Elastic wave device

Abstract

Provided is an elastic wave device with which it is possible to reduce the size of a filter device and to reduce resistance. This elastic wave device (10) is provided with: a piezoelectric layer (14); a first comb electrode (17) which is provided on the piezoelectric layer (14), has a first bus bar (22) and a plurality of first electrode fingers (25) each having one end connected to the first bus bar (22), and is connected to an input potential; a second comb electrode (18) which is provided on the piezoelectric layer (14), has a second bus bar (23) and a plurality of second electrode fingers (26), one end of which is connected to each of the second bus bars (23), and which are interleaved and inserted with the plurality of first electrode fingers (25), and which is connected to the output potential; and a reference potential electrode (19) having a plurality of third electrode fingers (27) provided on the piezoelectric layer (14) so as to be aligned with the first electrode fingers (25) and the second electrode fingers (26) in the direction in which the first electrode fingers (25) and the second electrode fingers (26) are aligned, and a plurality of connection electrodes (24) connecting adjacent third electrode fingers (27) to each other. At least a portion of the second electrode is provided between the first comb electrode (17) and the second comb electrode (18), and is connected to a reference potential. The order in which the first electrode fingers (25), the second electrode fingers (26), and the third electrode fingers (27) are arranged is an order in which the first electrode fingers (25), the third electrode fingers (27), the second electrode fingers (26), and the third electrode fingers (27) are arranged as one cycle from the first electrode fingers (25). The reference potential electrode (19) has a plurality of potential connection sections (first to third potential connection sections (28A to 28C)) connected to at least three locations of the reference potential.

Description

Technical Field

[0001] The present invention relates to elastic wave devices. Background Art

[0002] In the past, elastic wave devices were widely used in filters of portable phones and the like. In recent years, an elastic wave device using a thickness shear mode bulk wave as described in the following patent document 1 has been proposed. In this elastic wave device, a piezoelectric layer is provided on a support. A pair of electrodes is provided on the piezoelectric layer. The paired electrodes are opposed to each other on the piezoelectric layer and are connected to different potentials. By applying an alternating voltage between the electrodes, a thickness shear mode bulk wave is excited.

[0003] Prior Art Literature

[0004] Patent Literature

[0005] Patent Document 1: U.S. Patent No. 10491192 Summary of the invention

[0006] Problem that the invention aims to solve

[0007] The so-called elastic wave device is, for example, an elastic wave resonator, which is used in, for example, a ladder filter. In order to obtain good characteristics in the ladder filter, it is necessary to increase the electrostatic capacitance ratio between a plurality of elastic wave resonators. In this case, it is necessary to increase the electrostatic capacitance of some elastic wave resonators in the ladder filter.

[0008] In order to increase the capacitance of the elastic wave resonator, for example, the elastic wave resonator needs to be enlarged. Therefore, when the elastic wave resonator is used in a ladder-type filter, the ladder-type filter tends to be enlarged. In particular, a ladder-type filter having an elastic wave resonator with a small capacitance and utilizing a bulk wave in a thickness shear mode tends to be enlarged.

[0009] The inventors of the present invention have found that, when the elastic wave device is used in a filter device, a suitable filter waveform can be obtained without increasing the size thereof by configuring the elastic wave device to have the following structure: the structure in which an electrode connected to a reference potential is arranged between an electrode connected to an input potential and an electrode connected to an output potential.

[0010] However, the inventors of the present invention have found that in the above-mentioned structure, the resistance of the electrode connected to the reference potential tends to be high. Therefore, when the elastic wave device is used in a filter device, the resistance of the elastic wave device tends to be high.

[0011] An object of the present invention is to provide an elastic wave device capable of promoting miniaturization of a filter device and reducing electrical resistance.

[0012] Technical solutions to solve problems

[0013] The elastic wave device of the present invention comprises: a piezoelectric layer; a first comb-shaped electrode, arranged on the piezoelectric layer, having a first bus bar and a plurality of first electrode fingers, one end of which is respectively connected to the first bus bar, and connected to an input potential; a second comb-shaped electrode, arranged on the piezoelectric layer, having a second bus bar and a plurality of second electrode fingers, one end of which is respectively connected to the second bus bar, and connected to an output potential, wherein the plurality of second electrode fingers are interlaced with the plurality of first electrode fingers; and a reference potential electrode, having a plurality of third electrode fingers and a plurality of connecting electrodes for connecting adjacent third electrode fingers to each other, and at least a portion of which is arranged on the first bus bar. The first comb-shaped electrode and the second comb-shaped electrode are connected to a reference potential, and the plurality of third electrode fingers are respectively arranged on the piezoelectric layer so as to be arranged with the first electrode finger and the second electrode finger in the direction in which the first electrode finger and the second electrode finger are arranged. The order in which the first electrode finger, the second electrode finger and the third electrode finger are arranged is the order of the first electrode finger, the third electrode finger, the second electrode finger and the third electrode finger as one period when starting from the first electrode finger. The reference potential electrode has a plurality of potential connection parts connected to at least three locations of the reference potential.

[0014] Effects of the Invention

[0015] According to the present invention, it is possible to provide an elastic wave device that can promote miniaturization of a filter device and reduce electrical resistance. BRIEF DESCRIPTION OF THE DRAWINGS

[0016] Figure 1 It is a schematic front cross-sectional view of the elastic wave device according to the first embodiment of the present invention.

[0017] Figure 2 It is a schematic plan view of the elastic wave device according to the first embodiment of the present invention.

[0018] Figure 3 This is a schematic front cross-sectional view showing the vicinity of first to third electrode fingers in the first embodiment of the present invention.

[0019] Figure 4 It is a diagram showing the transmission characteristics and reflection characteristics of the elastic wave device according to the first embodiment of the present invention.

[0020] Figure 5 It is a schematic top view of an elastic wave device according to a reference example.

[0021] Figure 6 This is a diagram showing a mapping of the fractional bandwidth with respect to the Euler angle (0°, θ, ψ) of LiNbO 3 when d / p is made as close to 0 as possible.

[0022] Figure 7 It is a schematic plan view of an elastic wave device according to a second embodiment of the present invention.

[0023] Figure 8 It is a schematic plan view of an elastic wave device according to a third embodiment of the present invention.

[0024] Fig. 9 It is a schematic plan view of an elastic wave device according to a fourth embodiment of the present invention.

[0025] Fig.10 (a) is a schematic perspective view showing the appearance of an elastic wave device using a thickness shear mode bulk wave, Fig.10 (b) is a top view showing the electrode structure on the piezoelectric layer.

[0026] Fig.11 It is along Fig.10 (a) is a cross-sectional view of a portion taken along line AA.

[0027] Fig.12 (a) is a schematic front cross-sectional view for explaining Lamb waves propagating through a piezoelectric film of an elastic wave device. Fig.12 (b) is a schematic front cross-sectional view for explaining a thickness shear mode bulk wave propagating through a piezoelectric film in an elastic wave device.

[0028] Fig.13 It is a diagram showing the amplitude direction of the body wave in the thickness shear mode.

[0029] Fig.14 FIG. 1 is a diagram showing the resonance characteristics of an elastic wave device using bulk waves in a thickness shear mode.

[0030] Fig.15 : This is a diagram showing the relationship between d / p and the fractional bandwidth of the resonator when the distance between the centers of adjacent electrodes is defined as p and the thickness of the piezoelectric layer is defined as d.

[0031] Fig.16 This is a top view of an elastic wave device using bulk waves in a thickness shear mode.

[0032] Fig.17 FIG. 2 is a diagram showing the resonance characteristics of an elastic wave device according to a reference example in which spurious emission occurs.

[0033] Fig.18 This is a diagram showing the relationship between the fractional bandwidth and the phase rotation amount of the impedance of the spurious signal normalized by 180 degrees, which is the magnitude of the spurious signal.

[0034] Fig.19 It is a graph showing the relationship between d / 2p and the metallization ratio MR.

[0035] Fig. 20 This is a diagram showing a mapping of the fractional bandwidth with respect to the Euler angle (0°, θ, ψ) of LiNbO 3 when d / p is made as close to 0 as possible.

[0036] Fig.21 It is a front cross-sectional view of an elastic wave device having an acoustic multilayer membrane.

[0037] Fig. 22 This is a partially cutaway perspective view for explaining an elastic wave device using Lamb waves. DETAILED DESCRIPTION

[0038] Below, while referring to the attached Figure 1 The present invention will be clarified by describing specific embodiments of the present invention.

[0039] In addition, it should be noted that each embodiment described in this specification is an example, and some structures can be replaced or combined among different embodiments.

[0040] Figure 1 It is a schematic front cross-sectional view of the elastic wave device according to the first embodiment of the present invention. Figure 2 is a schematic plan view of the elastic wave device according to the first embodiment. Figure 1 It is along Figure 2 Schematic cross-sectional view of line II in FIG. Figure 2 In FIG. 1 , hatching is performed to indicate each electrode. Figure 2 In FIG. 1 , the reference potential symbol is used to schematically indicate that the reference potential electrode described later is connected to the reference potential. Figure 2 Similarly, in other schematic plan views, electrodes may be hatched and reference potential symbols may be used.

[0041] Figure 1 The elastic wave device 10 shown is configured to utilize the thickness shear mode. The elastic wave device 10 is an acoustic coupling filter. The structure of the elastic wave device 10 will be described below.

[0042] The elastic wave device 10 has a piezoelectric substrate 12 and a functional electrode 11. The piezoelectric substrate 12 is a substrate having piezoelectricity. Specifically, the piezoelectric substrate 12 has a supporting member 13 and a piezoelectric layer 14. In the present embodiment, the supporting member 13 includes a supporting substrate 16 and an insulating layer 15. The insulating layer 15 is provided on the supporting substrate 16. The piezoelectric layer 14 is provided on the insulating layer 15. However, the supporting member 13 may also be composed of only the supporting substrate 16. In addition, the supporting member 13 may not necessarily be provided.

[0043] The piezoelectric layer 14 has a first principal surface 14a and a second principal surface 14b. The first principal surface 14a and the second principal surface 14b face each other. Of the first principal surface 14a and the second principal surface 14b, the second principal surface 14b is located on the support member 13 side.

[0044] like Figure 2 As shown, the functional electrode 11 includes a pair of comb-shaped electrodes and a reference potential electrode 19. The reference potential electrode 19 is connected to a reference potential. Specifically, the pair of comb-shaped electrodes is a first comb-shaped electrode 17 and a second comb-shaped electrode 18. The first comb-shaped electrode 17 is connected to an input potential. The second comb-shaped electrode 18 is connected to an output potential.

[0045] The first comb-shaped electrode 17 and the second comb-shaped electrode 18 are provided on the first main surface 14a of the piezoelectric layer 14. The first comb-shaped electrode 17 has a first bus bar 22 and a plurality of first electrode fingers 25. One ends of the plurality of first electrode fingers 25 are respectively connected to the first bus bar 22. The second comb-shaped electrode 18 has a second bus bar 23 and a plurality of second electrode fingers 26. One ends of the plurality of second electrode fingers 26 are respectively connected to the second bus bar 23.

[0046] The first bus bar 22 and the second bus bar 23 face each other. The plurality of first electrode fingers 25 and the plurality of second electrode fingers 26 are interlaced with each other. The first electrode fingers 25 and the second electrode fingers 26 are alternately arranged in a direction orthogonal to the direction in which the first electrode fingers 25 and the second electrode fingers 26 extend.

[0047] The reference potential electrode 19 has a meandering shape. Specifically, the reference potential electrode 19 has a plurality of connection electrodes 24 and a plurality of third electrode fingers 27. The plurality of connection electrodes 24 and the plurality of third electrode fingers 27 are provided on the first principal surface 14a of the piezoelectric layer 14. Adjacent third electrode fingers 27 are connected to each other through the connection electrodes 24. By repeating this structure, the shape of the reference potential electrode 19 is set to a meandering shape.

[0048] More specifically, the plurality of third electrode fingers 27 extend in parallel with the plurality of first electrode fingers 25 and the plurality of second electrode fingers. The plurality of third electrode fingers 27 are respectively arranged to be arranged with the first electrode fingers 25 and the second electrode fingers 26 in the direction in which the first electrode fingers 25 and the second electrode fingers 26 are arranged. Therefore, the first electrode fingers 25, the second electrode fingers 26, and the third electrode fingers 27 are arranged in one direction.

[0049] Hereinafter, the direction in which the first electrode finger 25, the second electrode finger 26, and the third electrode finger 27 extend is referred to as the electrode finger extending direction, and the direction orthogonal to the electrode finger extending direction is referred to as the electrode finger orthogonal direction. When the direction in which the first electrode finger 25, the second electrode finger 26, and the third electrode finger 27 are arranged is referred to as the electrode finger arrangement direction, the electrode finger arrangement direction is parallel to the electrode finger orthogonal direction. In this specification, the first electrode finger 25, the second electrode finger 26, and the third electrode finger 27 are sometimes collectively and simply referred to as electrode fingers.

[0050] One third electrode finger 27 is located at one end in the orthogonal direction of the electrode fingers in the region where the plurality of electrode fingers are provided. The plurality of third electrode fingers 27 other than the one third electrode finger 27 are provided between the first electrode finger 25 and the second electrode finger 26. Therefore, in the present embodiment, the plurality of third electrode fingers 27 other than the one third electrode finger 27 are provided between the first comb-shaped electrode 17 and the second comb-shaped electrode 18. On the other hand, the one third electrode finger 27 is not located between the first comb-shaped electrode 17 and the second comb-shaped electrode 18. In the present embodiment, the one third electrode finger 27 is adjacent only to the second electrode finger 26 among the first electrode finger 25 and the second electrode finger 26.

[0051] Figure 3 This is a schematic front cross-sectional view showing the vicinity of first to third electrode fingers in the first embodiment.

[0052] The order in which the plurality of electrode fingers are arranged is the order of the first electrode finger 25, the third electrode finger 27, the second electrode finger 26, and the third electrode finger 27 as one cycle, when starting from the first electrode finger 25. Therefore, the order in which the plurality of electrode fingers are arranged is the first electrode finger 25, the third electrode finger 27, the second electrode finger 26, the third electrode finger 27, the first electrode finger 25, the third electrode finger 27, the second electrode finger 26, ... and so on. If the input potential is represented by IN, the output potential is represented by OUT, and the reference potential is represented by GND, and the order of the plurality of electrode fingers is represented as the order of the potentials to which they are connected, then it is represented as IN, GND, OUT, GND, IN, GND, OUT, ... and so on.

[0053] In the present embodiment, in a region where a plurality of electrode fingers are provided, an electrode finger located at one end in a direction orthogonal to the electrode finger is the third electrode finger 27. In this region, an electrode finger located at the other end in a direction orthogonal to the electrode finger is the first electrode finger 25. In addition, in this region, an electrode finger located at an end in a direction orthogonal to the electrode finger may be any type of electrode finger among the first electrode finger 25, the second electrode finger 26, and the third electrode finger 27.

[0054] like Figure 2As shown, the front ends of two adjacent third electrode fingers 27 on the first bus bar 22 side or the front ends on the second bus bar 23 side are connected by the connection electrodes 24. For example, the third electrode fingers 27 other than the two ends of the electrode fingers in the orthogonal direction among the plurality of third electrode fingers 27 are each connected to one connection electrode 24 at both the front ends on the first bus bar 22 side and the front ends on the second bus bar 23 side. The third electrode finger 27 is connected to the third electrode finger 27 adjacent to the left and right through each connection electrode 24. By repeating this structure, the shape of the reference potential electrode 39 is set to a meandering shape.

[0055] In the present embodiment, the third electrode finger 27 at one end of the plurality of third electrode fingers in the orthogonal direction is not disposed between the first comb-shaped electrode 17 and the second comb-shaped electrode 18. However, the entire reference potential electrode 19 may be disposed between the first comb-shaped electrode 17 and the second comb-shaped electrode 18. It is sufficient that at least a portion of the reference potential electrode 19 is disposed between the first comb-shaped electrode 17 and the second comb-shaped electrode 18.

[0056] exist Figure 2 In FIG. 1 , the reference potential electrode 19 is connected to the reference potential by the symbol of the reference potential. The portion of the reference potential electrode 19 connected to the reference potential is a potential connection portion. The reference potential electrode 19 has a plurality of potential connection portions. More specifically, the reference potential electrode 19 has three potential connection portions, namely, a first potential connection portion 28A, a second potential connection portion 28B, and a third potential connection portion 28C. In addition, the potential connection portion is not limited to three portions. The reference potential electrode 19 only needs to have at least three potential connection portions.

[0057] The first potential connection portion 28A is located at a third electrode finger 27 at one end of the plurality of third electrode fingers 27 in the direction orthogonal to the electrode fingers. More specifically, the first potential connection portion 28A is configured as a part of the third electrode finger 27. In addition, although not shown, at least one connection wiring is provided on the first main surface 14a of the piezoelectric layer 14. The first potential connection portion 28A is connected to the connection wiring. The first potential connection portion 28A is connected to the reference potential via the connection wiring.

[0058] As described above, the first potential connection portion 28A is located at the third electrode finger 27 at one end in the orthogonal direction of the electrode fingers among the plurality of third electrode fingers 27. On the other hand, the second potential connection portion 28B is located at the third electrode finger 27 at the other end in the orthogonal direction of the electrode fingers among the plurality of third electrode fingers 27. The second potential connection portion 28B is connected to the reference potential via a connection wiring.

[0059] The third potential connection portion 28C is located between two third electrode fingers 27 at both ends of the plurality of third electrode fingers 27 in the orthogonal direction of the electrode fingers of the reference potential electrode 19. Specifically, the third potential connection portion 28C is located at the connection electrode 24 that connects two third electrode fingers 27 at the center in the electrode finger arrangement direction among the plurality of third electrode fingers 27. More specifically, the connection electrode 24 connects the front end portions of the second bus bar 23 side of the adjacent third electrode fingers 27. The third potential connection portion 28C is configured as a part of the connection electrode 24. The third potential connection portion 28C is connected to the reference potential via a connection wiring.

[0060] In addition, the position of each potential connection part is not limited to the above. In this embodiment, the first potential connection part 28A, the second potential connection part 28B, and the third potential connection part 28C are connected to different connection wirings. However, a plurality of potential connection parts may be connected to the same connection wiring.

[0061] The connection wiring may be electrically connected to the external reference potential of the elastic wave device 10. For example, when the elastic wave device 10 is mounted on a mounting substrate, the connection wiring may be electrically connected to the mounting substrate via other wiring, electrode pads, a conductive adhesive, etc. In this case, the connection wiring may be electrically connected to the external reference potential via the mounting substrate, etc. In addition, the conductive adhesive is, for example, a bump or a conductive adhesive.

[0062] The elastic wave device 10 is an elastic wave resonator configured to utilize bulk waves in a thickness shear mode. Figure 2 As shown, the elastic wave device 10 has a plurality of excitation regions C. In the plurality of excitation regions C, body waves in the thickness shear mode and elastic waves in other modes are excited. Figure 2 In FIG, only two excitation areas C among multiple excitation areas C are shown.

[0063] A plurality of excitation regions C that are part of all the excitation regions C are regions where adjacent first electrode fingers 25 and third electrode fingers 27 overlap when viewed from the orthogonal direction of the electrode fingers, and are regions between the centers of adjacent first electrode fingers 25 and third electrode fingers 27. The remaining plurality of excitation regions C are regions where adjacent second electrode fingers 26 and third electrode fingers 27 overlap when viewed from the orthogonal direction of the electrode fingers, and are regions between the centers of adjacent second electrode fingers 26 and third electrode fingers 27. These excitation regions C are arranged in the orthogonal direction of the electrode fingers. In addition, the excitation region C is a region of the piezoelectric layer 14 defined based on the structure of the functional electrode 11.

[0064] The present embodiment is characterized in that it has the following structure. 1) A third electrode finger 27 of a reference potential electrode 19 is provided between the first electrode finger 25 of the first comb-shaped electrode 17 and the second electrode finger 26 of the second comb-shaped electrode 18. 2) The reference potential electrode 19 has a plurality of potential connection portions at at least three locations. Thus, when the elastic wave device 10 is used in a filter device, the filter device can be miniaturized and the resistance of the elastic wave device 10 can be reduced. This is described below.

[0065] pass Figure 4 An example of the transmission characteristic and reflection characteristic of the elastic wave device 10 is shown.

[0066] Figure 4 : is a diagram showing the transmission characteristic and reflection characteristic of the elastic wave device according to the first embodiment. Figure 4 The results of FEM (Finite Element Method) simulation are shown.

[0067] like Figure 4 As shown in FIG. 1 , it can be seen that a filter waveform can be appropriately obtained in one elastic wave device 10. The elastic wave device 10 is an acoustic coupling type filter. More specifically, Figure 2 As shown, the elastic wave device 10 includes an excitation region C located between the centers of the adjacent first electrode finger 25 and the third electrode finger 27, and an excitation region C located between the centers of the adjacent second electrode finger 26 and the third electrode finger 27. Elastic waves of multiple modes including a thickness shear mode bulk wave are excited in these excitation regions C. By coupling these modes, a filter waveform can be appropriately obtained in one elastic wave device 10.

[0068] When elastic wave device 10 is used as an elastic wave resonator in a filter device, a filter waveform can be appropriately obtained even if the number of elastic wave resonators constituting the filter device is one or less, thereby promoting miniaturization of the filter device.

[0069] Moreover, if Figure 2 As shown, in this embodiment, reference potential electrode 19 has three potential connection portions. This can reduce the resistance of reference potential electrode 19 and reduce the resistance of elastic wave device 10. The details are described below by comparing this embodiment with a reference example.

[0070] Figure 5The reference example shown is different from the first embodiment in the structure of the reference potential electrode 109. Specifically, the reference potential electrode 109 has only two potential connection portions 108. The reference potential electrode 109 has a plurality of connection electrodes 24 and a plurality of third electrode fingers 27 in the same manner as the first embodiment. The reference potential electrode 109 has a meandering shape. The two potential connection portions 108 are located at two third electrode fingers 27 at both ends of the plurality of third electrode fingers 27 in the orthogonal direction of the electrode fingers. In addition, the elastic wave device 100 of the reference example has a pair of comb-shaped electrodes. A portion of the reference potential electrode 109 is disposed between the pair of comb-shaped electrodes.

[0071] In the reference potential electrode 109, a plurality of connection electrodes 24 and a plurality of third electrode fingers 27 are included between the potential connection portions 108 of both sides. Therefore, the length of the portion between the potential connection portions 108 of two locations in the reference potential electrode 109 is long. Furthermore, a plurality of third electrode fingers 27 other than one and a plurality of connection electrodes 24 are provided between a pair of comb-shaped electrodes. Therefore, the widths of the plurality of third electrode fingers 27 and the plurality of connection electrodes 24 are narrow. Therefore, the resistance of the reference potential electrode 109 is high.

[0072] In addition, the length of the reference potential electrode 109 is the sum of the lengths of the plurality of third electrode fingers 27 and the lengths of the plurality of connecting electrodes 24. The so-called length of the third electrode finger 27 is the dimension of the third electrode finger 27 along the direction in which the electrode finger extends. The so-called length of the connecting electrode 24 is the dimension along the direction in which the connecting electrode 24 extends. In the reference example and the first embodiment, the direction in which the connecting electrode 24 extends is parallel to the direction orthogonal to the electrode fingers. The width of the third electrode finger 27 is the dimension of the third electrode finger 27 along the direction orthogonal to the electrode fingers. The width of the connecting electrode 24 is the dimension along the direction orthogonal to the direction in which the connecting electrode 24 extends. In the reference potential electrode in the present invention, the above-mentioned lengths and widths are also defined in the same way.

[0073] Compared with the reference example, Figure 2 In the first embodiment shown, the reference potential electrode 19 has three potential connection portions. Therefore, at least one potential connection portion is located between two third electrode fingers 27 at both ends in the orthogonal direction of the electrode fingers among the plurality of third electrode fingers 27 of the reference potential electrode 19. Thus, the length of the portion between the potential connection portions in the reference potential electrode 19 can be shortened.

[0074] More specifically, the reference potential electrode 19 has a portion between two potential connection parts. The portion between one potential connection part is a portion between the first potential connection part 28A and the third potential connection part 28C. The portion between the other potential connection part is a portion between the second potential connection part 28B and the third potential connection part 28C. Thus, the length of the portion between the potential connection parts can be shortened compared to a case where there is only one portion between the potential connection parts. Thus, the resistance of the reference potential electrode 19 can be reduced. Therefore, the resistance of the elastic wave device 10 can be reduced.

[0075] The resistance of elastic wave device 10 referred to here is a so-called series resistance. When elastic wave device 10 is used in a filter device, the series resistance of elastic wave device 10 is the resistance of elastic wave device 10 when elastic wave device 10 is connected in series with another element.

[0076] Hereinafter, the structure of the first embodiment will be described in more detail.

[0077] like Figure 2 As shown, two third electrode fingers 27 at both ends in the orthogonal direction of the electrode fingers among the plurality of third electrode fingers 27 are located outside in the orthogonal direction of the electrode fingers than the two ends in the orthogonal direction of the electrode fingers of the second comb-shaped electrode 18. The connection wirings connected to the first potential connection portion 28A and the second potential connection portion 28B pass outside in the orthogonal direction of the electrode fingers than the above-mentioned two ends of the second comb-shaped electrode 18.

[0078] On the other hand, the third potential connection portion 28C is located at a position inward of both ends of the first comb-shaped electrode 17 in the electrode finger orthogonal direction and both ends of the second comb-shaped electrode 18 in the electrode finger orthogonal direction. Here, in the first embodiment, the second bus bar 23 is divided into two divided bus bar portions 23A and a divided bus bar portion 23B. The divided bus bar portion 23A and the divided bus bar portion 23B are opposed to each other with a gap in the electrode finger orthogonal direction. The connection wiring connected to the third potential connection portion 28C passes between the divided bus bar portion 23A and the divided bus bar portion 23B and is connected to the reference potential.

[0079] The position of each potential connection portion is not particularly limited. However, the first potential connection portion 28A and the second potential connection portion 28B are preferably located at two third electrode fingers 27 at both ends of the electrode fingers in the orthogonal direction among the plurality of third electrode fingers 27. Thus, the connection wiring of the first potential connection portion 28A and the second potential connection portion 28B can be easily connected to the reference potential.

[0080] In addition, the connection wiring has a portion located outside the first comb-shaped electrode 17 and the second comb-shaped electrode 18 and does not contribute to the excitation of the elastic wave. Therefore, the width of the connection wiring can be increased. Therefore, the resistance of the connection wiring can be easily reduced.

[0081] like Figure 1 As shown, the support member 13 includes a support substrate 16 and an insulating layer 15. The piezoelectric substrate 12 is a stack of the support substrate 16, the insulating layer 15, and the piezoelectric layer 14. That is, when viewed from the direction in which the first principal surface 14a and the second principal surface 14b of the piezoelectric layer 14 are opposite, the piezoelectric layer 14 and the support member 13 overlap.

[0082] As the material of the support substrate 16, for example, semiconductors such as silicon, ceramics such as aluminum oxide, etc. can be used. As the material of the insulating layer 15, appropriate dielectrics such as silicon oxide or tantalum oxide can be used. The piezoelectric layer 14 is, for example, a lithium niobate layer such as a LiNbO3 layer or a lithium tantalate layer such as a LiTaO3 layer.

[0083] A recess is provided in the insulating layer 15. A piezoelectric layer 14 is provided on the insulating layer 15 so as to block the recess. Thus, a hollow portion is formed. The hollow portion is a cavity 10a. In the first embodiment, the supporting member 13 and the piezoelectric layer 14 are configured so that a part of the supporting member 13 and a part of the piezoelectric layer 14 are opposed to each other with the cavity 10a sandwiched therebetween. However, the recess in the supporting member 13 may also be provided throughout the insulating layer 15 and the supporting substrate 16. Alternatively, it may be that only the recess provided in the supporting substrate 16 is blocked by the insulating layer 15. The recess may also be provided in the piezoelectric layer 14. In addition, the cavity 10a may also be a through hole provided in the supporting member 13.

[0084] The cavity 10a is the acoustic reflection portion in the present invention. The acoustic reflection portion can effectively confine the energy of the elastic wave on the piezoelectric layer 14 side. The acoustic reflection portion can be provided at a position overlapping with at least a portion of the functional electrode 11 in the support member 13 when viewed from above. More specifically, when viewed from above, at least a portion of each of the first electrode finger 25, the second electrode finger 26, and the third electrode finger 27 can overlap with the acoustic reflection portion. When viewed from above, it is preferred that a plurality of excitation regions C overlap with the acoustic reflection portion.

[0085] In this specification, the term "top view" refers to a view from the Figure 1 The upper direction in FIG. 1 is viewed along the stacking direction of the support member 13 and the piezoelectric layer 14. Figure 1For example, the piezoelectric layer 14 side is the upper side of the support substrate 16 side and the piezoelectric layer 14 side. Furthermore, in this specification, looking down is synonymous with observing from the main surface facing direction. The so-called main surface facing direction is the direction in which the first main surface 14a and the second main surface 14b of the piezoelectric layer 14 are facing each other. More specifically, the main surface facing direction is, for example, the normal direction of the first main surface 14a.

[0086] In addition, the sound reflecting portion may be a sound reflecting film such as a sound multilayer film described later. For example, a sound reflecting film may be provided on the surface of the supporting member.

[0087] In the first embodiment, the center-to-center distances of the adjacent multiple pairs of first electrode fingers 25 and third electrode fingers 27 are the same as the center-to-center distances of the adjacent multiple pairs of second electrode fingers 26 and third electrode fingers 27. In this case, when the thickness of the piezoelectric layer 14 is d and the center-to-center distances of the adjacent electrode fingers are p, it is preferred that d / p is 0.5 or less. It is more preferred that d / p is 0.24 or less. Thus, a thickness shear mode body wave can be appropriately excited.

[0088] However, the distance between the centers of the adjacent first electrode fingers 25 and the third electrode fingers 27 and the distance between the centers of the adjacent second electrode fingers 26 and the third electrode fingers 27 may not be constant. In this case, it is preferred that the longest distance between the distance between the centers of the adjacent first electrode fingers 25 and the third electrode fingers 27 and the distance between the centers of the adjacent second electrode fingers 26 and the third electrode fingers 27 be set to p. In this case, it is preferred that d / p be less than 0.5, and it is more preferred that d / p be less than 0.24. In addition, the elastic wave device of the present invention may not necessarily be configured to utilize the thickness shear mode.

[0089] like Figure 2 As shown, when viewed from the orthogonal direction of the electrode fingers, the area where the adjacent first electrode finger 25 and the third electrode finger 27, or the adjacent second electrode finger 26 and the third electrode finger 27 overlap each other is the intersection area E. The intersection area E includes a plurality of excitation areas C. The elastic wave device according to the present invention can also be configured to utilize plate waves. In this case, the excitation area is the intersection area E. In addition, the intersection area E is an area of ​​the piezoelectric layer 14 defined based on the structure of the functional electrode 11.

[0090] The functional electrode 11 has the same structure as an IDT (Interdigital Transducer) electrode except for the reference potential electrode 19. The intersection region E can be regarded as a region where adjacent first electrode fingers 25 and second electrode fingers 26 overlap each other when viewed in the orthogonal direction of the electrode fingers.

[0091] Each connection electrode 24 of the reference potential electrode 19 is provided outside the electrode finger extending direction of the excitation region C. In the first embodiment, in the reference potential electrode 19, all adjacent third electrode fingers 27 are connected to each other through the connection electrodes 24. However, all adjacent third electrode fingers 27 may not necessarily be connected through the connection electrodes 24.

[0092] The second bus bar 23 has a split bus bar portion 23A and a split bus bar portion 23B. On the other hand, the first bus bar 22 is not split. In addition, the structure of the first bus bar 22 and the second bus bar 23 is not limited to the above. However, it is preferred that at least one of the first bus bar 22 and the second bus bar 23 is split and has a plurality of split bus bar portions. Thus, it is possible to set the connection wiring connected to the potential connection portion to pass through the structure between the split bus bar portions. Therefore, the connection wiring can be easily connected to the reference potential.

[0093] In the first embodiment, the piezoelectric layer 14 is a lithium niobate layer. Specifically, a rotated Y-cut LiNbO3 is used as the material of the piezoelectric layer 14. In this case, the fractional bandwidth of the elastic wave device 10 depends on the Euler angles (φ, θ, ψ) of the lithium niobate used for the piezoelectric layer 14. The so-called fractional bandwidth is represented by (|fa-fr| / fr)×100[%] when the resonant frequency is fr and the antiresonant frequency is fa.

[0094] The relationship between the fractional bandwidth of elastic wave device 10 and the Euler angles (φ, θ, ψ) of piezoelectric layer 14 was derived when d / p was made as close to 0 as possible. In addition, φ in the Euler angles was set to 0°.

[0095] Figure 6 This is a diagram showing a mapping of the fractional bandwidth with respect to the Euler angle (0°, θ, ψ) of LiNbO 3 when d / p is made as close to 0 as possible.

[0096] Figure 6 The region R indicated by the hatching is a region where a fractional bandwidth of at least 2% can be obtained. If the range of the region R is approximated, it becomes the range represented by the following equations (1), (2), and (3). In addition, when φ in the Euler angles (φ, θ, ψ) is within the range of 0°±10°, the relationship between θ and ψ and the fractional bandwidth is Figure 6 When the piezoelectric layer 14 is a lithium tantalate layer, the relationship between θ and ψ in the Euler angle (in the range of 0°±10°, θ, ψ) and the fractional bandwidth is also the same as Figure 6 The relationship shown is the same.

[0097] (within the range of 0°±10°, 0°~25°, arbitrary ψ) …Formula (1)

[0098] (Within the range of 0°±10°, 25°~100°, 0°~75°[(1-(θ-50) 2 / 2500)] 1 / 2 or 180°-75°[(1-(θ-50) 2 / 2500)] 1 / 2 ~180°) …Formula (2)

[0099] (0°±10°, 180°-40°[(1-(ψ-90) 2 / 8100)] 1 / 2 ~180°, any ψ) …Formula (3)

[0100] The range of the Euler angles of the above-mentioned formula (1), formula (2), or formula (3) is preferably 0. This allows the fractional bandwidth to be sufficiently widened. This allows the elastic wave device 10 to be suitably used in a filter device.

[0101] Figure 7 It is a schematic plan view of an elastic wave device according to a second embodiment.

[0102] The present embodiment differs from the first embodiment in that the reference potential electrode 39 has potential connection portions at five locations. The present embodiment also differs from the first embodiment in that the first bus bar 32 is divided into three divided bus bar portions 32A, divided bus bar portions 32B, and divided bus bar portions 32C. The divided bus bar portions 32A and the divided bus bar portions 32B are opposed to each other with a gap in the direction orthogonal to the electrode fingers. The divided bus bar portions 32B and the divided bus bar portions 32C are opposed to each other with a gap in the direction orthogonal to the electrode fingers. Except for the above aspects, the elastic wave device 30 of the present embodiment has the same structure as the elastic wave device 10 of the first embodiment. In addition, the second bus bar 23 has two divided bus bar portions 23A and a divided bus bar portion 23B in the same manner as in the first embodiment.

[0103] Specifically, the potential connection portions at the five locations of the reference potential electrode 39 are the first potential connection portion 38A, the second potential connection portion 38B, and the third potential connection portions at three locations. Specifically, the third potential connection portions at three locations are the third potential connection portion 38C, the third potential connection portion 38D, and the third potential connection portion 38E. The first potential connection portion 38A and the second potential connection portion 38B are located at two third electrode fingers 27 at both ends of the plurality of third electrode fingers 27 in the orthogonal direction of the electrode fingers.

[0104] On the other hand, the third potential connection portions at three locations are located in the portion between the two third electrode fingers 27 in the reference potential electrode 39. More specifically, the third potential connection portion 38C is located at the connection electrode 24 that connects the front end portions on the second bus bar 23 side of the adjacent third electrode fingers 27. On the other hand, the third potential connection portion 38D and the third potential connection portion 38E are located at the connection electrode 24 that connects the front end portions on the first bus bar 32 side of the adjacent third electrode fingers 27.

[0105] The third potential connection portion 38C is located between the third potential connection portion 38D and the third potential connection portion 38E. More specifically, the third potential connection portion 38C is located at the connection electrode 24 that connects two third electrode fingers 27 located in the center of the plurality of third electrode fingers 27 in the orthogonal direction. However, the positions of the potential connection portions are not limited to the above.

[0106] Specifically, each connection wiring connected to the first potential connection portion 38A and the second potential connection portion 38B passes outside both ends of the second comb-shaped electrode 18 in the electrode finger orthogonal direction.

[0107] The connection wiring connected to the third potential connection portion 38D passes between the divided bus bar portion 32A and the divided bus bar portion 32B of the first bus bar 32. The connection wiring connected to the third potential connection portion 38E passes between the divided bus bar portion 32B and the divided bus bar portion 32C of the first bus bar 32.

[0108] In the present embodiment, reference potential electrode 39 has five potential connection portions. This effectively shortens the length of the portion between the potential connection portions. This effectively reduces the resistance of reference potential electrode 39 and the resistance of elastic wave device 30.

[0109] Figure 8 It is a schematic plan view of an elastic wave device according to a third embodiment.

[0110] The present embodiment differs from the first embodiment in that, in reference potential electrode 49, some of third electrode fingers 27 among all adjacent third electrode fingers 27 are connected to each other via connection electrodes 24. Elastic wave device 40 of the present embodiment has the same structure as elastic wave device 10 of the first embodiment except for the above.

[0111] The reference potential electrode 49 has a plurality of electrode portions. Specifically, the plurality of electrode portions in the present embodiment are a first electrode portion 46A, a second electrode portion 46B, and a third electrode portion 46C. Each electrode portion includes a plurality of third electrode fingers 27 and at least one connecting electrode 24. The electrode portions are not connected to each other via the connecting electrode 24. That is, the reference potential electrode 49 is divided into a plurality of electrode portions.

[0112] Each electrode portion has a potential connection portion. Specifically, the first electrode portion 46A is located at one end of the electrode fingers in the orthogonal direction among the plurality of electrode portions. The first electrode portion 46A has a first potential connection portion 28A. The second electrode portion 46B is located at the other end of the electrode fingers in the orthogonal direction among the plurality of electrode portions. The second electrode portion 46B has a second potential connection portion 28B. The third electrode portion 46C is located between the first electrode portion 46A and the second electrode portion 46B. The third electrode portion 46C has a third potential connection portion 28C.

[0113] Each electrode portion is connected to a reference potential at each potential connection portion. More specifically, each electrode portion is connected to a reference potential via a connection wiring connected to each potential connection portion.

[0114] Each electrode portion has an end portion in the reference potential electrode 49. The end portion of the electrode portion is an end portion of the third electrode finger 27 that is not connected to the connection electrode 24, the connection wiring, or the like.

[0115] In the present embodiment, the reference potential electrode 49 is divided into a plurality of electrode portions. Therefore, the length of each electrode portion is short compared to the total length of all third electrode fingers 27 and all connecting electrodes 24 of the reference potential electrode 49. Therefore, in each electrode portion, the length from the potential connecting portion to the end of the electrode portion is short. Thus, the resistance of the reference potential electrode 49 can be effectively reduced, and the resistance of the elastic wave device 40 can be effectively reduced. In addition, the length of the electrode portion is the total length of all third electrode fingers 27 in the electrode portion and the length of all connecting electrodes 24.

[0116] Fig. 9 It is a schematic plan view of an elastic wave device according to a fourth embodiment.

[0117] The present embodiment is different from the second embodiment in that, in reference potential electrode 59, a portion of third electrode fingers 27 among all adjacent third electrode fingers 27 are not connected to each other via connection electrode 24. Specifically, reference potential electrode 59 is divided into five electrode portions. Except for the above, elastic wave device 50 of the present embodiment has the same structure as elastic wave device 30 of the second embodiment.

[0118] In addition, the first bus bar 32 and the second bus bar 23 each have a plurality of split bus bar portions similarly to the second embodiment. More specifically, the first bus bar 32 has three split bus bar portions 32A, a split bus bar portion 32B, and a split bus bar portion 32C. The second bus bar 23 has two split bus bar portions 23A and a split bus bar portion 23B.

[0119] The reference potential electrode 59 has five potential connection parts arranged in the same manner as in the second embodiment. Specifically, the five potential connection parts are a first potential connection part 38A, a second potential connection part 38B, and three third potential connection parts. Specifically, the three third potential connection parts are a third potential connection part 38C, a third potential connection part 38D, and a third potential connection part 38E.

[0120] In this embodiment, the reference potential electrode 59 has five electrode portions. Specifically, the five electrode portions in this embodiment are the first electrode portion 56A, the second electrode portion 56B, and three third electrode portions. Specifically, the three third electrode portions are the third electrode portion 56C, the third electrode portion 56D, and the third electrode portion 56E.

[0121] The first electrode portion 56A is located at one end of the electrode fingers orthogonal to the plurality of electrode portions. The first electrode portion 56A has a first potential connection portion 38A. The second electrode portion 56B is located at the other end of the electrode fingers orthogonal to the plurality of electrode portions. The second electrode portion 56B has a second potential connection portion 38B.

[0122] The third electrode portion 56C, the third electrode portion 56D, and the third electrode portion 56E are located between the first electrode portion 56A and the second electrode portion 56B. Furthermore, the third electrode portion 56C is located between the third electrode portion 56D and the third electrode portion 56E. The third electrode portion 56C has a third potential connection portion 38C. The third electrode portion 56D has a third potential connection portion 38D. The third electrode portion 56E has a third potential connection portion 38E.

[0123] Specifically, each connection wiring connected to the first potential connection portion 38A and the second potential connection portion 38B passes outside both ends of the second comb-shaped electrode 18 in the electrode finger orthogonal direction.

[0124] The connection wiring connected to the third potential connection portion 38C passes between the divided bus bar portion 23A and the divided bus bar portion 23B of the second bus bar 23. The connection wiring connected to the third potential connection portion 38D passes between the divided bus bar portion 32A and the divided bus bar portion 32B of the first bus bar 32. The connection wiring connected to the third potential connection portion 38E passes between the divided bus bar portion 32B and the divided bus bar portion 32C of the first bus bar 32.

[0125] The number of electrode portions of the reference potential electrode 59 and the number of potential connection portions are not limited to the above. Similarly, the number of divisions of the first bus bar 32 and the second bus bar 23 is not limited to the above.

[0126] In the present embodiment, the reference potential electrode 59 is divided into a plurality of electrode portions. Therefore, the length of each electrode portion is short compared to the total length of all third electrode fingers 27 and all connecting electrodes 24 of the reference potential electrode 59. Therefore, in each electrode portion, the length from the potential connecting portion to the end of the electrode portion is short. Thus, the resistance of the reference potential electrode 59 can be effectively reduced, and the resistance of the elastic wave device 50 can be effectively reduced.

[0127] The following describes the details of the thickness shear mode using an example in which the functional electrode is an IDT electrode. In addition, the IDT electrode does not have a third electrode finger. The "electrode" in the IDT electrode described later is equivalent to the electrode finger. The supporting member in the following example is equivalent to the supporting substrate in the present invention. In the following, the reference potential is sometimes recorded as the ground potential.

[0128] Fig.10 (a) is a schematic perspective view showing the appearance of an elastic wave device using a thickness shear mode bulk wave, Fig.10 (b) is a top view showing the electrode structure on the piezoelectric layer. Fig.11 It is along Fig.10 (a) is a cross-sectional view of a portion taken along line AA.

[0129] The elastic wave device 1 has a piezoelectric layer 2 containing LiNbO3. The piezoelectric layer 2 may also contain LiTaO3. The cutting angle of LiNbO3 and LiTaO3 is Z-cut, but it may also be a rotated Y-cut or X-cut. The thickness of the piezoelectric layer 2 is not particularly limited, but in order to effectively excite the thickness shear mode, it is preferably greater than 40nm and less than 1000nm, and more preferably greater than 50nm and less than 1000nm. The piezoelectric layer 2 has a first principal surface 2a and a second principal surface 2b facing each other. An electrode 3 and an electrode 4 are provided on the first principal surface 2a. Here, electrode 3 is an example of a "first electrode", and electrode 4 is an example of a "second electrode". In Fig.10 (a) and Fig.10In (b), a plurality of electrodes 3 are connected to a first bus bar 5. A plurality of electrodes 4 are connected to a second bus bar 6. The plurality of electrodes 3 and the plurality of electrodes 4 are interlaced with each other. The electrodes 3 and 4 have a rectangular shape and a length direction. In a direction perpendicular to the length direction, the electrode 3 and the adjacent electrode 4 are opposed to each other. The length directions of the electrodes 3 and 4, and the directions perpendicular to the length directions of the electrodes 3 and 4 are all directions intersecting with the thickness direction of the piezoelectric layer 2. Therefore, it can also be said that the electrode 3 and the adjacent electrode 4 are opposed to each other in a direction intersecting with the thickness direction of the piezoelectric layer 2. In addition, the length directions of the electrodes 3 and 4 can also be perpendicular to the length directions of the electrodes 3 and 4. Fig.10 (a) and Fig.10 The direction perpendicular to the length direction of the electrodes 3 and 4 shown in (b) is changed. Fig.10 (a) and Fig.10 In (b), the electrodes 3 and 4 may be extended in the direction in which the first bus bar 5 and the second bus bar 6 extend. In this case, the first bus bar 5 and the second bus bar 6 are Fig.10 (a) and Fig.10 (b) becomes extended in the direction in which electrodes 3 and 4 extend. Moreover, a pair of structures in which an electrode 3 connected to one potential and an electrode 4 connected to another potential are adjacent are provided with multiple pairs in a direction orthogonal to the length direction of the electrodes 3 and 4. Here, the so-called electrode 3 and electrode 4 are adjacent does not mean that the electrodes 3 and electrode 4 are arranged in direct contact, but refers to the case where the electrodes 3 and electrode 4 are arranged with a gap. In addition, when the electrode 3 and electrode 4 are adjacent, no electrode connected to the signal electrode and the ground electrode, including other electrodes 3 and 4, is arranged between the electrode 3 and the electrode 4. The number of pairs does not need to be an integer pair, and can also be 1.5 pairs, 2.5 pairs, etc. The center-to-center distance between the electrodes 3 and 4, that is, the spacing, is preferably in the range of 1 μm or more and 10 μm or less. In addition, the width of the electrodes 3 and 4, that is, the size of the opposing direction of the electrodes 3 and 4, is preferably in the range of 50 nm or more and 1000 nm or less, and more preferably in the range of 150 nm or more and 1000 nm or less. The center-to-center distance between electrodes 3 and 4 is the distance connecting the center of the dimension (width dimension) of electrode 3 in a direction perpendicular to the length direction of electrode 3 and the center of the dimension (width dimension) of electrode 4 in a direction perpendicular to the length direction of electrode 4.

[0130] In addition, in the elastic wave device 1, since a Z-cut piezoelectric layer is used, the direction perpendicular to the length direction of the electrodes 3 and 4 becomes a direction perpendicular to the polarization direction of the piezoelectric layer 2. In the case where a piezoelectric body with another cutting angle is used as the piezoelectric layer 2, this is not limited to this. Here, the so-called "orthogonal" is not limited to the case of being strictly orthogonal, and can also be approximately orthogonal (the angle formed by the direction perpendicular to the length direction of the electrodes 3 and 4 and the polarization direction is, for example, within the range of 90°±10°).

[0131] A support member 8 is stacked on the second main surface 2b side of the piezoelectric layer 2 via an insulating layer 7. The insulating layer 7 and the support member 8 have a frame-like shape. Fig.11 As shown, there are through holes 7a and 8a. Thus, a cavity 9 is formed. The cavity 9 is provided so as not to hinder the vibration of the excitation region C of the piezoelectric layer 2. Therefore, the support member 8 is stacked on the second main surface 2b via the insulating layer 7 at a position that does not overlap with the portion where at least one pair of electrodes 3 and 4 is provided. In addition, the insulating layer 7 may not be provided. Therefore, the support member 8 may be directly or indirectly stacked on the second main surface 2b of the piezoelectric layer 2.

[0132] The insulating layer 7 includes silicon oxide. However, in addition to silicon oxide, suitable insulating materials such as silicon nitride oxide and alumina can also be used. The supporting member 8 includes Si. The surface orientation of Si in the surface on the side of the piezoelectric layer 2 can be (100), (110), or (111). It is preferred that the Si constituting the supporting member 8 has a high resistance with a resistivity of more than 4 kΩcm. However, the supporting member 8 can also be composed of suitable insulating materials or semiconductor materials.

[0133] As the material of the supporting member 8, for example, piezoelectrics such as aluminum oxide, lithium tantalate, lithium niobate, and quartz, various ceramics such as alumina, magnesium oxide, sapphire, silicon nitride, aluminum nitride, silicon carbide, zirconium oxide, cordierite, mullite, steatite, and forsterite, dielectrics such as diamond and glass, and semiconductors such as gallium nitride can be used.

[0134] The plurality of electrodes 3, 4 and the first bus bar 5 and the second bus bar 6 include a suitable metal or alloy such as Al or AlCu alloy. In the elastic wave device 1, the electrodes 3, 4 and the first bus bar 5 and the second bus bar 6 have a structure in which an Al film is stacked on a Ti film. Alternatively, a bonding layer other than a Ti film may be used.

[0135] During driving, an AC voltage is applied between the plurality of electrodes 3 and the plurality of electrodes 4. More specifically, an AC voltage is applied between the first bus bar 5 and the second bus bar 6. Thus, it is possible to obtain a resonance characteristic of a body wave utilizing a thickness shear mode excited in the piezoelectric layer 2. In addition, in the elastic wave device 1, when the thickness of the piezoelectric layer 2 is set to d and the center distance between any adjacent electrodes 3 and 4 among the plurality of pairs of electrodes 3 and 4 is set to p, d / p is set to be less than 0.5. Therefore, the body wave of the thickness shear mode can be effectively excited, and a good resonance characteristic can be obtained. More preferably, d / p is less than 0.24, in which case, a better resonance characteristic can be obtained.

[0136] In the elastic wave device 1, since it has the above-mentioned structure, even if the number of pairs of electrodes 3 and 4 is reduced in order to achieve miniaturization, the Q value is not likely to decrease. This is because even if the number of electrode fingers in the reflectors on both sides is reduced, the propagation loss is small. In addition, the number of electrode fingers can be reduced because the body wave of the thickness shear mode is utilized. Fig.12 (a) and Fig.12 (b) explains the difference between the Lamb wave used in the elastic wave device and the bulk wave of the thickness shear mode.

[0137] Fig.12 (a) is a schematic front cross-sectional view for explaining Lamb waves propagating in a piezoelectric film of an elastic wave device as described in Japanese Patent Gazette No. 2012-257019. Here, the wave propagates in the piezoelectric film 201 as indicated by the arrow. Here, in the piezoelectric film 201, the first principal surface 201a and the second principal surface 201b are opposed to each other, and the thickness direction connecting the first principal surface 201a and the second principal surface 201b is the Z direction. The X direction is the direction in which the electrode fingers of the IDT electrode are arranged. Fig.12 As shown in (a), for Lamb waves, the waves propagate in the X direction as shown in the figure. Since it is a plate wave, although the piezoelectric film 201 vibrates as a whole, the waves propagate in the X direction, so reflectors are arranged on both sides to obtain resonance characteristics. Therefore, wave propagation loss occurs, and when miniaturization is sought, that is, when the number of pairs of electrode fingers is reduced, the Q value decreases.

[0138] In contrast, Fig.12As shown in (b), in the elastic wave device 1, the vibration displacement is in the thickness shear direction, so the wave generally propagates in the direction connecting the first main surface 2a and the second main surface 2b of the piezoelectric layer 2, that is, the Z direction, and resonates. That is, the X-direction component of the wave is significantly smaller than the Z-direction component. Moreover, the resonance characteristics can be obtained by the propagation of the wave in the Z direction, so even if the number of electrode fingers of the reflector is reduced, it is not easy to cause propagation loss. Furthermore, even if the number of pairs of electrode pairs including electrodes 3 and 4 is reduced in order to promote miniaturization, it is not easy to cause a decrease in the Q value.

[0139] In addition, if Fig.13 As shown in FIG. 1 , the amplitude directions of the thickness shear mode body wave are opposite in the first region 451 included in the excitation region C of the piezoelectric layer 2 and in the second region 452 included in the excitation region C. Fig.13 , a body wave is schematically shown when a voltage is applied between the electrode 3 and the electrode 4 so that the electrode 4 has a higher potential than the electrode 3. The first region 451 is a region between the first principal surface 2a and the imaginary plane VP1 which is orthogonal to the thickness direction of the piezoelectric layer 2 and divides the piezoelectric layer 2 into two parts in the excitation region C. The second region 452 is a region between the imaginary plane VP1 and the second principal surface 2b in the excitation region C.

[0140] As described above, elastic wave device 1 includes at least one pair of electrodes including electrode 3 and electrode 4 , but since waves are not propagated in the X direction, there is no need to have more than one pair of electrodes including electrode 3 and electrode 4 . In other words, at least one pair of electrodes is sufficient.

[0141] For example, the electrode 3 is an electrode connected to the signal potential, and the electrode 4 is an electrode connected to the ground potential. However, the electrode 3 may be connected to the ground potential, and the electrode 4 may be connected to the signal potential. In the elastic wave device 1, at least one pair of electrodes is an electrode connected to the signal potential or an electrode connected to the ground potential as described above, and no floating electrode is provided.

[0142] Fig.14 It is shown Fig.11 The resonance characteristics of the elastic wave device are shown in FIG. 1. The design parameters of the elastic wave device 1 that obtain the resonance characteristics are as follows.

[0143] Piezoelectric layer 2: LiNbO3 with Euler angles (0°, 0°, 90°), thickness = 400nm.

[0144] When observed in a direction orthogonal to the length direction of electrode 3 and electrode 4, the length of the overlapping area of ​​electrode 3 and electrode 4, i.e., the excitation area C, is 40 μm, the number of pairs of electrodes including electrodes 3 and 4 is 21 pairs, the center distance between electrodes is 3 μm, the width of electrodes 3 and 4 is 500 nm, and d / p is 0.133.

[0145] Insulating layer 7: a silicon oxide film having a thickness of 1 μm.

[0146] Supporting member 8: Si.

[0147] The length of the excitation region C refers to the dimension of the excitation region C along the length direction of the electrodes 3 and 4 .

[0148] In elastic wave device 1 , the inter-electrode distances of the electrode pairs including electrodes 3 and 4 are set equal in a plurality of pairs. In other words, electrodes 3 and electrodes 4 are arranged at equal intervals.

[0149] according to Fig.14 It is clear that good resonance characteristics with a fractional bandwidth of 12.5% ​​are obtained even though there is no reflector.

[0150] When the thickness of the piezoelectric layer 2 is d and the center-to-center distance between the electrodes 3 and 4 is p, as described above, in the elastic wave device 1, d / p is 0.5 or less, and more preferably 0.24 or less. Fig.15 This is explained.

[0151] and got Fig.14 A plurality of elastic wave devices are obtained by changing d / p in the same manner as the elastic wave devices having the resonance characteristics shown. Fig.15 3 is a diagram showing the relationship between d / p and the fractional bandwidth of a resonator as an elastic wave device.

[0152] according to Fig.15 It is clear that if d / p>0.5, even if d / p is adjusted, the fractional bandwidth is less than 5%. In contrast, when d / p≤0.5, if d / p is changed within this range, the fractional bandwidth can be made greater than 5%, that is, a resonator with a high coupling coefficient can be constructed. In addition, when d / p is less than 0.24, the fractional bandwidth can be increased to more than 7%. In addition, if d / p is adjusted within this range, a resonator with a wider fractional bandwidth can be obtained, and a resonator with a higher coupling coefficient can be realized. Therefore, it can be seen that by setting d / p to less than 0.5, a resonator with a high coupling coefficient that utilizes the body wave of the thickness shear mode mentioned above can be constructed.

[0153] Fig.16 1 is a top view of an elastic wave device using a thickness shear mode bulk wave. In the elastic wave device 80, a pair of electrodes including an electrode 3 and an electrode 4 are provided on the first principal surface 2a of the piezoelectric layer 2. Fig.16K in is the crossing width. As described above, in the elastic wave device of the present invention, the number of pairs of electrodes may be 1 pair. In this case, as long as the above d / p is 0.5 or less, a thickness shear mode body wave can be effectively excited.

[0154] In the elastic wave device 1, it is preferable that the metallization ratio MR of the excitation region C, which is an area where any adjacent electrodes 3 and 4 overlap when viewed in the opposite direction relative to the adjacent electrodes 3 and 4, satisfies MR≤1.75(d / p)+0.075. In this case, the stray can be effectively reduced. Fig.17 as well as Fig.18 Let me explain this. Fig.17 : is a reference diagram showing an example of the resonance characteristics of the elastic wave device 1. Spurious signals indicated by arrow B appear between the resonance frequency and the antiresonance frequency. In addition, d / p=0.08 is assumed, and the Euler angles of LiNbO3 are assumed to be (0°, 0°, 90°). In addition, the metallization ratio MR is assumed to be 0.35.

[0155] Reference Fig.10 (b) will explain the metallization ratio MR. Fig.10 In the electrode structure of (b), when focusing on a pair of electrodes 3 and 4, it is assumed that only the pair of electrodes 3 and 4 are provided. In this case, the portion surrounded by the single-dot chain line becomes the excitation region C. The so-called excitation region C refers to the region where the electrode 3 overlaps with the electrode 4, the region where the electrode 4 overlaps with the electrode 3, and the region between the electrode 3 and the electrode 4 where the electrode 3 overlaps with the electrode 4 when the electrode 3 and the electrode 4 are observed in the direction perpendicular to the length direction of the electrodes 3 and 4, that is, in the opposite direction. Moreover, the area of ​​the electrodes 3 and 4 in the excitation region C relative to the area of ​​the excitation region C becomes the metallization ratio MR. That is, the metallization ratio MR is the ratio of the area of ​​the metallized portion to the area of ​​the excitation region C.

[0156] When a plurality of pairs of electrodes are provided, the ratio of the total area of ​​the metallized portions included in all the excitation regions to the area of ​​the excitation regions may be defined as MR.

[0157] Fig.18 This is a diagram showing the relationship between the fractional bandwidth and the phase rotation amount of the impedance of the spurious signal normalized by 180 degrees when a plurality of spurious wave resonators are formed according to the structure of the spurious wave device 1. In addition, the film thickness of the piezoelectric layer and the size of the electrode were variously changed to adjust the fractional bandwidth. In addition, Fig.18 These are the results obtained when a piezoelectric layer including Z-cut LiNbO 3 is used, but the same tendency is observed when a piezoelectric layer having other cut angles is used.

[0158] exist Fig.18 In the area surrounded by ellipse J, the spurious signal increases to 1.0. Fig.18 It is clear that if the fractional bandwidth exceeds 0.17, that is, if it exceeds 17%, even if the parameters constituting the fractional bandwidth are changed, large spurs with a spurious level of 1 or more will appear in the passband. Fig.17 As shown in the resonance characteristics, large spurious signals indicated by arrow B appear in the frequency band. Therefore, the fractional bandwidth is preferably 17% or less. In this case, spurious signals can be reduced by adjusting the film thickness of the piezoelectric layer 2, the dimensions of the electrodes 3 and 4, and the like.

[0159] Fig.19 : is a graph showing the relationship between d / 2p, metallization ratio MR, and fractional bandwidth. In the above-mentioned elastic wave device, various elastic wave devices having different d / 2p and MR were constructed, and the fractional bandwidths were measured. Fig.19 The hatched portion on the right side of the dotted line D is an area where the fractional bandwidth is less than 17%. The boundary between the hatched area and the unhatched area is represented by MR=3.5(d / 2p)+0.075. That is, MR=1.75(d / p)+0.075. Therefore, it is preferred that MR≤1.75(d / p)+0.075. In this case, it is easy to make the fractional bandwidth less than 17%. More preferably, Fig.19 The region to the right of MR=3.5(d / 2p)+0.05 is indicated by the dashed line D1 in FIG. That is, as long as MR≤1.75(d / p)+0.05, the fractional bandwidth can be reliably set to 17% or less.

[0160] Fig. 20 This is a diagram showing a mapping of the fractional bandwidth with respect to the Euler angle (0°, θ, ψ) of LiNbO 3 when d / p is made as close to 0 as possible. Fig. 20 The multiple regions R shown in the figure and indicated by hatching are regions where a fractional bandwidth of 2% or more can be obtained. In addition, when φ among the Euler angles (φ, θ, ψ) is within the range of 0°±5°, the relationship between θ and ψ and the fractional bandwidth is Fig. 20 When the piezoelectric layer contains lithium tantalate (LiTaO3), the relationship between θ and ψ in the Euler angle (in the range of 0°±5°, θ, ψ) and BW is also the same as Fig. 20 The relationship shown is the same.

[0161] Therefore, as long as the Euler angles (φ, θ, ψ) of lithium niobate or lithium tantalate constituting the piezoelectric layer are within the range of 0°±5°, and θ and φ are Fig. 20 The fractional bandwidth can be sufficiently expanded within the range of any of the plurality of regions R shown, which is preferable.

[0162] Fig.21 It is a front cross-sectional view of an elastic wave device having an acoustic multilayer membrane.

[0163] In the elastic wave device 81, an acoustic multilayer film 82 is stacked on the second main surface 2b of the piezoelectric layer 2. The acoustic multilayer film 82 has a stacked structure of low acoustic impedance layers 82a, 82c, 82e having relatively low acoustic impedance and high acoustic impedance layers 82b, 82d having relatively high acoustic impedance. When the acoustic multilayer film 82 is used, the bulk wave of the thickness shear mode can be confined in the piezoelectric layer 2 even without using the cavity 9 in the elastic wave device 1. In the elastic wave device 81, by making the above-mentioned d / p less than 0.5, the resonance characteristics of the bulk wave based on the thickness shear mode can be obtained. In addition, in the acoustic multilayer film 82, the number of stacked low acoustic impedance layers 82a, 82c, 82e and high acoustic impedance layers 82b, 82d is not particularly limited. It is sufficient as long as at least one high acoustic impedance layer 82b, 82d is arranged on the side away from the piezoelectric layer 2 compared with the low acoustic impedance layers 82a, 82c, 82e.

[0164] The low acoustic impedance layers 82a, 82c, 82e and the high acoustic impedance layers 82b, 82d can be made of appropriate materials as long as the above-mentioned acoustic impedance relationship is satisfied. For example, silicon oxide or silicon oxynitride can be used as the material of the low acoustic impedance layers 82a, 82c, 82e. In addition, alumina, silicon nitride or metal can be used as the material of the high acoustic impedance layers 82b, 82d.

[0165] Fig. 22 This is a partially cutaway perspective view for explaining an elastic wave device using Lamb waves.

[0166] The elastic wave device 91 has a support substrate 92. A recessed portion opened on the upper surface is provided on the support substrate 92. A piezoelectric layer 93 is stacked on the support substrate 92. Thus, a cavity 9 is formed. An IDT electrode 94 is provided on the piezoelectric layer 93 above the cavity 9. Reflectors 95 and 96 are provided on both sides of the IDT electrode 94 in the direction of propagation of elastic waves. Fig. 22 In the figure, the outer periphery of the cavity 9 is shown by a dotted line. Here, the IDT electrode 94 has a first bus bar 94a, a second bus bar 94b, a plurality of first electrode fingers 94c, and a plurality of second electrode fingers 94d. The plurality of first electrode fingers 94c are connected to the first bus bar 94a. The plurality of second electrode fingers 94d are connected to the second bus bar 94b. The plurality of first electrode fingers 94c and the plurality of second electrode fingers 94d are interlaced with each other.

[0167] In the elastic wave device 91, Lamb waves as plate waves are excited by applying an alternating electric field to the IDT electrode 94 on the cavity 9. Furthermore, since the reflectors 95 and 96 are provided on both sides, resonance characteristics based on the Lamb waves can be obtained.

[0168] As described above, the elastic wave device of the present invention can also utilize plate waves. Fig. 22 In the example shown, the Figure 1 The main surface of the first main surface 14a of the piezoelectric layer 14 shown in the figure is provided with an IDT electrode 94, a reflector 95, and a reflector 96. On the other hand, in the elastic wave device of the present invention, a pair of comb-shaped electrodes and a plurality of third electrode fingers are provided on the first main surface 14a. In the case where the elastic wave device of the present invention uses plate waves, a pair of comb-shaped electrodes and a plurality of third electrode fingers, and the above-mentioned reflectors 95 and 96 are provided on the first main surface 14a of the piezoelectric layer 14 in the first embodiment to the fourth embodiment. In this case, the reflectors 95 and 96 only need to sandwich a pair of comb-shaped electrodes and a plurality of third electrode fingers in the direction orthogonal to the electrode fingers.

[0169] In the elastic wave devices of the first to fourth embodiments, for example, an acoustic reflection film may be provided between the support member and the piezoelectric layer. Fig.21 The acoustic multilayer membrane 82 shown. Specifically, the supporting member and the piezoelectric layer may be arranged so that at least a portion of the supporting member and at least a portion of the piezoelectric layer are opposed to each other with the acoustic multilayer membrane 82 sandwiched therebetween. In this case, in the acoustic multilayer membrane 82, it is sufficient to alternately stack low acoustic impedance layers and high acoustic impedance layers. The acoustic multilayer membrane 82 may also be an acoustic reflection portion in an elastic wave device.

[0170] In the elastic wave devices of the first to fourth embodiments utilizing bulk waves in the thickness shear mode, as described above, d / p is preferably 0.5 or less, and more preferably 0.24 or less. This can provide better resonance characteristics.

[0171] Furthermore, in the excitation region of the elastic wave device in the first to fourth embodiments using the thickness shear mode body wave, as described above, MR ≤ 1.75 (d / p) + 0.075 is preferably satisfied. More specifically, when the metallization ratio of the first electrode finger and the third electrode finger, and the second electrode finger and the third electrode finger with respect to the excitation region is MR, MR ≤ 1.75 (d / p) + 0.075 is preferably satisfied. In this case, spurious emission can be suppressed more reliably.

[0172] Description of Reference Numerals

[0173] 1…Elastic wave device

[0174] 2…Piezoelectric layer

[0175] 2a, 2b...1st main surface, 2nd main surface

[0176] 3, 4…Electrode

[0177] 5, 6…1st bus bar, 2nd bus bar

[0178] 7…Insulation layer

[0179] 7a…Through hole

[0180] 8…Supporting member

[0181] 8a…Through hole

[0182] 9…Hollow

[0183] 10…Elastic wave device

[0184] 10a…Cavity

[0185] 11…Functional electrode

[0186] 12…Piezoelectric substrate

[0187] 13…Supporting member

[0188] 14…Piezoelectric layer

[0189] 14a, 14b…1st main surface, 2nd main surface

[0190] 15…Insulation layer

[0191] 16...Support base plate

[0192] 17, 18…1st comb-shaped electrode, 2nd comb-shaped electrode

[0193] 19…Reference potential electrode

[0194] 22, 23…1st bus bar, 2nd bus bar

[0195] 23A, 23B...divided bus bar portion

[0196] 24…Connect the electrodes

[0197] 25 to 27…1st electrode finger to 3rd electrode finger

[0198] 28A, 28B ... first potential connection portion, second potential connection portion

[0199] 28C…Third potential connection part

[0200] 30…Elastic wave device

[0201] 32…Bus bar 1

[0202] 32A~32C…Divided bus bar section

[0203] 38A, 38B ... first potential connection portion, second potential connection portion

[0204] 38C~38E…Third potential connection part

[0205] 39…Reference potential electrode

[0206] 40…Elastic wave device

[0207] 46A to 46C ... first electrode portion to third electrode portion

[0208] 49…Reference potential electrode

[0209] 50…Elastic wave device

[0210] 56A, 56B ... first electrode portion, second electrode portion

[0211] 56C~56E…Third electrode part

[0212] 59…Reference potential electrode

[0213] 80, 81…Elastic wave device

[0214] 82…Acoustic multilayer film

[0215] 82a, 82c, 82e…low acoustic impedance layer

[0216] 82b, 82d…high acoustic impedance layer

[0217] 91…Elastic wave device

[0218] 92…Support base plate

[0219] 93…Piezoelectric layer

[0220] 94…IDT electrode

[0221] 94a, 94b…1st bus bar, 2nd bus bar

[0222] 94c, 94d ... first electrode finger, second electrode finger

[0223] 95, 96…Reflector

[0224] 100…Elastic wave device

[0225] 108…Potential connection

[0226] 109…reference potential electrode

[0227] 201…Piezoelectric film

[0228] 201a, 201b…1st main surface, 2nd main surface

[0229] 451, 452…1st area, 2nd area

[0230] C…Incentive Area

[0231] E…Intersection Area

[0232] R…Region

[0233] VP1…imaginary plane.

Claims

1. An elastic wave device comprising: Piezoelectric layer; a first comb-shaped electrode, disposed on the piezoelectric layer, having a first bus bar and a plurality of first electrode fingers each having one end connected to the first bus bar, and connected to an input potential; A second comb-shaped electrode, disposed on the piezoelectric layer, having a second bus bar and a plurality of second electrode fingers, one end of each of which is connected to the second bus bar, connected to an output potential, wherein the plurality of second electrode fingers are interlaced with the plurality of first electrode fingers; and A reference potential electrode having a plurality of third electrode fingers and a plurality of connection electrodes for connecting adjacent third electrode fingers to each other, and at least a portion of the third electrode fingers is arranged between the first comb-shaped electrode and the second comb-shaped electrode and connected to a reference potential, wherein the plurality of third electrode fingers are respectively arranged on the piezoelectric layer so as to be arranged with the first electrode finger and the second electrode finger in the direction in which the first electrode finger and the second electrode finger are arranged, The first electrode finger, the second electrode finger, and the third electrode finger are arranged in the order of the first electrode finger, the third electrode finger, the second electrode finger, and the third electrode finger forming one cycle, starting from the first electrode finger. The reference potential electrode has a plurality of potential connection portions connected to at least three locations of a reference potential.

2. The elastic wave device according to claim 1, wherein: When the direction orthogonal to the direction in which the first electrode finger, the second electrode finger and the third electrode finger extend is set as the electrode finger orthogonal direction, the multiple potential connection portions of the reference potential electrode are located at two third electrode fingers at both ends of the electrode fingers in the orthogonal direction among the multiple third electrode fingers and the portion between the two third electrode fingers.

3. The elastic wave device according to claim 1 or 2, wherein: In the reference potential electrode, all adjacent third electrode fingers are connected to each other via the connection electrode.

4. The elastic wave device according to claim 1 or 2, wherein: In the reference potential electrode, a portion of the third electrode fingers among all the adjacent third electrode fingers are not connected to each other via the connection electrode, and the reference potential electrode has a plurality of electrode portions that are not connected to each other via the connection electrode. Each of the electrode portions has the potential connecting portion.

5. The elastic wave device according to any one of claims 1 to 4, wherein: The elastic wave device further includes a connection wiring provided on the piezoelectric layer and connected to the potential connection portion of the reference potential electrode. The potential connection portion is connected to a reference potential via the connection wiring. At least one of the first bus bar and the second bus bar is divided and has a plurality of divided bus bar portions. The connection wiring passes between the divided bus bar portions.

6. The elastic wave device according to any one of claims 1 to 5, wherein: The elastic wave device is configured to utilize plate waves.

7. The elastic wave device according to any one of claims 1 to 5, wherein: The elastic wave device is configured to utilize thickness shear mode body waves.

8. The elastic wave device according to any one of claims 1 to 5, wherein: The elastic wave device further includes a supporting member stacked on the piezoelectric layer. In a plan view along the stacking direction of the support member and the piezoelectric layer, an acoustic reflection portion is formed at a position of the support member overlapping with the plurality of first electrode fingers, the plurality of second electrode fingers, and the plurality of third electrode fingers, When the longest distance between the centers of the adjacent first and third electrode fingers and the center distance between the adjacent second and third electrode fingers is represented by p, and the thickness of the piezoelectric layer is represented by d, d / p is equal to or less than 0.

5.

9. The elastic wave device according to claim 8, wherein: d / p is less than 0.

24.

10. The elastic wave device according to claim 8 or 9, wherein: The sound reflecting portion is a cavity, and the supporting member and the piezoelectric layer are arranged so that a part of the supporting member and a part of the piezoelectric layer face each other with the cavity interposed therebetween.

11. The elastic wave device according to claim 8 or 9, wherein: The acoustic reflection portion is an acoustic reflection membrane including a high acoustic impedance layer with relatively high acoustic impedance and a low acoustic impedance layer with relatively low acoustic impedance, and the supporting member and the piezoelectric layer are configured so that at least a portion of the supporting member and at least a portion of the piezoelectric layer are opposite to each other with the acoustic reflection membrane sandwiched therebetween.

12. The elastic wave device according to any one of claims 8 to 11, wherein: When a direction orthogonal to the direction in which the first electrode finger, the second electrode finger, and the third electrode finger extend is set as the electrode finger orthogonal direction, an area where adjacent first electrode fingers and third electrode fingers overlap with each other in the electrode finger orthogonal direction and an area between the centers of adjacent first electrode fingers and third electrode fingers, and an area where adjacent second electrode fingers and third electrode fingers overlap with each other in the electrode finger orthogonal direction and an area between the centers of adjacent second electrode fingers and third electrode fingers are excitation areas, When the metallization ratio of the first electrode finger and the third electrode finger and the second electrode finger and the third electrode finger with respect to the excitation region is MR, MR≦1.75(d / p)+0.075 is satisfied.

13. The elastic wave device according to any one of claims 1 to 12, wherein: The piezoelectric layer includes lithium tantalate or lithium niobate.

14. The elastic wave device according to claim 13, wherein: The Euler angles (φ, θ, ψ) of lithium niobate or lithium tantalate constituting the piezoelectric layer are within the range of the following formula (1), formula (2) or formula (3), (within the range of 0°±10°, 0°~25°, arbitrary ψ) …Formula (1) (Within the range of 0°±10°, 25°~100°, 0°~75°[(1-(θ-50) 2 / 2500)] 1 / 2 or 180°-75°[(1-(θ-50) 2 / 2500)] 1 / 2 ~180°) …Formula (2) (0°±10°, 180°-40°[(1-(ψ-90) 2 / 8100)] 1 / 2 ~180°, arbitrary ψ) …Equation (3).

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