Micromechanical devices comprising n-type doping agents

Abstract

The invention concerns a micromechanical device and method of manufacturing thereof. The device comprises an oscillating or deflecting element made of semiconductor material comprising n-type doping agent and excitation or sensing means functionally connected to said oscillating or deflecting element. According to the invention, the oscillating or deflecting element is essentially homogeneously doped with said n-type doping agent. The invention allows for designing a variety of practical resonators having a low temperature drift.

Claims

1. A micromechanical bulk acoustic wave (BAW) device comprising an oscillating or deflecting element which is a resonator element made of semiconductor material comprising n-type doping agent, and an excitation or sensing means functionally connected to said resonator element and comprising transducer means for exciting a resonance mode to the resonator B element, wherein the oscillating or deflecting element is essentially homogeneously doped with said n-type doping agent having a doping concentration sufficient to set the temperature coefficient of frequency (TCF) of the resonator to 5 ppm/ C. at 25 C.

2. The micromechanical device according to claim 1, wherein the oscillating or deflecting element comprises the n-type doping agent in an average concentration of at least 1.0*10.sup.19 cm.sup.3.

3. The micromechanical device according to claim 1, wherein the minimum dimension of the oscillating or deflecting element is 5 m or more.

4. The micromechanical device according to claim 3, wherein the resonator element is adapted to resonate in a length or width extensional mode or in-plane or out-of plane flexural mode and the concentration of the n-type doping agent at least 1.6*10.sup.19 cm.sup.3.

5. The micromechanical device according to claim 3, wherein the resonator element is a beam.

6. The micromechanical device according to claim 5, wherein the beam is manufactured on a 100-plane wafer or on a 110-plane wafer, the main axis of the beam being oriented along the [110] direction of the semiconductor material, or on a 110-plane wafer so that the main axis of the beam is along a direction that is obtained by rotating the beam within the plane by 20 to 50 degrees from the [110] direction towards the [100] direction of the semiconductor material, and adapted to resonate in a torsional mode.

7. The micromechanical device according to claim 1, wherein the resonator element is adapted to resonate in a shear mode and the concentration of the n-type doping agent is at least 1.1*10.sup.19 cm.sup.3.

8. The micromechanical device according to claim 1, wherein the resonator element is adapted to resonate in a square extensional mode and the concentration of the n-type doping agent is at least 2*10.sup.19 cm.sup.3.

9. The micromechanical device according to claim 1, wherein the oscillating or deflecting element comprises a silicon crystal structure and the n-type doping agent is phosphorus, arsenic or antimony.

10. The micromechanical device according to claim 1, wherein the resonator element is free from pn-junctions.

11. The micromechanical device according to claim 1, wherein the doping concentration of the n-type doping agent is sufficient to set the temperature coefficient of frequency (TCF) of the resonator element to 0 ppm/ C. at 25 C.

12. The micromechanical device according to claim 1, wherein the resonator element comprises a body, which can be divided into at least one mass element and at least one spring, and the resonator element is adapted to resonate in a resonance mode in which one or more springs experience torsion.

13. The micromechanical device according to claim 12, wherein the orientation of the at least one spring is along the [100] crystal direction.

14. The micromechanical device according to claim 1, wherein the resonator element comprises a body, which can be divided into at least one mass element and at least one spring, and the resonator element is adapted to resonate in a resonance mode in which one or more springs experience flexure and/or extension.

15. The micromechanical device according to claim 1, wherein the resonator element comprises a plate.

16. The micromechanical device according to claim 15, wherein the resonator element comprises a plate which can be divided into a plurality of similar subsquares.

17. The micromechanical device according to claim 15, wherein the resonator element is adapted to resonate in a Lame resonance mode or in a square extensional (SE) resonance mode.

18. The micromechanical device according to claim 15, wherein the resonator element comprises a rectangular plate manufactured on a 100-plane wafer, the sides of the plate coinciding with the [100] directions of the crystal of the semiconductor material of the resonator element.

19. The micromechanical device according to claim 1, wherein the resonator element is a first resonator element and wherein the micromechanical device further comprises, at least one second resonator element mechanically coupled with the first resonator element, said first and second resonator elements having different contributions to the overall temperature coefficient of frequency (TCF) of the resonator.

20. The micromechanical device according to claim 1, wherein the resonator element comprises a homogeneously doped silicon plate or beam having a thickness of at least 4 m and at least one lateral dimension of at least 50 m, and the transducer means is adapted to produce a shear, square extensional, width extensional or flexural plate bulk acoustic wave mode or extensional, flexural or torsional beam bulk acoustic wave mode to the resonator element.

21. The micromechanical device according to claim 1, wherein the resonator element is a beam.

22. A method of manufacturing a micromechanical bulk acoustic wave (BAW) device, comprising the steps of: providing a semiconductor wafer comprising a homogeneously n-doped device layer, processing the semiconductor wafer to form a resonator element from the n-doped device layer, the element being capable of deflection or oscillation, providing excitation or sensing means functionally connected to said resonator element for exciting a resonance mode to the resonator element or sensing the resonance frequency or degree of deflection of the element wherein the resonator element is doped with a n-type doping agent having a doping concentration sufficient to set the temperature coefficient of frequency (TCF) of the resonator element to 5 ppm/ C. at 25 C.

23. A micromechanical device comprising an oscillating or deflecting resonator element made of silicon having a crystal orientation that deviates less than 30 from a direction that maximizes the temperature coefficient of frequency of the device, and comprising an n-type doping agent, the resonator element being essentially homogeneously doped with said n-type doping agent, and excitation or sensing means functionally connected to said resonator element and comprising transducer means for exciting a resonance mode to the resonator element, wherein the resonator element is adapted to resonate in a shear mode and the concentration of the n-type doping agent is at least 1.1*10.sup.19 cm.sup.3.

24. The micromechanical device according to claim 23, wherein the resonator element is free from pn-junctions.

25. The micromechanical device according to claim 23, wherein the resonator element comprises a plate.

26. The micromechanical device according to claim 25, wherein the resonator element comprises a plate which can be divided into a plurality of similar subsquares.

27. The micromechanical device according to claim 23, wherein the resonator element is a first resonator element, the micromechanical device further comprising at least one second resonator element mechanically coupled with the first resonator element, said first and second resonator elements having different contributions to an overall temperature coefficient of frequency (TCF) of the resonator.

28. The micromechanical device according to claim 23, wherein said deviation of the crystal orientation of said resonator element from said direction that maximizes the temperature coefficient of frequency of the device, is achieved by rotating the resonator element from an optimal direction on the lateral plane of said resonator element.

29. The micromechanical device according to claim 23, wherein the excitation or sensing means comprise a piezoelectric thin film excitation means or electrostatic excitation means.

30. The micromechanical device according to claim 23, wherein the device is adapted to be actuated without bias current.

31. The micromechanical device according to claim 23, wherein the minimum dimension of the oscillating or deflecting element is 5 m or more.

32. A micromechanical device comprising an oscillating or deflecting resonator element made of silicon having a crystal orientation that deviates less than 30 from a direction that maximizes the temperature coefficient of frequency of the device, and comprising an n-type doping agent, the resonator element being essentially homogeneously doped with said n-type doping agent, and excitation or sensing means functionally connected to said resonator element and comprising transducer means for exciting a resonance mode to the resonator element, wherein the resonator element is adapted to resonate in a square extensional mode and the concentration of the n-type doping agent is at least 2*10.sup.19 cm.sup.3.

33. The micromechanical device according to claim 32, wherein the minimum dimension of the oscillating or deflecting element is 5 m or more.

34. The micromechanical device according to claim 32, wherein the resonator element is free from pn-junctions.

35. The micromechanical device according to claim 32, wherein the resonator element comprises a plate.

36. The micromechanical device according to claim 35, wherein the resonator element comprises a plate which can be divided into a plurality of similar subsquares.

37. The micromechanical device according to claim 32, wherein the resonator element is a first resonator element, the micromechanical device further comprising at least one second resonator element mechanically coupled with the first resonator element, said first and second resonator elements having different contributions to the overall temperature coefficient of frequency (TCF) of the resonator.

38. The micromechanical device according to claim 32, wherein said deviation of the crystal orientation of said resonator element from said direction that maximizes the temperature coefficient of frequency of the device, is achieved by rotating the resonator element from the optimal direction on the lateral plane of said resonator element.

39. The micromechanical device according to claim 32, wherein the excitation or sensing means comprise a piezoelectric thin film excitation means or electrostatic excitation means.

40. The micromechanical device according to claim 32, wherein the device is adapted to be actuated without bias current.

Description

BRIEF DESCRIPTION OF THE DRAWINGS

(1) FIGS. 1a and 1b illustrate top and cross section views of electrostatically actuated plate resonators fabricated on SOI or CSOI wafer.

(2) FIGS. 2a and 2b illustrate top and cross section views of piezoelectrically actuated plate resonators fabricated on SOI or CSOI wafer.

(3) FIG. 3 shows the thermal dependency of the Lam mode frequency, when the plate resonator is aligned in such a way that the plate sides coincide with the [100] directions.

(4) FIG. 4 shows the thermal dependency of the SE mode frequency, when the plate resonator is aligned in such a way that the plate sides coincide with the [100] directions.

(5) FIG. 5 shows a piezoelectrically actuated Lam resonator array.

(6) FIG. 6 illustrates the temperature coefficient of a flexural/length extensional beam resonator for all possible beam orientations relative to the silicon crystal.

(7) FIGS. 7-15 illustrate exemplary modeshapes taking advantage of n-doping for adjusting the temperature drift, and the dependency of respective simulated TCF values on relevant design parameters:

(8) FIG. 7: Plate resonator in a Lam/face-shear mode

(9) FIG. 8: Plate resonator in a square extensional mode

(10) FIG. 9: Plate resonator in flexural modes

(11) FIG. 10: Plate resonator in a width extensional mode

(12) FIG. 11: Beam resonator in length-extensional 1.sup.st order mode

(13) FIG. 12: Beam resonator in length-extensional 3.sup.rd harmonic mode

(14) FIG. 13: Beam resonator in an in-plane flexural mode

(15) FIG. 14: Beam resonator in an out-of plane flexural mode

(16) FIG. 15: Beam resonator in a torsional mode

(17) FIG. 16 shows an exemplary compound resonator.

(18) FIGS. 17a-17c show the silicon elastic constant c.sub.11, c.sub.12 and c.sub.44 sensitivities as a function of n-dopant concentration.

(19) FIG. 18 shows the silicon elastic matrix term c.sub.11-c.sub.12 sensitivity as a function of n-dopant concentration.

(20) FIG. 19 shows the location of two square extensional modes, a Lam mode and a flexural beam mode in a (Q/P, R/P)-plane.

DETAILED DESCRIPTION OF EMBODIMENTS

(21) According to one aspect thereof, the present in invention presents the idea and applications of heavy n-type (e.g. by phosphorous) doping of silicon for compensating for the temperature drift of resonance frequency of silicon MEMS resonators. As will be shown below, the invention can be applied for resonators with various geometries, such as beam resonators and plate resonators and their combinations, various wave types, such as longitudinal and shear bulk acoustic waves (BAWs), and various resonance modes, including torsional, flexural and extensional modes.

(22) Many specific examples of these variations are given below after a short discussion of general aspects of n-doping, applicable to several or all of the specific examples.

(23) The temperature drift of silicon with standard doping levels, if no special measures taken, is in the order of 30 ppm/ C. According to typical embodiments of the invention, the temperature drift of the n-doped silicon resonator is 20-+20, in particular 5-+5 ppm/ C., depending i.a. on the doping concentration, crystal orientation, modeshape, geometrical design and interfering material structures of the resonator. It is common that TCF curves are determined by fitting an N-order (typically N=3) polynomial function to measured temperature drift data at the temperature of 25 C. Unless otherwise mentioned, the temperature drift (or TCF) values cited herein are given at 25 C. The values may differ from that in other temperatures due to nonlinearity of the temperature/frequency curve, which may occur, as will be discussed later in more detail.

(24) According to one embodiment, the resonator element comprises an n-doped silicon crystal essentially free (within normal purity levels) from p-type doping agents, yielding a TCF of 3 ppm/ C. or higher.

(25) According to one embodiment, the n-type doping is homogeneous within ordinary manufacturing tolerances throughout the resonator thickness. Thus, there are no interfaces of differently behaving materials inside the resonator element.

(26) The experimental and theoretical investigation described in the following shows that by n-type doping the compensation of modes characterized by mainly by the c.sub.11-c.sub.12 stiffness term, in contrast to previously known modes characterized by c.sub.44 stiffness term and temperature compensated by p doping.

(27) According to one embodiment, the resonator element comprises a silicon crystal doped with phosphorus to a relatively heavy doping concentration of 10.sup.18 . . . 10.sup.20 cm.sup.3. Such concentration is sufficient for the manufacture of close to zero (TCF 3 ppm/C) temperature drift resonators operating in the square extensional (SE) plate mode dominated by the c.sub.11 stiffness constant. Such concentration is also sufficient for manufacturing close to zero (TCF 3 ppm/ C.) temperature drift resonators operating in the c.sub.11-c.sub.12 characterized Lam mode, with the additional possibility to achieve TCF >0 ppm/C in order to account for the negative temperature drift effect of additional thin film layers such as the electrodes and the piezolayer.

(28) The resonator element can comprise a crystal grown, for example, epitaxially or by the Czochralski method. Suitable methods are presented e.g. in Handbook of Semiconductor Silicon Technology, Edited by: O'Mara, W. C.; Herring, R. B.; Hunt, L. P. 1990 William Andrew Publishing/Noyes.

(29) Actuation of the Resonator

(30) The micromechanical resonator according to the invention can be actuated with transducer means known per se. According to one embodiment the transducer means comprise a piezoelectric actuator element. According to another embodiment, the transducer means comprise electrostatic actuator means.

(31) FIGS. 1a and 1b show the basic design of an electrostatically actuated MEMS resonator manufactured on a silicon substrate 18. The device layer on top of the substrate comprises a resonator element 16 separated from the surroundings, i.e. electrode layer 10, of the device layer and from the substrate by a gap 12. Anchors (not shown) hold the resonator element 16 in place. On at lateral sides of the resonator element 16, there are electrodes 14. When an alternating actuation voltage is coupled over the gap 12 from the electrode layer 10 to electrodes 14, the resonator element 16 can be set to oscillate.

(32) FIGS. 2a and 2b show the basic design of a piezoelectrically actuated MEMS resonator manufactured on a silicon substrate 28. The resonator element 26 is provided with a superimposed piezoelectric layer 27. By applying a voltage over the piezoelectric layer 27, e.g. from a conducting electrode (not shown) arranged on top of it to the resonator element itself, the piezoelectric layer subjects a force also to the resonator element 26.

(33) The present invention can generally be applied in both electrostatically and piezoelectrically actuated resonators.

(34) Of particular importance is a temperature compensated square extensional (SE) mode resonator that can be easily excited using a piezoelectic layer and an electrode layer deposited on top of the n-doped resonator element. As shown below, it has been observed that the SE mode has a close to zero TCF (1 ppm/ C.). The SE mode can be excited electrostatically as well.

(35) According to one embodiment, the resonator comprises an n-doped resonator element (TCF >0) and a piezoelectric layer (for example AlN) and electrode layer so as to form a piezoactivated MEMS resonator. The piezoelectric layer and electrode layer are known to have a negative TCF. However, the overall TCF of the resonator is close to zero because of overcompensation of the TCF of the resonator element as such. This arrangement is particularly suitable for temperature compensated SE mode resonators.

(36) With reference to FIG. 5, the Lam mode can also be excited using a piezoelectric actuator, for example, as disclosed in FI 20105849 by providing at least two resonator elements 50A, 50B laterally with respect to each other as an array and at least one piezoelectric transducer element 52 between the resonator elements 50A, 50B and coupled to the resonator elements. Other plate resonator modes within the scope of the present invention, like the Face-Shear mode or the Wine glass mode can be excited similarly, too.

(37) According to one embodiment the resonator comprises an array of overcompensated (TCF >0) resonator elements 50A, 50B and an undercompensated piezoelectric actuator 52 coupled to the resonator elements 50A, 50B. The overall TCF of the resonator is thus between the TCF's of its individual components and may be designed to be close to zero.

(38) In the example of FIG. 5, there is shown a two-plate Lam resonator array. However, as extensively discussed in FI 20105849, the relevant contents of which are incorporated herein by reference, the array can be two-dimensional and comprise a plurality of resonator plates and piezotransducers in a variety of geometries.

(39) Resonator Geometry

(40) According to one embodiment, the resonator element is a beam. The term beam generally refers to resonator elements whose in-plane aspect ratio (length vs. width) is at least 5. Typically, the aspect ratio is at least 10.

(41) According to one embodiment, the resonator element is a plate. The plate can be rectangular. For example, it may be a square plate. The plate can also be a polygonal, circular or elliptical. The aspect ratio of a plate (any lateral dimension vs. thickness) is greater than 5.

(42) Depending on the desired resonance mode and level of temperature compensation, the crystal orientation of the resonator is varied with respect to its side orientations, anchoring points and/or actuation means. In the following, the preferred resonance geometries and resonance modes are discussed in the theoretically most optimal crystal orientation, i.e. orientation maximizing the TCF. However, as will be discussed with reference to FIG. 6, deviation from this optimal orientation may be utilized for adjusting the TCF.

(43) The following discussion is divided into paragraphs according to the geometry of the resonator element. First, resonators comprising an essentially two-dimensional plate are discussed. Then, essentially one-dimensional beam resonators are discussed. Finally, some generalizations of the resonators are introduced and more complex geometries and variations taking advantage of the invention are briefly discussed.

(44) Anchoring of the resonators can be carried out at any suitable location allowing the desired resonance mode to emerge into the resonator element. Typically, the anchors are intended not to significantly contribute to the operation of the resonance element and are not considered to be part of the resonance element. However, as will be discussed later, there are some special designs in which the anchors are designed to be beams that have a considerable effect on the resonance properties and, in particular, TCF of the resonator. In this case, the anchors are part of the resonator element.

(45) Plate Resonators

(46) Shear Mode Plate Resonators

(47) A Lam mode excited to a square silicon plate, in which the crystal orientation in the silicon wafer is (100) and the plate sides are oriented along [100] directions, is an example of a pure shear mode resonator which can take full advantage of n-doping. In this configuration the resonance frequency of a Lam mode is proportional to sqrt(c.sub.11-c.sub.12).

(48) As an example, the Lam modeshape of a plate having dimensions (lengthwidthheight) of 320 m*320 m*10 m is illustrated in FIG. 7a. The Lam mode appears at 10 MHz.

(49) In addition to this basic Lam modeshape, also higher-order variants of the Lam mode are covered by the invention. The first order mode is comprised of two diagonally propagating shear waves in a square plate. In higher order modes this condition is satisfied in each square subplate the square plate is virtually divided in. Higher order Lam modes are more extensively discussed in FI 20105849. A shear wave of similar character is excited to a plate resonator, which is rotated 45 degrees within the plane of the resonator. This mode is called the face-shear mode, and it is of almost pure shear character. The mode shape of the face-shear mode is shown in FIG. 7b. The resonance frequency is 9 MHz.

(50) Finite element model simulations show that when a plate is rotated within the (100) plane the mode gradually changes from Lam mode to face-shear mode, and the resonance frequency is simultaneously decreased from 10 MHz to 9 MHz. The linear TCF for various n-dopant concentrations can be calculated using FEM simulations and stiffness matrix element temperature sensitivities from theory.

(51) The evolution from the Lam mode TCF to the face-shear mode TCF as the plate is rotated within (100) plane is shown in FIG. 7c. 0/90 degrees in-plane rotation angle corresponds to a plate with sides aligned with the [100] directions: at this orientation the Lam mode occurs. At 45 degrees rotation angle the sides are aligned with the [110] directions and the face-shear mode appears. At intermediate angles the mode gradually transfers from Lam mode to face-shear mode.

(52) In (110) plane modes that are similar to the above mentioned Lame/face-shear modes can be observed, but they are not as pure as in (100) plane in terms of their sqrt(c.sub.11-c.sub.12) dependency; hence we refer to these as pseudo-Lam mode and pseudo face-shear mode in the following. This is because (110) plane is less symmetric than the (100) plane. At 0/90 degrees in-plane rotation one side of the plate is aligned along [100] while the other is oriented along [110]. In the same manner, at 45 degrees in-plane rotation angle the diagonals are oriented along [100] and [110]. FIG. 7d shows the TCF values for the pseudo-Lam mode (0/90 degrees in-plane rotation), pseudo face-shear mode (45 degrees) and the mixture modes at intermediate angles.

(53) The TCF values in FIGS. 7c and 7d have been calculated by FEM simulation: the sensitivity of the resonance frequency with respect to elastic matrix elements c.sub.11, c.sub.12 and c.sub.44 has been extracted from modal simulations and the linear TCF values have been calculated with various n-dopant concentrations using the elastic matrix element temperature sensitivities from the theory (data points labeled with VTT theor.). Data points labeled with Bourgeois n low/Bourgeois p low are reference curves that represent silicon with relatively weak n-dopant/p-dopant concentration (see C. Bourgeois et al., Design of resonators for the determination of the temperaturecoefficients of elastic constants of monocrystalline silicon, in Frequency Control Symposium, 1997, Proceedings of the 1997 IEEE International, 1997, 791-799) (corresponding Si resistivities are 4 Ohm*m and 0.05 Ohm*m for Bourgeois p low and Bourgeois n low, correspondingly). A constant contribution of +1.3 ppm/ C. from thermal expansion has been assumed for all data (see chapters Theoretical model for TCF of a resonator and Theoretical model for the stiffness matrix element temperature sensitivities). The given description of the calculated data concerns any data plots of the similar type in the treatment below unless stated otherwise. In the FIGS. 8a and 8b the line labeled with theor. approx. is the calculation for an ideal shear mode, whose frequency is proportional to sqrt(c.sub.11-c.sub.12); elastic matrix element temperature sensitivities from theory with n-dopant concentration n=5*10.sup.19 cm.sup.3 have been used. Experimental data point is shown in FIG. 7c with legend VTT experimental, n=5e19.

(54) Following observations can be made from the data in FIGS. 7c and 7d: The TCF for simulated Lam mode coincides with the theor. approx. curve, which confirms the validity of the simulation; the form of Lam mode resonance frequency is analytically known and the frequency is proportional to sqrt(c.sub.11-c.sub.12). With n-dopant concentration n=5*10.sup.19 cm.sup.3 the Lam-mode, the face shear-mode and the modes at intermediate angles are overcompensated with TCF >+13 ppm/ C. For n-dopant concentration n=2*10.sup.19 cm.sup.3 the same holds with TCF>+6 ppm/ C. The concentration at which TCF is zero for the Lam mode is approximately n=1.2*10.sup.19 cm.sup.3. The experimental data point coincides well with the simulated data. In the (110) plane the quasi-Lam, quasi face-shear modes and the modes at intermediate angles have TCFs above zero for n=5*10.sup.19 cm.sup.3 and n=2*10.sup.19 cm.sup.3, although due to lack of symmetry the within the (110) plane the modes are not pure shear modes. The TCF of any of the modes can be zeroed by selecting the concentration level suitably at a value n<2*10.sup.19 cm.sup.3. The typical silicon TCF of approximately 30 ppm/ C. is observed with relatively low n- or p-doping.

(55) The Lam mode is important because its temperature drift can be overcompensated with well realizable dopant levels. Experimentally it has been shown that a TCF of even +18 ppm/ C. can be achieved in such resonator.

(56) The invention is not limited to simple square plate Lam mode plate resonators, but also more complex variants which can be theoretically divided into subsquares, can be realized. The principle is the same as that more extensively discussed in the Finnish patent application No. 20105849 of the same applicant.

(57) Square Extensional (SE) Mode Plate Resonators

(58) SE mode excited to a square silicon plate, in which the crystal orientation in the silicon wafer is (100) and the plate sides are oriented along [100] directions, represents an example of an extensional mode resonator which can take advantage of n-doping. The SE mode at 13 MHz is illustrated in FIG. 8a for a similar plate resonator as was discussed in the previous example of a Lam or face-shear mode resonator.

(59) FIG. 8b illustrates the calculated temperature coefficients of the SE mode, when the plate resonator is rotated within the (100) plane. 0/90 degrees in-plane rotation angle corresponds to plate sides aligned with [100] directions, 45 degrees corresponds to side alignment with [110].

(60) Observations are: The TCF attains its highest value for plate with sides aligned with [100] directions. Experimental data agrees reasonably well with simulated data. The observed offset of approximately 1 ppm/C between experiment and simulation can be attributed to corner anchors. (Fine-tuning of the system with anchors or a central hole in the resonator element are discussed later.) Again, with low level of p- or n-doping the typical silicon TCF of about 30 ppm/ C. is reproduced. Point labelled as SE100 in FIG. 19 indicates that the SE mode of a plate whose sides are aligned with the [100] directions fullfills the temperature compensation criterion. Similar plate with alignment with [110] does not meet the criterion.

(61) Flexural Mode Plate Resonators

(62) Flexural saddle modes that exist in square-shaped plate resonators are strongly c.sub.11-c.sub.12 characterized modes, when aligned suitably with the crystal axis. Saddle modes are characterized by the saddle surface shape of the resonance mode, which results in two intersecting nodal lines (sets of locations with no displacement) through the resonator body (for more discussion on saddle modes, see FI 20105851). Two types of saddle modes in plates of dimensions (width*length*thickness) of 320 m*320 m*10 m are shown in FIGS. 9a and 9b.

(63) According to simulations, saddle mode of FIG. 9a is has maximal (above zero, assuming sufficiently high n dopant concentration (>1.2*10.sup.19/cm.sup.3)) TCF when the plate is fabricated to (100) plane and the plate sides are aligned to [100] directions or
the plate is fabricated to (110) plane and one side of the plate is along [100] direction and other side is along [110] direction

(64) Saddle mode of FIG. 9b is has maximal (above zero) TCF when the plate is fabricated to (100) plane and the plate diagonals are aligned to [100] directions, or the plate is fabricated to (110) plane and one side of the diagonals is along [100] direction and other diagonal is along [110] direction.

(65) As discussed in more detail in FI 20105851, saddle modes are not restricted to rectangular plates, and modes of similar character exist, e.g., in disk-shaped plates as well.

(66) In addition to the saddle modes discussed above, other flexural (bending) resonance modes, which can be temperature compensated by n-doping, can be excited to a plate resonator. These modes can be can be characterized as being similar modes as the out-of-plane flexural modes of a beam resonator (discussed later in a separate section). A plate resonator can be described as a beam whose lateral dimensions (height, width) are close or equal to each other.

(67) Width Extensional Plate Resonators

(68) It turns out that when the length of one of the sides of a square plate resonator is varied, the mode shape changes gradually from the square extensional mode (FIG. 8a) to a width extensional (WE) mode (FIG. 10a), a width extensional mode of a 320*680*10 m.sup.3 resonator at 12 MHz, where the in-plane extension occurs in the direction defined by the smaller side.

(69) Interestingly, the WE mode is more susceptible to temperature compensation by n-doping than the SE-mode. FIG. 10b shows the simulated TCFs of the resonance modes, which evolve from WE mode to SE mode, and again back to WE mode, as one side of the resonator is changed from 230 m to 680 m and the other side is kept at 320 m. The SE mode at side length of 320 m has the lowest TCF, and TCF is increased as the side length aspect ratio departs from 1. The graph suggests a WE mode can be temperature compensated at an n-dopant concentration of about 2.3*10.sup.19 cm.sup.3, when the ratio of the sides is more than 2:1.

(70) Beam Resonators

(71) Approximation of the Frequency Vs. Elastic Matrix Elements Relationship of the Extensional/Flexural Resonance of a Beam Resonator

(72) The Young's modulus for one dimensional stretching/shrinking of an element of a material is given by Y1D=T/S, where T is the stress along the stretching/shrinking direction and S is the related strain. We assume that no stresses affect the material element in the directions that are perpendicular to T. If we assume that the material has cubic crystal symmetry, the stress vs. strain relationship is given by the matrix equation [T]=[c][S], where [T] and [S] are 61 stress and strain matrices, respectively, and [c] is the 66 elastic matrix with three independent elements c.sub.11, c.sub.12 and c.sub.44. Solving Y1D for stretch along the [100] crystal axes yields a result Y1D=c.sub.112*c.sub.12.sup.2/(c.sub.11+c.sub.12).sup.2.

(73) The bending stiffness and the extensional stiffness of a beam are proportional to Y1D. Therefore the resonance frequency of a flexural (bending) resonance mode or an extensional resonance mode is proportional to sqrt(Y1D).

(74) Point labeled as Y1D in FIG. 19 indicates that extensional/flexural resonance modes belong to the class of resonance modes that can be temperature compensated with n-doping. FIG. 6 shows the value of the linear TCF calculated from Y1D in all possible crystal orientations of a cubic crystal. The calculation is based on theory and a n-dopant concentration of n=5*10.sup.19 cm.sup.3 has been assumed. TCF is maximized along [100] directions. This approximative result suggests that a flexural or extensional resonance of a beam can be temperature compensated when the beam is oriented along a [100] direction, or does not remarkably deviate from a [100] direction. The simulated examples below provide further proof for this.

(75) Torsional beams with rectangular cross sections are discussed in more detail below. However, the invention can be generalized to beams with non-rectangular cross-sections (e.g. circle or ellipse), and to even beams, whose cross section is varied along the length of the beam (e.g. a tapering beam).

(76) Extensional Mode Beam Resonators

(77) A beam shaped body has a length extensional resonance where the resonance is characterized by the contraction/extension of the resonator. The resonance frequency is approximately given by f=sqrt(Y1D//2L, where Y1D is the Young's modulus for 1D stretch defined above, is the resonator density and L is the resonator length. As suggested above the resonance can be temperature compensated by n-doping when the resonator length dimension is aligned along the [100] crystal direction (or when the deviation from the [100] direction is small). The resonator can be fabricated on any wafer plane.

(78) Example of a beam resonator extensional resonance mode is shown in FIG. 11a. The dimensions (lengthwidthheight) of the beam are 320 m*5 m*10 m.

(79) FIGS. 11b and 11c (see legend in FIG. 11c) illustrate how the linear TCF changes as the beam resonator of FIG. 11a is rotated within the plane (100) or within plane (110). For (100) plane (FIG. 11b) in-plane rotation angles 0 or 90 correspond to beam being aligned along [100] direction while 45 degrees corresponds to alignment with [110]. For (110) plane (FIG. 11c) in-plane rotation angle 0 corresponds to [100] alignment and 90 degrees corresponds to alignment with [110]. The curve labeled with theor. approx. is the calculation based on Y1D, with elastic matrix element temperature sensitivities from the theory with n-dopant concentration n=5*10.sup.19 cm.sup.3. All other data labels are as described in the above discussion related to Lam/face shear modes.

(80) Following observations can be made: For dopant concentration n=5*10.sup.19 cm.sup.3 resonators aligned with the [100] direction are overcompensated with TCF >10 ppm/ C. For dopant concentration n=5*10.sup.19 cm.sup.3 TCF is zeroed for approximately 20 degrees deviation from [100]. For the optimal direction, near zero TCF is attained at an approximate concentration of n=1.6*10.sup.19 cm.sup.3. The approximation based on Y1D and the simulated data agree well with each other. The typical silicon TCF of approximately 30 ppm/ C. is obtained with relatively low n or p doping. For each rotation angle between about 0 . . . 25, there is an optimal doping concentration between about n=1.6*10.sup.19 . . . 5*10.sup.19 cm.sup.3.

(81) In addition to the first order length extensional mode illustrated in FIG. 11a, higher order length extensional modes can be temperature compensated in a similar manner. FIG. 12a illustrates the mode shape of the third-order extensional mode of the same resonator as discussed above. Instead of one nodal point at the center the resonance mode has three nodal points along its length axis.

(82) FIGS. 12b and 12c illustrate the calculated temperature coefficients for the third-order length-extensional mode. Observations are essentially similar as for the first-order length-extensional mode.

(83) Flexural Mode Beam Resonators

(84) A beam similar to as referred above in connection with length extensional modes can be excited also to a flexural mode (including in-plane flexure and out-of plane flexure). A flexural mode resonance is proportional to sqrt(Y1D), and therefore it can be temperature compensated by n-doping when it is aligned along the [100] crystal direction in the same manner as the length extensional resonator from the above discussion.

(85) FIG. 13a illustrates the lowest-order in-plane flexural resonance mode of the beam resonator from previous examples.

(86) FIGS. 13b and 13c illustrate the calculated temperature coefficients for the first-order in-plane flexural mode. The results are very similar to what they were for the length-extensional resonance mode. In particular, for each rotation angle between about 0 . . . 25, there is an optimal doping concentration between about n=1.6*10.sup.19 . . . 5*10.sup.19 cm.sup.3.

(87) In addition to the first order flexural mode illustrated in FIG. 13a, higher-order modes can be used too. As an example of such a mode, an illustration of a higher-order out-of-plane flexural mode is shown in FIG. 14a. The resonator dimensions are again the same as in the previous examples.

(88) FIGS. 14b and 14c illustrate the calculated temperature coefficients for this higher-order out-of-plane flexural mode. Temperature compensation appears to work in the same manner as in the above examples. For each rotation angle between about 0 . . . 25, there is an optimal doping concentration between about n=1.8*10.sup.19 . . . 5*10.sup.19 cm.sup.3. It can be noted that the contribution to resonance frequency from other than (c.sub.11-c.sub.12) terms is increased a bit for a higher-order mode, when compared to previous examples, and therefore e.g. maximum overcompensation with n=5*10.sup.19 cm.sup.3 is slightly smaller. From the same reason the minimum n-dopant concentration with which TCF can be zeroed is lower: according to the simulation this lower limit is at approximately n=1.8*10.sup.19 cm.sup.3.

(89) Torsional Mode Beam Resonators

(90) FIG. 15a shows the mode shape of a first-order torsional resonance of a beam with dimensions (length*width*height) of 320 m*40 m*10 m. The torsion axis is defined by the length dimension of the beam. A torsional resonance is strongly dependent on (c.sub.11-c.sub.12) when the torsion axis is oriented along the [110] axis and the larger of the beam cross section dimensions is also oriented along [110] (this ties the smaller cross section dimension to be aligned with [100]).

(91) Conditions for the torsional beam cross section aspect ratio and the needed n-doping concentration are more accurately quantified by simulation results presented in FIGS. 15b, 15c and 15d, where the TCF of the first order torsional mode at different n-dopant concentration levels has been simulated as a function of the thickness of beam (length and width of the beam being identical to that in FIG. 15a).

(92) FIG. 15b shows the simulation result for the beam that is fabricated on a 110-plane so that its length is along the [110] direction.

(93) FIG. 15c shows the simulation result for the beam that is fabricated on a 100-plane so that its length is along the [110] direction.

(94) FIG. 15d shows the simulation result for the beam that is fabricated on a 110-plane so that its length is along the direction that is obtained by rotating the beam within the plane by 35 degrees from the [110] direction towards [100].

(95) Observations are: In cases shown by FIGS. 15b and 15c TCF=0 is attained at n-dopant concentration n=5*10.sup.19 cm.sup.3 at approximately the thickness of 40 microns, which is the case at which beam thickness is equal to beam width. Thus, temperature compensation is possible for beam fabricated to (100) plane, whose length is along [110] direction and whose width is larger than, or approximately equal to, the height (thickness) a beam fabricated to (110) plane, whose length is along [110] direction and whose height (thickness) is larger than, or approximately equal to, the width. FIGS. 15b and 15c indicate that the lowest n-dopant concentration at which temperature compensation still is possible (with extreme cross sectional aspect ratios) is approximately n=1.3*10.sup.19 cm.sup.3. FIG. 15d shows that for a beam fabricated to (110) plane there is an intermediate angle, at which the TCF dependence on the beam cross section aspect ratio is minimized. This direction appears to be 20-50 degrees, in particular near 35 degrees tilt from the [110] direction towards [100]. TCF being independent of the cross-sectional aspect ratio is advantageous in practice since it provides roboustness against process variations and allows more freedom for the device designer (e.g. a device may contain multiple torsional springs with different cross sectional aspect ratios, and all those springs have similar effect to the TCF).

(96) Simulations were done at 5 degree steps, and the cases with 30/40 degrees tilt were inferior to the presented case. Optimal tilt direction is expected to occur between 33 and 37 degrees. Importantly, an intermediate angle producing similar aspect-ratio independence is not found for beams fabricated to the (100) plane.

(97) In addition to the first order torsional mode illustrated in FIG. 15a, and discussed in the context of FIGS. 15b-d, TCF behavior of higher-order torsional modes is of similar character, and can be used as well.

(98) Generalizations and Variations

(99) The abovementioned principles and resonator structures can be applied in various ways in order to achieve more complex resonator entities. Thus, the geometry of the resonator can be designed to meet the needs of a particular application and still the temperature compensation be adjusted to the desired level. For example, additional mass-loading elements can be brought to plate or beam resonators in order to adjust the resonance frequency of the resonator. Flexural-mode mass-loaded resonators per se are more extensively discussed in the Finnish patent application No. 20105851 of the same applicant.

(100) Any compound resonator, which can be divided into mass element(s) and spring(s) can be temperature compensated with n-doping, when the resonance mode is such that the at least some of the spring(s) experience extension or bending (flexure), and simultaneously the springs and their orientation with respect to the crystal fullfill the conditions presented above in the context of extensional/flexural mode beam resonators, experience torsion, and simultaneously the spring(s) and its/their dimension(s) fullfill the conditions presented above in the context of torsional mode beam resonators.

(101) It should be noted that a compound resonator may contain of multiple springs, and the individual springs may independently experience extension, bending or torsion.

(102) FIG. 16 shows a simple exemplary compound resonator design, which can be divided into a spring and a mass. This system has resonance modes, where the spring experiences extensional, flexural or torsional oscillation, and, therefore these resonance modes may be temperature compensated by n-doping when the spring alignment with the crystal and the spring dimensioning are selected correctly: For extensional/flexural modes the sufficient condition (assuming a correct level of doping) is that the main axis of the beam is oriented along the [100] crystal direction, for torsional modes the orientation conditions are more stringent as discussed above.

(103) Discussion in chapter Shear mode plate resonators was restricted to (pseudo) Lam modes and (pseudo) face-shear modes of square plate resonators. As appreciated by a skilled person, the resonator body does not have to be of square shape to be able to resonate in a shear mode whose frequency would be (c.sub.11-c.sub.12)-characterized. For example, the so called Wine glass resonance mode of a circular resonator plate is a (c.sub.11-c.sub.12)-characterized shear mode that can be temperature compensated. Even further, allowing the geometry to deviate from the perfect square or disk shapes to more asymmetric geometries gradually changes the strongly (c.sub.11-c.sub.12)-characterized shear modes to modes with weaker (c.sub.11-c.sub.12) dependence, which, however, due to the overcompensation capability with n-doping can still be temperature compensated with a suitable n-dopant concentration.

(104) In general, although such modifications can bring non-idealities to the system and decrease the temperature compensation effect of n-doping of the resonator element with respect to simple geometries, the desired compensation level may still be well achievable due to the overcompensation capability with n-doping.

(105) Optimization and Practical Implementations of n-Doped Resonators

(106) As is apparent from the above discussion, many resonators designs can be overcompensated with the aid of the invention. This fact indicates that there is some slack to trade off when optimizing the resonator overall performance. The aim typically is to have the overall temperature compensation close to zero. This could be achieved, for example, by optimizing the frequency vs. temperature behavior by: Adjusting the doping concentration of the n-dopant suitably, Inclusion of additional dopants, typically to a total amount of less than 50%, in particular 1-49%, typically less than 30%, of the number of all dopant atoms. The additional dopants may be of n- or p-type, or both. Choosing the angle of the resonator element with respect to the silicon crystal suitably. Any deviation from the optimal angle will cause the TCF to drop. Thus, by deviating overcompensated resonators from the optimal direction axis (typically by rotating the resonator at the lateral plane) the TCF can be adjusted to the desired level. Rotation angle may be e.g. 130. Providing to the resonator structure additional parts, optionally with negative TCFs. Thus, the overall TCF of the resonator could be tuned to zero by suitable selection of the materials and the resonator design. The additional parts may comprise, for example, additional mass elements which are part of the resonator element, or anchors or transducer elements generally not considered to be part of the resonator element. A piezoactuated SE resonator and a Lam resonator array described in more detail below are examples of such design.

(107) Special Features

(108) As discussed in our previous patent application PCT/FI2010/050935, the effect of manufacturing tolerances of BAW resonators can be minimized by providing at least one void to the resonanor element. In FEM simulations it has been found that a central void provided to an n-doped resonator can also increase the TCF of the resonator. For example, in the case of a SE mode square plate resonator of dimensions 32032010 m.sup.3, over +2 ppm/ C. increase in TCF was observed, when a central void of 100 micrometers diameter was created in the center of the plate (assuming n-dopant density of 5*10.sup.19 cm.sup.3). Similar behaviour can be expected for other modes, too.

(109) Consequently, according to one aspect of the invention, the n-doped resonator element comprises at least one void, typically in the form of a recess or through-hole in the resonator element. Preferably, the void is in the form of a closed-loop trench. Typically, the void is provided in the middle of the resonator element but it may also be located in non-central position or there may be an array of symmetrically or non-symmetrically arranged voids.

(110) According to another aspect of the invention, the anchors of the resonator plate are designed to be beams that have a considerable influence on the temperature compensation properties of the resonator. In this context, the term considerable influence means that they affect the TCF of the resonator by at least 2 ppm/ C.

(111) According to one still another aspect, the resonator device comprises at least two separate resonator elements which have a TCF difference, preferably of the order of 30-50 ppm/ C. At least one or both of the resonator elements may by n-doped. According to a preferred embodiment, both resonator elements are n-doped, typically with the same concentration, but their crystal orientations differ by 45. For example, there could be two Lam resonators, one having a TCF of 30 ppm/ C. and the other a TCF of +18 ppm/ C. Measurement of both of the two resonator elements can be used for temperature compensation utilizing the TCF difference. This kind of method is disclosed in more detail in U.S. Pat. No. 7,145,402.

(112) Theoretical Model for TCF of a Resonator

(113) The frequency of a resonator can be given in a generalized form by

(114) $f=\frac{1}{L}\sqrt{\frac{c}{}},$
where c is the generalized stiffness of the material (which takes into account the resonance mode, resonator geometry and its orientation with respect to crystal), is the material density and L is the generalized dimension of the resonator.

(115) When the temperature changes the resonance frequency changes due to change in material parameters and in the dimensions of the resonator. The temperature coefficient of the resonance frequency

(116) $T{C}_f=\frac{1}{f}\frac{f}{T}$
depends on material parameters on the following way:
TC.sub.f=TC.sub.v,
where is the linear coefficient of thermal expansion taking into account the elongation of the resonator and TC of acoustic velocity is

(117) $T{C}_v=\frac{1}{v}\frac{v}{T}=\frac{1}{2}\left(T{C}_c-T{C}_{}\right)=\frac{1}{2}\left(T{C}_c+3\right),\mathrm{leading}\mathrm{to}$$T{C}_f=\frac{1}{2}\left(T{C}_c+\right)$

(118) Usually the clearly dominating effect is the first term, i.e. thermal coefficient of stiffness TC.sub.c, whereas thermal expansions effect is much smaller. If one is able to modify TC.sub.c of the resonating material strongly enough, temperature stable resonators can be realised. The following experimental verification of the model shows that by heavy n-doping of silicon this can be achieved.

(119) Theoretical Model for the Stiffness Matrix Element Temperature Sensitivities

(120) To further demonstrate the feasibility and to understand the operation of the invention, the authors have developed a theoretical model. The model utilizes the free electron contribution to the elastic constants of silicon using the many-valley approach by Keyes (R. W. Keyes, Solid State Physics, Vol. 20, 1967). The model contains a single fitting parameter, the deformation potential. The deformation potential parameter was fitted to the data published by Hall (Electronic Effect in the Elastic Constants of n-Type Silicon, Physical Review, vol 161 (2), pp. 756-761, 1967) using the data points at temperature range T=100 . . . 308 K.

(121) FIGS. 17a-c show the temperature sensitivities of the elastic matrix parameters for n-dopant level n=0.5*10.sup.19 cm.sup.3 . . . 10*10.sup.19 cm.sup.3 (dc.sub.ij/dT units are in Pa/C). It was assumed that in case of n-doping, the sensitivity of the c.sub.44 elastic matrix term is unaffected and the data provided by Hall was used for estimating the c.sub.44 temperature sensitivity. For c.sub.11 and c.sub.12, temperature sensitivities were obtained from this theory.

(122) The theory, as applied on the present inventive structure and bulk acoustic wave modes was found to agree with experimental data at a reasonably good level (see experimental section below). When TCFs have been calculated using the theory results it has always been assumed that the thermal expansion coefficient is that of normal (non-doped or weakly doped) silicon, i.e., =2.6 ppm/ C. in the general TCF theory above.

(123) Definition of c.sub.11-c.sub.12 Characterized Modes

(124) The expressions that the frequency of a particular resonance mode is characterized or dominated by matrix element terms (c.sub.11-c.sub.12), or mainly dependent on c.sub.11-c.sub.12 are clarified in the following.

(125) The theoretical model presented in above is able to predict the elastic constant temperature sensitivities dc.sub.ij/dT as a function of the n-dopant concentration. For minimizing the temperature dependence of a resonator it would be desirable that the temperature sensitivity of a constant would be zero at some dopant concentration level. This appears not to be the case for the constants c.sub.11, c.sub.12 and c.sub.44, but, when investigating the difference of the terms c.sub.11 and c.sub.12 we see that the sensitivity d(c.sub.11-c.sub.12)/dT is zero at the dopant concentration of approximately n=1.2*10.sup.19 cm.sup.3, see FIG. 18.

(126) This result suggests that it is possible to temperature compensate a resonator, if the resonator geometry, its orientation with respect to the crystal, and the resonance mode in question is such that the generalized stiffness (see chapter Theoretical model for TCF of a resonator) c is proportional to c.sub.11-c.sub.12. Such a mode is, for example, the Lam mode of a plate resonator.

(127) By c is proportional to c.sub.11-c.sub.12 is meant the following: Assume that the generalized stiffness c can be expressed as a linear polynomial, c=P*c.sub.11+Q*c.sub.12+R*c.sub.44, where P, Q and R are constants. The polynomial may be refactored as:
c=P*(c.sub.11-c.sub.12)+Q*c.sub.12+R*c.sub.44[eq1],
where the new constant we have Q=Q+P.

(128) By generalized stiffness c proportional to c.sub.11-c.sub.12 is meant that constant P is nonzero and that Q=R=0.

(129) However, the temperature compensation property can be applied to a wider class of resonators than to just those for which the above discussed strict condition is fulfilled. As shown in FIG. 18, the sensitivity d(c.sub.11-c.sub.12)/dT attains positive values at concentrations above n=1.2*10.sup.19 cm.sup.3. At the same time, dc.sub.12/dT and dc.sub.44/dT stay negative (see chapter Theoretical model for TCF of a resonator). Therefore, it is possible to temperature compensate a resonance mode, whose generalized stiffness c equation has nonzero Q and R factors: the positive effect from d(c.sub.11-c.sub.12)/dT is cancelled out from contribution(s) from dc.sub.12/dT and/or dc.sub.44/dT for a resonator with a suitable mode and orientation with the crystal and an optimal n-dopant concentration. Such non-pure (c.sub.11-c.sub.12)modes are, e.g., the flexural and extensional resonance modes of a beam resonator. Detailed examples are presented elsewhere in this document.

(130) In general, the generalized stiffness c of a resonance mode does not have to be a linear function of c.sub.ij'stypically for non-pure shear modes the function is not linear (see Approximation of the frequency vs. elastic matrix elements relationship of the extensional/flexural resonance of a beam resonator). Since the relative changes of stiffness dc.sub.ij/c.sub.ij are always small for in the context of this invention, a linear expansion of the generalized stiffness can be used (linear expansion is done at the point [c.sub.11, c.sub.12, c.sub.44]=[166, 64, 80] Gpa, which represents the stiffness terms of ordinary (essentially non-doped) silicon, can be used since the absolute value of silicon stiffness is not greatly affected by doping). The polynomial approximation for the generalized stiffness change dc can be written as
dc=P*(dc.sub.11-dc.sub.12)+Q*dc.sub.12+R*dc.sub.44.

(131) The frequency of a resonator is proportional to sqrt(c) (see Theoretical model for TCF of a resonator). Because of the small magnitude of the relative changes dc.sub.ij/c.sub.ij, and that of the generalized stiffness dc/c, we can linearize the relation for the frequency change as well and obtain
df=(constant)*(P*(dc.sub.11-dc.sub.12)Q*dc.sub.12R*dc.sub.44),
which has a factor of the same polynomial form as has been described above.

(132) When expressed in terms of the temperature sensitivities the relation reads as
df/dT=(constant)*(P*d(c.sub.11-c.sub.12)/dT+Q*dc.sub.12/dT+R*dc.sub.44/dT).

(133) A mode that can be temperature compensated has df/dT >=0 at some n-dopant concentration level. A condition for a resonance mode that can be temperature compensated is thus given by the linear inequality
d(c.sub.11-c.sub.12)/dT+Q/P*dc.sub.12/dT+R/P*dc.sub.44/dT0.

(134) A numerical estimate can be derived: from FIG. 18 we find the maximum value [d(c.sub.11-c.sub.12)/dT].sub.max=3.5 MPa/C, and from FIGS. 17b and 17c the following minimum values are found: [dc.sub.12/dT].sub.min=7.9 MPa/C and [dc.sub.44/dT]=4.4 MPa/C. Defining x=Q/P and y=R/P the inequality is cast into form
y7.9/4.4*x+3.5/4.4 or approximately y1.8*x+0.8.

(135) Assuming an error margin of 5% to the values calculated by the (see Theoretical model for TCF of a resonator) we arrive at inequality
y1.8*x+1.

(136) Thus, all points (Q/P, R/P) that fall under the line 1.8*x+1 represent modes that can be temperature compensated.

(137) In conclusion, a mode can be temperature compensated with n-doping, when its linearized frequency change df (as a function of the changes of matrix elements dc.sub.ij), which can be written as
df=P*dc.sub.11+Q*dc.sub.12+R*dc.sub.44,
or
df=P*(dc.sub.11-dc.sub.12)+Q*dc.sub.11+R*dc.sub.44,
where Q=Q+P
has coefficients P, Q and R, which fulfill the inequality R/P<1.8*Q/P+1.

(138) FIG. 19 shows where the modes discussed in connection with the various embodiments are located in the (Q/P, R/P)-plane. The modes that can be temperature compensated fall into the shaded region. Label Y1D refers to the approximation for the flexural/extensional mode of a beam, when the beam is oriented along the [100] direction (see Approximation of the frequency vs. elastic matrix elements relationship of the extensional/flexural resonance of a beam resonator).

(139) Experimental Verification

(140) The effect of homogeneous n-doping on single-crystal silicon MEMS resonator temperature coefficients has been experimentally tested. Plate resonators were fabricated on SOI wafers, whose device layer was n-doped with phosphorus to concentration of 510.sup.19 cm.sup.3. Devices were characterized at temperature range of T=40 . . . 80 C., and the temperature vs. resonance frequency curves were extracted.

(141) A total of four different resonance modes in two different resonator types were characterized. The tested resonator types were plate resonator of dimensions (lengthwidthheight) 320 m*320 m*10 m with plate sides aligned with [100] directions, fabricated on (100) silicon wafer, and similar resonator as above but rotated 45 degrees in the plane, i.e. with sides aligned with the [110] directions.

(142) For both resonator types the square extensional resonance mode and the Lam resonance mode were characterized.

(143) The frequency vs. temperature data from the measurements are shown in FIGS. 3a, 3b, 4a and 4b, and the extracted linear TCFs are summarized in Table 1 along with the predictions from theory.

(144) TABLE-US-00001 TABLE 1 SE 110 SE Lame Lame aligned 100 aligned 110 aligned 100 aligned measured TCF, 6.0 1.0 29.0 18.0 n = 5 * 10.sup.19 cm.sup.3 VTT theory 5.1 0.2 26.4 18.3 n = 5 * 10.sup.19 cm.sup.3

(145) The most important observations were: 1. The Lame-mode of an [100] aligned plate was found to be overcompensated with linear TCF +18 ppm/ C. 2. The SE resonance mode of a [100] aligned plate had near-zero TCF of 1 ppm 1 C. 3. The TCF of the Lame-mode of an [110] aligned plate was very little modified by n-doping 4. The predictions from the theory presented herein appear to have a good agreement with the experimental data.

(146) It can be seen that the graphs are not fully linear but there is a noticeable second-order term (constant b) present. It is expected that the curves can be linearized by e.g. using additional dopant in the silicon crystal.

(147) The description of embodiments, theory and experiments above and the attached drawings are for illustrative purposes only and are not intended to limit the invention, whose scope is defined in the following claims. The claims should be interpreted in their full scope taking equivalents into account.