Abstract
The invention provides a microelectromechanical resonator device comprising a support structure and a resonator manufactured on a (100) or (110) semiconductor wafer, wherein the resonator is suspended to the support structure and comprises at least one beam being doped to a doping concentration of 1.1*10.sup.20 cm.sup.3 or more with an n-type doping agent and is being capable of resonating in a length-extensional, flexural resonance or torsional mode upon suitable actuation. In particular, the doping concentration and angle of the beam are chosen so as to simultaneously produce zero or close to zero second order TCF, and even more preferably zero or close to zero first and second order TCFs, for the resonator in said resonance mode, thus providing a temperature stable resonator.
Claims
1. A microelectromechanical resonator device comprising, a support structure, a resonator suspended to the support structure and having at least one beam, the beam; having a longitudinal axis, being doped to a doping concentration with an n-type doping agent, and being capable of resonating in a length-extensional, flexural or torsional resonance mode, an actuator for exciting said length-extensional, flexural or torsional resonance mode into the resonator, wherein the support structure and the resonator are manufactured from a (100) oriented semiconductor wafer or a (110) oriented semiconductor wafer, wherein said doping concentration is at least 1.1*10.sup.20 cm.sup.3, and wherein the longitudinal axis of the beam is oriented at an angle of 1710 degrees with respect to the [100] crystal direction of the wafer if said resonance mode is length-extensional or flexural, and at an angle of 035 degrees with respect to the [110] crystal direction of the wafer if said resonance mode is torsional.
2. The resonator device according to claim 1, wherein the resonator includes only one beam.
3. The resonator device according to claim 2, wherein the resonator is essentially comprised of said beam.
4. The resonator device according to claim 1, wherein the resonator includes at least two beams having longitudinal axes at an angle with respect to each other.
5. The resonator device according to claim 1, wherein the resonator includes at least two beams whose longitudinal axes are oriented at said angle, and the whole resonator has an axis of symmetry which is essentially parallel with the [100] direction if said resonance mode is length-extensional or flexural or [110] direction if said resonance mode is torsional.
6. The resonator device according to claim 4, wherein the beams are connected to each other from their ends.
7. The resonator device according to claim 6, further comprising three, four or more beams arranged in ring formation.
8. The resonator device according to claim 4, wherein the at least two beams are from their first ends connected to a common base portion having an axis of symmetry essentially parallel with the [100] direction if said resonance mode is length-extensional or flexural or [110] direction if said resonance mode is torsional, and extend from the common base portion at equal angles with respect to the [100] or [110] direction, respectively, but at opposite sides of said direction in the plane of the wafer.
9. The resonator device according to claim 4, wherein the beams intersect each other and are suspended to the support structure at their zone of intersection.
10. The resonator device according to claim 1, wherein the resonator comprises an axis of symmetry which is perpendicular to the longitudinal direction of the beam.
11. The resonator device according to claim 1, wherein said resonance mode is length-extensional.
12. The resonator device according to claim 1, wherein said resonance mode is flexural, the flexural plane being in the wafer plane and/or out of the wafer plane.
13. The resonator device according to claim 11, wherein said angle is 178 degrees with respect to the [100] crystal direction.
14. The resonator device according to claim 1, wherein the resonance mode is torsional.
15. The resonator device according to claim 14, wherein an out-of-plane aspect ratio of the at least one beam is less than 2.
16. The resonator device according to claim 14, wherein the beam is manufactured in (110) wafer plane and the longitudinal axis of the beam is oriented at 035 degrees with respect to the [110] crystal direction, or the beam is manufactured in (100) wafer plane and the longitudinal axis of the beam is oriented at 020 degrees with respect to the [110] crystal direction.
17. The resonator device according to claim 14, wherein the beam is manufactured in (110) wafer plane and its out-of-plane aspect ratio is 1.25 or less, or the beam is manufactured in (100) wafer plane and its out-of-plane aspect ratio is 0.9 or less.
18. The resonator device according to claim 14, wherein the angle of the beam deviates from the [110] crystal direction by at least 5 degrees.
19. The resonator device according to claim 1, wherein said doping concentration is at least 1.2*10.sup.20 cm.sup.3.
20. The resonator device according to claim 1, wherein the in-plane aspect ratio of the at least one beam is 5:1 or more.
21. The resonator device according to claim 1, wherein the first order temperature coefficient of frequency (TCF.sub.1) of the resonator is 1 ppm/C or less at at least the one temperature and a second order temperature coefficient of frequency (TCF.sub.2) is 12 ppb/C.sup.2 or less at said temperature.
22. The resonator device according to claim 1, wherein said actuator comprises a piezoelectric actuator acoustically coupled with the beam.
Description
BRIEF DESCRIPTION OF THE DRAWINGS
(1) FIGS. 1A and 1B illustrate beam resonator geometries according to selected embodiments of the invention.
(2) FIGS. 1C, 1D, 1G and 1H illustrate tuning fork resonator geometries according to selected embodiments of the invention.
(3) FIGS. 1E and 1F illustrate ring resonator geometries according to selected embodiments of the invention.
(4) FIGS. 2A-2E illustrate behavior of beams according to various embodiments of the invention in different resonance modes.
(5) FIG. 3A illustrates a graph of total frequency drift vs. temperature for a WE mode resonator with optimized lateral aspect ratio at different doping concentrations.
(6) FIG. 3B shows a graph of second order TCF vs. doping concentration for a WE resonator whose linear TCF is made zero by optimal design.
(7) FIGS. 3C and 3D show TCF.sub.1=0 and TCF.sub.1=0 curves of a [100]-oriented beam in length-extensional 1.sup.st order and 3.sup.rd order resonances, respectively, in a plane defined by doping concentration and rotation angle of the beam.
(8) FIGS. 3E and 3F show TCF.sub.1=0 and TCF.sub.2=0 curves of a [100]-oriented beam in flexural 1.sup.st order and 5.sup.th order resonances, respectively, in a plane defined by doping concentration and rotation angle of the beam.
(9) FIGS. 3G and 3H show TCF.sub.1=0 and TCF.sub.2=0 curves of a [110]-oriented beam manufactured on a (100) wafer and (110) wafer, respectively, in torsional resonance in a plane defined by doping concentration and out-of-plane aspect ratio of the beam.
(10) FIG. 3I shows graphs of temperature coefficients of the elastic parameters c.sub.ij of silicon as a function of doping concentration n.
(11) FIG. 3J 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>=10e19 cm.sup.3) were first discovered in connection with the present invention.
(12) FIGS. 3K and 3L show TCF.sub.1=0 and TCF.sub.2=0 curve of a 1.sup.st order torsional mode beam in (110) plane and (100) plane, respectively, as graphs with axes defined by optimal out-of-plane aspect ratio of the beam and deviation (n) of the beam length from [110] direction, when the doping concentration is 1.11*10.sup.20 cm.sup.3.
DETAILED DESCRIPTION OF EMBODIMENTS
(13) FIG. 1A shows a resonator consisting of a single beam 10A manufactured on a (100) wafer and having a length L and width W. The beam 10A is tilted with respect to the [100] crystal direction by an angle . The tilted beam 10A can be anchored to the surrounding support structure (not shown) from an anchoring point at one longitudinal end thereof or in the middle of one or both longitudinal sides, depending on the intended resonance mode to be excited. Preferably anchoring takes place at one or more nodal or quasinodal points of the resonance mode at the perimeter of the beam 10A.
(14) A drawback in the resonator of FIG. 1A is that it is not symmetric with respect to the [100] direction, whereby parasitic resonance modes may arise in the beam along with the main resonance mode. These parasitic modes may decrease the quality factor of the resonator, and may even affect the TCF characteristics of the main resonance mode.
(15) The resonator of FIG. 1B avoids the drawback. It comprises two tilted beams 11A, 11B crossing each other such that the resonator is symmetric with respect to the [100] crystal direction. Both beams 11A, 11B are tilted by angle towards opposite directions. Herein, crossing of the beams 11A, 11B takes place in the middle of the beams, whereby the design is also symmetric with respect to the transverse direction, but geometries asymmetric with respect to that direction are possible too. Anchoring preferably takes place at the crossing zone of the beams 11A, 11B, leaving all four beam ends capable of resonating in a flexural, torsional or length-extensional mode in accordance with the invention.
(16) According to a still further embodiment there is provided a mass element, such as rectangular, circular or elliptical mass element, at one or both ends of the beams 10A or 11A and/or 11B. Generally speaking, a mass element is an element having a width larger than the width of the beam. In such configuration, the tilted beam acts as a length-extensional, flexural or torsional spring between the substrate and the mass element, and its orientation with respect to the crystal contributes to minimizing the TCF.sub.1 and TCF.sub.2 of the resonator. A mass element can be added also to any other configuration comprising a tilted beam having a free end suitable for the mass element.
(17) FIG. 1C shows a tilted tuning fork design comprising a base beam 10C tilted by angle with respect to the [100] crystal direction. The base beam 10C has two branches attached to it, namely a first fork beam 12A and a second fork beam 12B. The first and second fork beams 12A, 12B are both tilted at the same angle and have a constant distance between them. As the fork beams 12A, 12B are suspended to the base beam 10C on one end thereof, they are capable of resonating in flexural, torsional or length-extensional modes in accordance with the invention.
(18) Like in the design of FIG. 1A, the resonator of FIG. 1C is also not symmetric with respect to the [100] direction, and therefore non-optimal because of potential parasitic resonance modes excited.
(19) FIG. 1D shows a resonator with improved tuning fork geometry. Like above, the resonator comprises a base beam 10D and two fork beams 13A, 13B attached thereto. Now the base beam is oriented along the [100] direction and the fork beams 13A, 13B at an angle of with respect to that direction, but at opposite sides thereof. This means that the fork beams 13A, 13B protrude away from each other, with an angle of 2 between them. This design completely symmetrizes the resonator with respect to the [100] direction, thereby avoiding the arising of parasitic resonances. Still, both fork beams 13A, 13B are at an angle with respect to the crystal, thereby allowing for simultaneous zeroing of TCF.sub.1 and TCF.sub.2.
(20) In the illustrated design, there is a transversely oriented section between the base beam 10D and the fork beams 13A, 13B, keeping the fork beams 13A, 13B separate from each other even close to the base beam 10D. This is, however, not absolutely necessary, but the fork beams 13A, 13B may also be located closer to each other.
(21) In one variation of the symmetrized tuning fork geometry, the fork beams are tilted at the same angles towards each other, such that their distance from each other comes shorter towards their ends.
(22) FIG. 1G shows a two-ended tuning fork resonator. There is a base beam 10G having two branched ends 16A, 16B with parallel fork beams, each end therefore corresponding to those described above in relation to FIG. 1C.
(23) A crystal-symmetrized two-ended tuning fork resonator is shown in FIG. 1H. It comprises a [100]-aligned base beam 10H and two non-parallel fork beams at both ends 17A, 17B thereof. The fork beams are arranged pairwise in mirror symmetric configuration with respect to the [100] axis.
(24) FIG. 1E shows a ring resonator embodiment. The ring resonator 10E comprises four beams 14A-D which are connected to each other from their ends such that a closed ring is formed. Two opposing beams 14A, 14C are oriented at an equal angle with respect to the [100] crystal direction, whereas the two remaining beams 14B, 14D are at right angles with respect to those beams 14A, 14C.
(25) As is the case with the designs of FIGS. 1A and 1C, the tilted beam portions allow for simultaneous zeroing of TCF.sub.1 and TCF.sub.2 in one or more resonance modes, but non-symmetry with respect to the crystal may cause parasitic resonances to emerge, decreasing the overall performance of the resonator.
(26) A ring resonator can also be symmetrized, as shown in FIG. 1F. The resonator 10F comprises again four beams 15A-D connected to each other from their ends. Two opposing beams 15A, 15C of the ring formed are directed at opposite angles with respect to the [100] direction. The remaining beams 15B, 15D are oriented at 90 degrees in-plane angle with respect to the [100] direction.
(27) Ring resonators can be suspended at their central portion, in particular center-of-mass, using one or more anchor elements (not shown) extending from the support structure to the ring. Thus, the final structure resembles a cartwheel.
(28) The invention is not limited to the geometric designs discussed above but there exists also other variations. In addition, the beams need not be exactly rectangular but may also be curved, involve width changes or have round corners, to mention some examples. In one embodiment, the resonator surface is free from trenches or perforations. However, the presence of such features is generally not excluded.
(29) All the resonator geometries described above allow the tilted beams to be driven into flexural or length-extensional modes, in which case the doping concentration of 1.2*10.sup.20 cm.sup.3 or more, at least at the zone of the actual resonator, and a tilt angle of 1710 degrees in accordance with the invention can be used to minimize the overall TCF of the resonator device. At least some of the geometries are usable also for torsional-mode beam resonators, when the [100] direction is replaced with [110] direction. Typically, the whole resonator is homogeneously doped to the same concentration. The resonator can for example be manufactured from a doped single-crystal wafer.
(30) FIG. 2A illustrates length-extensional resonance of a tilted beam 21. The solid lines represent the outer shape of the beam in contracted or normal (resting) state and dashed lines the shape of the beam in extended state.
(31) FIG. 2B illustrates in-plane flexural resonance of a tilted beam 22. The solid lines represent the outer shape of the beam in normal (resting) state and dashed lines the shape of the beam under flexure.
(32) The modes of FIGS. 2A and 2B are applicable for the tilted (and also non-tilted) beams of FIGS. 1A-1D, 1G and 1H.
(33) FIGS. 2C and 2D illustrate two in-plane flexural resonance modes of a ring beam resonator 23. In the mode of FIG. 2C, all beams bent inwards (towards the center of the ring) at the same time, whereas in the mode of FIG. 2D, the beams bend pairwise in opposite directions (inward and outwards). These modes are applicable for the resonators shown in FIGS. 1E and 1F.
(34) FIG. 2E illustrates an example of a torsional resonance mode, where a mass element is suspended between two torsional beams, which attach the resonator to the substrate. These beams are oriented along the dashed line, whose deviation from the 110 direction is given by .
(35) By choosing torsional beam dimensions (aspect ratio of the cross section of the beam), angle and n-doping concentration suitably, not only TCF.sub.1 of the resonator is zeroed but also TCF.sub.2 is zeroed, giving the resonator a wide stable temperature range of operation.
(36) FIG. 3A illustrates total frequency drift f 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. Taking into account practical factors that cause slow saturation of the TCF.sub.2 behavior when the doping concentration is increased, the TCF.sub.2 is zeroed in practice at 1.3*10.sup.20 or higher. Assuming that the second-order TCF grows monotonously when doping gets larger than 11*10.sup.19, and that the linear TCF gets smaller simultaneously, 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.
(37) FIG. 3A shows that a total frequency drift of less than 50 ppm over the industrial range of 40 . . . 85 C 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. 3B on the other hand supports that a drift of less than 25 ppm is feasible.
(38) The experimental data of FIGS. 3A and 3B has been measured for SE/WE mode branch, but flexural/torsional modes provide the same kind of behavior, because temperature dependence of a correctly oriented beam resonator according to the present invention is very similar to that of a WE mode. Data from flexural modes was not measured (and the WE modes are measured instead) since in practical atmospheric-pressure characterization measurements, high-frequency (>10 MHz) WE modes have relatively high Q values of the order of 10000. Flexural modes (typically <1 MHz.) suffer from strong gas damping, and therefore they cannot be measured in open air.
(39) The qualitative results shown in the following figures and explained below for the various modeshapes within the scope to the invention further demonstrate the feasibility of the invention.
(40) FIG. 3C shows zero TCF.sub.1 and TCF.sub.2 curves of a simple 1.sup.st order length-extensional beam resonator (according to FIG. 1A) when the n-doping concentration and rotation angle are varied. As can be seen, both TCF.sub.1 and TCF.sub.2 are zeroed simultaneously at about 1.2*10.sup.20 cm.sup.3 concentration and 17 degrees angle. For other geometries herein disclosed, optimal combination can be found when the doping concentration is 1.2*10.sup.20 cm.sup.3 or higher and the angle is 7-27 degrees with respect to the [100] direction of the wafer.
(41) FIG. 3C represents an optimal situation for a doped silicon resonator element. In practice, additional materials which must 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 lower angle. For a rectangular beam in the length-extensional mode, the optimal point in practice can be for example at a concentration of 1.3*10.sup.20 cm.sup.3 or higher, and the angle being 7-20 degrees, such as 7-17 degrees.
(42) FIG. 3D shows a graph similar to that of FIG. 3C but for a 3.sup.rd order length-extensional mode. The form of the curve is similar to that of FIG. 3C and the above considerations apply. This shows that the invention is not limited to 1.sup.st order resonance modes.
(43) FIGS. 3E and 3F show TCF.sub.1=TCF.sub.2=0 curves for 1.sup.st and 5.sup.th order flexural mode resonators. The behavior is analogous to that of length-extensional modes as far as the optimal angle and doping concentration are concerned.
(44) The examples described above are presented for resonators manufactured on a (100) wafer. The same principles of design can be used for resonators manufactured on (110) wafers.
(45) FIGS. 3G and 3H show curves analogous to those of FIG. 3C-3F but for a torsional mode beam resonator in two different wafer planes, i.e, (110) and (100), respectively, and using the out-of-plane cross sectional aspect ratio of the beam (beam height/beam width) at the vertical axis. Similar behavior applies to resonator geometries which comprise a torsional spring or a multitude of torsional springs, such as that illustrated in FIG. 2E. It can be seen that the optimal TCF point is found in both cases at about a concentration of 1.11*10.sup.20 cm.sup.3 or higher and when the out-of-plane aspect ratio is close to 1 (but not exactly 1). Zeroing the TCFs simultaneously in the torsional resonance mode is not as angle-sensitive as in the LE or flexural modes: if the beam orientation deviates from [110] within of 35 . . . 35 degrees, an optimal aspect ratio can be used to obtain TCF.sub.1=TCF.sub.2=0. These optimal configurations are illustrated in FIGS. 3K and 3L as a function of the deviation angle from [110]. It is seen that the aspect ratio is approximately 1 (about 1.2 or 0.85, depending on the wafer plane) for a beam aligned with [110], and that it is decreased when the orientation is deviated from [110].
(46) It should also be noted that in the case of torsional beams, the optimal solution never occurs with the combination of aspect ratio 1 and angle 0, but with other parameter combinations. To give some further details on how the curves of FIGS. 3C-3H are generated, FIG. 3I shows the temperature coefficients of the elastic parameters 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.11a.sub.11-12c.sup.0.sub.11-12)/c.sup.0.sub.12, and a similar equation holds for b.sub.12. 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. 3J, 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 FIGS. 3C-3H, the interpolated/extrapolated values represented by the dashed curves of FIG. 3I have been used.
(47) The fit at carrier concentration 0<n<7.5*10.sup.19 cm.sup.3 is based on a third order polynomial fit to the data points at carrier concentration 0<n<7.5*10.sup.19 cm.sup.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. 3C-3H 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 FIGS. 3C-3H hold fairly well.
(48) FIG. 3J 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. 3I, where the behavior of the b.sub.11-12 term has been extrapolated.
(49) It should be understood that the invention covers a large number of different resonance frequencies, doping concentrations, geometrical configurations (including lateral shape and thickness) of the resonator, only some of which are exemplified in the drawings or in the present description. Common to them is that the resonator comprises a beam section, the properties of which are within the claimed doping concentration and angle ranges and the beam section therefore working together with potential other parts of the resonator to minimize the overall temperature dependency of the resonance frequency of the resonator. 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 specific parameter combinations that suits his needs.
(50) 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.
(51) 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.
(52) 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.
(53) 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.
(54) 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.
(55) 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.
(56) 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.
