersect, moves towards higher concentration and higher aspect ratio. For plate resonators in the width-extensional mode, the optimal point in practice is at a concentration of 1.2*10.sup.20 cm.sup.−3 or higher, in particular 1.4*10.sup.20 cm.sup.−3 or higher and the aspect ratio being 1.2-1.5.

(18) To give some further details on how the curves of FIG. 2C are generated, FIG. 2D shows temperature coefficients of the elastic parameters c.sub.ij as a function of carrier concentration n. The first, second and third column represent the constant terms c.sup.0.sub.ij, linear coefficients a.sub.ij, and second-order coefficients b.sub.ij at T=25° C., respectively. c.sup.0.sub.11-12, a.sub.11-12 and b.sub.11-12 are shorthands for the coefficients of c.sub.11-c.sub.12. The dependent coefficient a.sub.12 is readily evaluated as a.sub.12=(a.sub.11c.sup.0.sub.11−a.sub.11-12c.sup.0.sub.11-12)/c.sup.0.sub.12, and a similar equation holds for b12. Data points at carrier concentration below 7.5*10.sup.19 cm.sup.−3 represent data from literature (Jaakkola et al, “Determination of doping and temperature dependent elastic constants of degenerately doped silicon from MEMS resonators,” IEEE Transactions on Ultrasonics, Ferroelectrics, and Frequency Control. IEEE. Vol. 61 (2014) No: 7, 1063-1074). Data points for 1.sup.st and 2.sup.nd order coefficients a.sub.11-12 and b.sub.11-12 at carrier concentration 10*10.sup.19 cm.sup.−3 and 11*10.sup.19 cm.sup.−3 are shown with circles as well; these data points are based on recent measurement results of the applicant, shown in FIG. 2E, and importantly show the property of values of b.sub.11-12 following the positive slope that starts from a dopant concentration below 5*10.sup.19 cm.sup.−3. In calculations for producing results of FIG. 2C, the interpolated/extrapolated values represented by the dashed curves of FIG. 2D have been used.

(19) The fit at carrier concentration 0<n<7.5*10.sup.19 cm-3 is based on a third order polynomial fit to the data points at carrier concentration 0<n<7.5*10.sup.19 cm-3 for all nine terms shown in the plot. The fit of a.sub.11-12 and b.sub.11-12 at carrier concentration n>=7.5*10.sup.19 cm.sup.−3 is based on a linear fit to the three data points available on this range. For other terms except a.sub.11-12 and b.sub.11-12 the values are assumed to stay at the same level as the experimental data at n=7.5*10.sup.19 cm.sup.−3. Hence, for these cases, the dashed line is horizontal for n≦7.5*10.sup.19 cm.sup.−3. Reason for this choice was that no experimental data exists for other than terms a.sub.11-12 and b.sub.11-12 at carrier concentrations above 7.5*10.sup.19 cm.sup.−3. As a result, the results of FIGS. 2C-2E are not expected to be quantitatively perfectly accurate, but they do demonstrate the existence of optimal configurations where TCF.sub.1 and TCF.sub.2 can be zeroed simultaneously. Also, as the main terms contributing to the temperature coefficients of the resonance modes discussed in this document are a.sub.11-12 and b.sub.11-12, it is justified to assume that predictions of FIG. 2C hold fairly well.

(20) FIG. 2E shows the experimental data measured for a Lame-mode resonator, which is aligned with the [100] crystalline direction so that its modal frequency is dependent solely on the elastic parameter difference term c.sub.11-c.sub.12. Data points for doping concentration n<7.5*10.sup.19 cm.sup.−3 are from literature (Jaakkola et al, “Determination of doping and temperature dependent elastic constants of degenerately doped silicon from MEMS resonators,” IEEE Transactions on Ultrasonics,Ferroelectrics, and Frequency Control. IEEE. Vol. 61 (2014) No: 7, 1063-1074), but the two data points with the highest doping concentration have not previously been published. Based on the experimental data, it can be expected that the 2.sup.nd order TCF of the [100]-aligned Lame mode resonator attains even more positive values at higher dopant levels. This has indeed been assumed in FIG. 2D, where the behavior of the b.sub.11-12 term has been extrapolated.

(21) It should be understood that the invention covers a large number of different resonance frequencies, doping concentrations, geometrical configurations (including shape, thickness and orientation) of the resonator, only some of which are exemplified in the drawings or in the present description. Common to them is that the resonator is capable of resonating the WE mode, has a relatively high n-doping concentration and a wide temperature-stable operating frequency range. It should be noted that is not possible to cover all possible parameter combinations in detail, but using the principles herein disclosed a skilled person is able to find a parameter combination that suits his needs.

(22) In addition to the n-type dopant, there may be p-type dopant present in the resonator. There may for example be a homogeneous p-type background doping in the crystal.

(23) The actuator of the present micromechanical resonator can be for example a piezoelectric actuator or electrostatic actuator, or any other actuator suitable for exciting resonance modes known per se. According to one embodiment, the actuator comprises a piezoelectric actuator positioned on top of the resonating element. The piezoelectric actuator may comprise for example an aluminum nitride (AlN) layer and a molybdenum electrode. In configurations symmetrized along a symmetry axis, two or more actuators may also be placed symmetrically with respect to that axis in order to maintain symmetry of the resonator device. Both piezoelectric and electrostatic actuators are known per se and applicable to the present resonator design by a skilled person and not discussed herein in further detail.

(24) It is to be understood that the embodiments of the invention disclosed are not limited to the particular structures, process steps, or materials disclosed herein, but are extended to equivalents thereof as would be recognized by those ordinarily skilled in the relevant arts. It should also be understood that terminology employed herein is used for the purpose of describing particular embodiments only and is not intended to be limiting.

(25) Reference throughout this specification to “one embodiment” or “an embodiment” means that a particular feature, structure, or characteristic described in connection with the embodiment is included in at least one embodiment of the present invention. Thus, appearances of the phrases “in one embodiment” or “in an embodiment” in various places throughout this specification are not necessarily all referring to the same embodiment.

(26) As used herein, a plurality of items, structural elements, compositional elements, and/or materials may be presented in a common list for convenience. However, these lists should be construed as though each member of the list is individually identified as a separate and unique member. Thus, no individual member of such list should be construed as a de facto equivalent of any other member of the same list solely based on their presentation in a common group without indications to the contrary. In addition, various embodiments and example of the present invention may be referred to herein along with alternatives for the various components thereof. It is understood that such embodiments, examples, and alternatives are not to be construed as de facto equivalents of one another, but are to be considered as separate and autonomous representations of the present invention.

(27) Furthermore, the described features, structures, or characteristics may be combined in any suitable manner in one or more embodiments. In the following description, numerous specific details are provided, such as examples of lengths, widths, shapes, etc., to provide a thorough understanding of embodiments of the invention. One skilled in the relevant art will recognize, however, that the invention can be practiced without one or more of the specific details, or with other methods, components, materials, etc. In other instances, well-known structures, materials, or operations are not shown or described in detail to avoid obscuring aspects of the invention.

(28) While the forgoing examples are illustrative of the principles of the present invention in one or more particular applications, it will be apparent to those of ordinary skill in the art that numerous modifications in form, usage and details of implementation can be made without the exercise of inventive faculty, and without departing from the principles and concepts of the invention. Accordingly, it is not intended that the invention be limited, except as by the claims set forth below.