Two-port acoustic wave sensor device

Abstract

An acoustic wave sensor device comprises a quartz material layer surface; arranged along a first axis, a first interdigitated transducer disposed over the planar surface of the quartz material layer, a first reflection structure disposed over the planar surface of the quartz material layer, and a second reflection structure disposed over the planar surface of the quartz material layer; and arranged along a second axis, a second interdigitated transducer disposed over the planar surface of the quartz material layer, a third reflection structure disposed over the planar surface of the quartz material layer, and a fourth reflection structure disposed over the planar surface of the quartz material layer; and wherein the first axis and the second axis are inclined to each other by a finite angle.

Claims

1. An acoustic wave sensor device, comprising: a quartz material layer comprising a planar surface; arranged along a first axis, a first interdigitated transducer disposed over the planar surface of the quartz material layer, a first reflection structure disposed over the planar surface of the quartz material layer, and a second reflection structure disposed over the planar surface of the quartz material layer; and arranged along a second axis, a second interdigitated transducer disposed over the planar surface of the quartz material layer, a third reflection structure disposed over the planar surface of the quartz material layer, and a fourth reflection structure disposed over the planar surface of the quartz material layer; and wherein: the first axis and the second axis are inclined to each other by a finite angle; and the planar surface of the quartz material layer is defined by a crystal cut of a quartz material of the quartz material layer with angles in the range of 14 to 24, in the range of 25 to 45 and in the range of +8 to +28.

2. The acoustic wave sensor device of claim 1, wherein the quartz material layer is a bulk substrate.

3. The acoustic wave sensor device of claim 2, further comprising a bulk substrate, and wherein the quartz material layer is disposed over the bulk substrate.

4. The acoustic wave sensor device of claim 3, wherein the bulk substrate comprises a silicon or sapphire bulk substrate.

5. The acoustic wave sensor device of claim 1, wherein at least one of the first, second, third and fourth reflection structures comprises or consists of a Bragg mirror.

6. The acoustic wave sensor device of claim 5, wherein the first reflection structure consists of a first Bragg mirror, the second reflection structure consists of a second Bragg mirror, the third reflection structure consists of a third Bragg mirror and the fourth reflection structure consists of a fourth Bragg mirror, and wherein all of the first, second, third and fourth Bragg mirrors have the same number and/or lengths of electrodes.

7. The acoustic wave sensor device of claim 5, wherein the first reflection structure consists of a first Bragg mirror, the second reflection structure consists of a second Bragg mirror, the third reflection structure consists of a third Bragg mirror and the fourth reflection structure consists of a fourth Bragg mirror and electrodes of each of the first, second, third and fourth Bragg mirrors are, respectively, a) connected to each other; or b) grounded; or c) neither connected to each other nor grounded.

8. The acoustic wave sensor device of claim 1, wherein the first and second reflection structures are located adjacent to the first interdigitated transducer and the third and fourth reflections structures are located adjacent to the second interdigitated transducer.

9. The acoustic wave sensor device of claim 1, wherein: a first resonance cavity is disposed between the first interdigitated transducer and the first reflection structure and a second resonance cavity is disposed between the first interdigitated transducer and the second reflection structure; and a third resonance cavity is disposed between the second interdigitated transducer and the third reflection structure and a fourth resonance cavity is disposed between the second interdigitated transducer and the fourth reflection structure.

10. The acoustic wave sensor device of claim 9, wherein: an upper surface of the second resonance cavity comprises a physical and/or chemical modification as compared to an upper surface of the first resonance cavity; and/or an upper surface of the fourth resonance cavity comprises a physical and/or chemical modification as compared to an upper surface of the third resonance cavity.

11. The acoustic wave sensor device of claim 10, wherein the physical and/or chemical modification comprises a metallization layer or passivation layer disposed on the upper surface of the second resonance cavity and/or fourth resonance cavity.

12. The acoustic wave sensor device of claim 9, wherein extension lengths of the first resonance cavity and the second resonance cavity differ from each other.

13. The acoustic wave sensor device of claim 1, wherein the first and second interdigitated transducers are connected in series or in parallel to each other.

14. The acoustic wave sensor device of claim 1, wherein the acoustic wave sensor device is a passive surface acoustic wave sensor device configured for sensing an ambient parameter selected from one of a temperature, chemical species, strain, pressure, torque of a rotating axis, and acceleration or frequency vibration of a vibrating part.

15. The acoustic wave sensor device of claim 1, wherein q is in the range of 17 to 22, is in the range of 30 to 40 and is in the range of +10 to +25.

16. The acoustic wave sensor device of claim 15, wherein q is in the range of 19 to 21, is in the range of 33 to 39 and is in the range of +15 to +25.

17. The acoustic wave sensor device of claim 1, wherein the first and second interdigitated transducers have a same number of electrodes and/or a same metallization ratio and/or a same aperture and/or a same tapering and/or lengths of electrodes.

Description

BRIEF DESCRIPTION OF THE DRAWINGS

(1) Additional features and advantages of the present disclosure will be described with reference to the drawings. In the description, reference is made to the accompanying figures that are meant to illustrate preferred embodiments of the present disclosure. It is understood that such embodiments do not represent the full scope of the present disclosure.

(2) FIG. 1 represents an example of a surface acoustic wave sensor device according to the art.

(3) FIG. 2 represents a principle sketch illustrating a two-port acoustic wave sensor device according to an embodiment of the present disclosure.

(4) FIG. 3 represents a principle sketch illustrating a two-port acoustic wave sensor device according to an embodiment of the present disclosure.

(5) FIG. 4 represents a principle sketch illustrating a two-port acoustic wave sensor device according to an embodiment of the present disclosure.

(6) FIG. 5 represents a principle sketch illustrating a two-port acoustic wave sensor device according to an embodiment of the present disclosure.

(7) FIG. 6 represents a principle sketch illustrating a two-port acoustic wave sensor device comprising resonance cavities formed between transducers and mirrors according to an embodiment of the present disclosure.

(8) FIG. 7 represents a principle sketch illustrating a two-port acoustic wave sensor device comprising resonance cavities formed between transducers and mirrors according to an embodiment of the present disclosure.

(9) FIG. 8 represents a principle sketch illustrating a two-port acoustic wave sensor device comprising resonance cavities formed between transducers and mirrors according to an embodiment of the present disclosure.

(10) FIG. 9 represents a principle sketch illustrating a two-port acoustic wave sensor device comprising resonance cavities formed between transducers and mirrors according to an embodiment of the present disclosure.

(11) FIG. 10 represents a sketch illustrating coordinates and angles of a piezoelectric plate.

(12) FIG. 11 represents a sketch illustrating coordinates and angles of a triple-rotated crystal cut.

DETAILED DESCRIPTION

(13) The present disclosure provides acoustic wave sensors, in particular, passive SAW sensors, that are characterized by a high signal-to-noise ratio, sensitivity and reliability, in particular, robustness against environmental influences and residual stresses not resulting from variations of the measurand, and high accuracy of differential measurements. These advantages are, particularly, achieved by using a piezoelectric quartz material layer providing resonance cavities that is characterized by a plane surface resulting from a crystal cut defined at angles in the range of 14 to 24, in the range of 25 to 45 and in the range of +8 to +28, in particular, in the range of 17 to 22, in the range of 30 to 40 and in the range of +10 to +25, and more particularly, in the range of 19 to 21, in the range of 33 to 39 and in the range of +150 to +250 according to the IEEE 176 1949 Standards on Piezoelectric Crystals, 1949 from Dec. 12, 1949.

(14) With respect to temperature measurements, for example, the obtainable resonance frequency sensitivity allows for a differential measurement sensitivity of more than 1 ppm per Kelvin together with second order TCF sensitivity (absolute value) smaller than 10 ppb.Math.K.sup.2 and even less than 5 ppb.Math.K.sup.2, thus guarantying a quasi-linear frequency-temperature variation over extended temperature ranges (typically 100 K ranges). The acoustic wave sensors can be interrogated by any interrogators that are configured to determine a response spectrum from an interrogated acoustic wave sensor. The interrogated acoustic wave sensor can, for example, be a resonator device, for example, a differential SAW sensor. It goes without saying that the present disclosure can be implemented in any devices employing acoustic wave sensors or dielectric resonators, RLC circuits, etc.

(15) The interrogation device (also called unit) interrogating one of the inventive acoustic wave sensor devices may comprise a transmission antenna for transmitting a RF interrogation signal to the sensor device and a reception antenna for receiving a RF response signal from the sensor device. The RF interrogation signal transmitted by the transmission antenna may be generated by a signal generator that may comprise a RF synthesizer or controlled oscillator as well as optionally some signal shaping module providing a suitable frequency transposition and/or amplification of the signal to be transmitted by the transmission antenna. The RF interrogation signal generated by the signal generator may be a pulsed or burst signal with a frequency selected according to the resonance frequency of the acoustic wave sensor device. It is noted that the emission antenna and the reception antenna may be the same antenna. In this case, the emission and reception processes should be synchronized with each other, for example, by means of a suitably controlled switch.

(16) Furthermore, the interrogation device may comprise a processing means connected to the reception antenna. The processing means may comprise filtering and/or amplification means and be configured for analyzing the RF response signal received by the reception antenna. For example, the sensor device operates at a resonance frequency of 434 MHz or 866 MHz or 915 MHz or 2.45 GHz (the ISM bands).

(17) The interrogation device may transmit a long RF pulse and after the transmission has been stopped, the resonance cavities of the sensor device discharge at their resonant eigenfrequencies with time constants T equal to Q.sub.f/F wherein F is the central frequency and Q.sub.f is the quality factor of the resonance, Q.sub.f corresponding to the ratio between the resonance central frequency and the width at half maximum of the band pass used in the interrogation process. For instance, Q.sub.f corresponds to the resonance quality factor estimated on the real part of the resonator admittance (the conductance) when the latter is designed to operate at the resonance. Spectral analysis performed by the processing means of the interrogation device allows for calculating the resonator frequency/frequencies and, thereby, the sensing of an ambient parameter. The received RF response signal may be mixed by the processing means with RF interrogation signal according to the so-called I-Q protocol as known in the art to extract the real and imaginary parts (in-phase components I=Y cos and quadrature components Q=Y sin with the signal amplitude Y and the signal phase ) from which the modulus and phase can then be derived.

(18) FIGS. 2 to 5 illustrate exemplary embodiments of an inventive surface acoustic wave (SAW) sensor device. The sensor devices 20, 30, 40 and 50 shown in FIGS. 2 to 5 are formed using a quartz material layer Q as a piezoelectric layer. Either the quartz material layer Q is a quartz bulk substrate or a quartz layer formed over some non-piezoelectric bulk substrate, for example, a Si substrate. The quartz material layer Q may be part of a piezo-electric-on-insulator (POI) substrate. The quartz layer may be bonded to the non-piezoelectric bulk substrate by means of a (dielectric) bonding layer, as, for instance, a silicon oxide layer. A so-called trap-rich layer (e.g., polycrystalline silicon) can be present at the interface with the non-piezoelectric bulk substrate.

(19) The quartz material layer Q comprises an upper operating planar surface and the planar surface of the quartz material layer Q is defined by a crystal cut of a quartz material of the quartz material layer with angles in the range of 14 to 24, in the range of 25 to 45 and in the range of +8 to +28, the angles being defined in accordance with the IEEE 176 1949 Standards on Piezoelectric Crystals, 1949 from Dec. 12, 1949. It is this particular cut family that at least partially provides the advantages mentioned and described above.

(20) A SAW sensor device according to an embodiment of the present disclosure, for example, the SAW sensor device 20, 30, 40 and 50 shown in FIGS. 2 to 5, comprises a first resonator R1 comprising a first interdigitated (comb) transducer T1 and a first Bragg mirror M1 and a second Bragg mirror M2 and a second resonator R2 comprising a second interdigitated (comb) transducer T2 and a third Bragg mirror M3 and a fourth Bragg mirror M4. The transducers T1 and T2 may be connected to an antenna (not shown in FIG. 2) for receiving an electromagnetic wave and converting the electromagnetic wave into a surface acoustic wave that, after reflection by the mirrors, is sensed again and converted back into an RF signal that in course in transmitted by the antenna (or another antenna) as an RF response signal to a reader.

(21) The first and second resonators R1 and R2 may be made similar to each other. Particularly, the first and second transducers T1 and T2 may have the same number of electrodes and/or the same metallization ratio and/or the same aperture and/or the same tapering and/or lengths of electrodes. The first and second transducers T1 and T2 may have different periods (separation distances between individual electrodes) in order to achieve significant differences in the resonance frequencies of the first and second resonators R1 and R2.

(22) According to the present disclosure, the first and second resonators R1 and R2 are tilted with respect to each other, i.e., the first interdigitated transducer T1 formed over the planar surface of the quartz material layer Q, the first Bragg mirror M1 formed over the planar surface of the quartz material layer Q, and the second Bragg mirror M2 formed over the planar surface of the quartz material layer Q are arranged along a first axis and the second interdigitated transducer T2 formed over the planar surface of the quartz material layer Q, the third Bragg mirror M3 formed over the planar surface of the quartz material layer Q and the fourth Bragg mirror M4 formed over the planar surface of the quartz material layer Q are arranged along a second axis and the first axis and the second axis are inclined to each other by a finite angle. The first axis corresponds to an axis at an angle 1, the angle 1 defining the direction of propagation of the acoustic wave along the axis X being defined by rotation by an angle 1 of the axis X of the quartz substrate. The second axis corresponds to an axis at an angle 2, the angle 2 defining the direction of propagation of the acoustic wave along the axis X being defined by rotation by an angle 2 of the axis X of the quartz substrate. The angle 1 and 2 are comprised in the range of +8 to +28 of the angle defined for the quartz substrate. The finite angle between the first and second axes may be in the range of 1 to 10, in particular, 1 to 6, more particularly, 2 to 4, and results in different propagation directions of the generated (surface) acoustic waves that in course result in different resonance frequencies of the first and second resonators R1 and R2.

(23) The surface acoustic wave sensor devices 20, 30, 40 and 50 shown in FIGS. 2 to 5 may operate at Bragg conditions with wavelengths of the excited surface acoustic waves of some multiples of the pitches of the comb electrodes of the comb transducers T1 and T2. When operation is performed at Bragg conditions the comb transducers themselves substantially functions as mirrors. However, in operation situations in that the reflection coefficient of the transducers T1 and T2 is not strong enough to allow for a clear enough separation between the individual resonances it is advantageous to split each of the transducers T1 and T2 into two parts and arrange an additional mirror between, respectively. Improvement of the cavity resonance separation by means of the split transducer and the additional mirror is particularly useful for operating with Rayleigh or more generally elliptically polarized waves.

(24) It is noted that the electrodes of the first and second transducers T1 and T2 may be made of or comprise AlCu. The use of materials with relatively high atomic numbers like, for instance, molybdenum or gold or platinum or tungsten may allow for larger reflection coefficients. It is, furthermore, noted that the configurations of the acoustic wave sensor devices 20, 30, 40 and 50 shown in FIGS. 2 to 5 may comprise tapered transducers T1 and T2 with lateral extensions of the electrodes varying along the lengths of the transducers T1 and T2 to suppress transverse wave modes. Further, supplementary mass loads may be provided at the edges of the electrodes in order to suppress transverse wave modes.

(25) According to different embodiments the resonators R1 and R2 (transducers T1 and T2) can be connected in series or parallel to each other. In the embodiment shown in FIG. 2, the resonators R1 and R2 of the acoustic wave sensor device 20 are connected parallel to each other and the electrodes of each of the mirrors M1, M2, M3 and M4 are connected with each other (short-circuited). In the embodiment shown in FIG. 3, the resonators R1 and R2 of the acoustic wave sensor device 30 are connected parallel to each other and the mirrors M1, M2, M3 and M4 are connected to ground. In the embodiment shown in FIG. 4, the resonators R1 and R2 of the acoustic wave sensor device 40 are connected parallel to each other and the electrodes of each of the mirrors M1, M2, M3 and M4 are not connected with each other. In the embodiment shown in FIG. 5, the resonators R1 and R2 of the acoustic wave sensor device 50 are connected in series to each other and the electrodes of each of the mirrors M1, M2, M3 and M4 are connected with each other. In FIGS. 2 to 5, an electromagnetic wave received by the transducers T1 and T2 (i.e., an interrogation signal) for the generation of acoustic waves is indicated by E1 and a back-converted acoustic wave signal (i.e., a response signal) is indicated by S1.

(26) The configurations shown in FIGS. 2 to 5 allow, for example, for accurately sensing temperatures based on variations of the resonance frequencies of the resonators R1 and R2 and, thus, the differences in resonance frequencies, depending on the actual temperature.

(27) In the embodiments shown in FIGS. 2 to 5 the mirrors M1, M2, M3 and M4 are arranged adjacent to the transducers T1 and T2. According to other embodiments resonance cavities are formed between the first transducer T1 and the first and second mirrors M1 and M2, respectively, and/or resonance cavities are formed between the second transducer T2 and the third and fourth mirrors M3 and M4, respectively.

(28) FIG. 6 shows an exemplary embodiment similar to the one shown in FIG. 2. Different from the configuration shown in FIG. 2, in the embodiment shown in FIG. 6 an acoustic wave sensor device 60 comprises a first resonance cavity with length g1 formed between the first transducer T1 and the first mirror M1 and a second resonance cavity with length g2 formed between the first transducer T1 and the second mirror M2. Similarly, a third resonance cavity with length g3 is formed between the second transducer T2 and the third mirror M3 and a fourth resonance cavity with length g4 is formed between the second transducer T2 and the fourth mirror M4. Alternatively, the first resonator R1 or the second resonator R2 only has the resonance cavities with lengths g1 and g2 or g3 and g4, respectively.

(29) In principle, the upper surface of the resonance cavity between the first transducer T1 and the first mirror M1 may comprise a physical and/or chemical modification as compared to the upper surface of the resonance cavity between the first transducer T1 and the second mirror or the other way round. Similarly, the upper surface of the resonance cavity between the second transducer T2 and the third mirror M3 may comprise a physical and/or chemical modification as compared to the upper surface of the resonance cavity between the second transducer T2 and the fourth mirror M4 or the other way round. All or some of the cavities with lengths g1, g2, g3 and g4 may differ from each other with respect to physical and/or chemical modifications and/or the extension lengths g1, g2, g3 and g4.

(30) There is a variety of means for providing the physical and/or chemical modifications in order to achieve propagating wave modes that exhibit differential parametric sensitivities. These means, for example, include realization of the physical and/or chemical modifications by the formation of a metallization layer and/or passivation layer and/or local doping. A metallization layer of some 100 nm thickness may be formed on the region of the resonance cavity of length g1, for example; no metallization layer may be formed on the resonance cavity of length g2. The metallization layer may be formed of the same material as the electrodes of the transducers T1 and T2 and/or the Bragg mirrors M1, M2, M3 and M4.

(31) When the same material is used for the metallization and the formation of the comb transducers T1 and T3 and electrodes of the Bragg mirror structures M1, M2, M3 and M4, all of these elements can be deposited in the same deposition process. In other embodiments, a different material is used for the metallization. For example, one metallization layer or passivation layer of one material is formed on a first resonance cavity and another metallization layer or passivation layer of another material is formed on a second resonance cavity of one or each of the resonators R1 and R2. According to another example, a positive-temperature shifting material, for example, SiO.sub.2 or Ta.sub.2O.sub.5, is formed on one of the resonance cavities and a negative-temperature shifting material, for example, Si.sub.3N.sub.4 or AlN, or no additional material is formed on the other one of the resonance cavities of one or each of the resonators R1 and R2.

(32) Passivation may be realized by forming a passivation layer made of or comprising Si.sub.3N.sub.4, Al.sub.2O.sub.3 or AlN. According to other embodiments, material layers can be formed on both resonance cavities. Moreover, material layers formed on one or more of the resonance cavities may have inhomogeneous thicknesses along the direction of propagation of the acoustic waves. Further, multi-layers may be formed on one or more of the resonance cavities. In this context, it should be noted that, in general, provision of a material layer on a resonance cavity may result in a reduction of the phase velocity of acoustic waves due to mass loading effects, particularly, if layers of a material of a high atomic number, as Pt, Au or W, are used. This effect can be compensated by adding a layer exhibiting a relatively high acoustic velocity, for example, AlN, Si.sub.3N.sub.4, Al.sub.2O.sub.3, adjacent to the quartz material layer. The resonance cavities exhibit different sensitivities to measurands due to the provided different resonance characteristics caused by different treatments of the surfaces and, thus, allow for differential measurements.

(33) Alternatively or additionally, the physical and/or chemical modification may comprise a recess of the surface of one of the resonance cavities with respect to the surface of the other of the resonance cavities of one or each of the resonators R1 and R2.

(34) The configurations shown in FIGS. 3 to 5 may, according to alternative embodiments, also show resonance cavities in the first and/or second resonators as described above. For example, FIG. 7 shows an acoustic wave sensor device 70 similar to the acoustic wave sensor device 30 shown in FIG. 3 but with the first and second resonators R1 and R2 comprising resonance cavities, FIG. 8 shows an acoustic wave sensor device 80 similar to the acoustic wave sensor device 40 shown in FIG. 4 but with the first and second resonators R1 and R2 comprising resonance cavities, and FIG. 9 shows an acoustic wave sensor device 90 similar to the acoustic wave sensor device 50 shown in FIG. 5 but with the first and second resonators R1 and R2 comprising resonance cavities.

(35) In the above-described embodiment shown in FIGS. 2 to 5, Bragg mirrors M1, M2, M3 and M4 are provided as reflection structures. However, according to alternative embodiments one or more of the Bragg mirrors may be replaced by side/edge reflection structures for pure shear mode guidance. Thereby, very compact configurations can be achieved in that the Bragg reflection is replaced by a flat surface reflection without any energy loss or mode conversion. Configurations with side/edge reflection structures for pure shear mode guidance are particularly useful for sensing ambient parameters in liquids. Shear waves are very suitable for in-liquid probing. Particularly, highly coupled modes (>5%) together with high-k materials (with a dielectric constant k larger than 30, for example) are attractive for in-liquid applications. According to other embodiments, one or more reflection structures are realized in the form of short reflectors comprising not more than three electrodes.

(36) It is, furthermore, noted that simple resonance cavities may be replaced by cascaded resonance cavities comprising multiple mirror electrode structures. The spectral distance between the two resonances as well as the coupling coefficient of the resonances can be controlled by the number of the mirror electrode structures and resonance sub-cavities.

(37) All previously discussed embodiments are not intended as limitations but serve as examples illustrating features and advantages of the present disclosure. It is to be understood that some or all of the above described features can also be combined in different ways.

(38) In the present disclosure, crystal cuts are defined in accordance with the IEEE 176 1949 Standards on Piezoelectric Crystals, 1949 from Dec. 12, 1949. In that standard, a crystal cut for SAW applications is uniquely defined by three angles, namely and defining the rotation of the crystal according a reference configuration of the crystal and a propagation direction defined in the plane (, ) that indicates the direction toward which the waves are propagating and hence the position of the transducer capable to launch the waves. Y and X denote crystalline axes considered as references for the definition of the initial state of the crystal plate. The first one is the axis normal to the plate whereas the second axis lies along the length of the plate. The plate is assumed to be rectangular, characterized by its length 1, its width w and its thickness t (see FIG. 10). The length/is lying along the crystalline axis X, the width w is along the Z axis and the thickness t along the Y axis considering the given (YX) axis system. Note that the case of (YXwlt)/0/0/0 actually coincides to the configuration shown in FIG. 10.

(39) Assuming now that none of the angles is zero, the general case of a triple-rotation or triply-rotated cut is considered. In that situation, the quartz crystal has a cutting plane (X, Z) defined with respect to the cutting plane (X, Z) and in a reference system (X, Y, Z), where X, Y, Z are crystallographic axes of quartz, a direction of propagation of the waves being defined along an axis X, a first cutting plane (X, Z) being defined by rotation by an angle Y about the axis Z of the plane (X, Z) so as to define a first reference system (X, Y, Z) with an axis Z that is the same as the axis Z, a second cutting plane (X, Z) being defined by rotation by an angle about the axis X of the plane (X, Z) so as to define a second reference system (X, Y, Z) with the axis X being the same as the axis X, the direction of propagation along the axis X being defined by rotation by an angle of the axis X, in the plane (X, Z) about the axis Y, as shown in FIG. 11.

(40) Some symmetry rules are recalled hereafter for quartz. Quartz is a trigonal crystal of class 32. Therefore, it is characterized by a ternary axis, i.e., the Z axis around which one can establish the relation:
(YXw)/=(YXw)/+120

(41) The two other axes are binary and therefore the following symmetry relations hold:
(YXl)/=(YXl)/+180, (YXt)/=(YXt)/+180

(42) For simple geometrical reasons, it is easy to demonstrate that the following set of axes are equivalent:
(YXwlt)/+/+/+=(YXwlt)//+/

(43) Actually, assuming that the upper face is identified by the plus sign for Y (the face where the surface wave is assumed to propagate), the bottom face of the plate is obtained by changing the sign to minus. Considering that the symmetry operation does not change the sign of one would assume that the direction of Z on the bottom side is unchanged but actually it is rotated by 180. Therefore, to recover the top surface situation, it is mandatory to apply a 180 rotation on y, which actually is equivalent to a sign change. Note that for crystal cuts without rotation around Z (=0), the following symmetry is effective: (YXlt)/+/+=(YXlt)/+/.