System And Method For Extraction Of Piezoelectric Constants Electrically

Abstract

Activity of piezoelectric material dimension and electrical properties can be changed with an applied stress. These variations are translated to a change in capacitance of the structure. Use of capacitance-voltage measurements for the extraction of double piezoelectric thin film material deposited at the two faces of a flexible steel sheet is described. Piezoelectric thin film materials are deposited using RF sputtering techniques. Gamry analyzer references 3000 is used to collect the capacitance-voltage measurements from both layers. A developed algorithm extracts directly the piezoelectric coefficients knowing film thickness, applied voltage, and capacitance ratio. The capacitance ratio is the ratio between the capacitances of the film when the applied field in antiparallel and parallel to the poling field direction, respectively. Piezoelectric bulk ceramic is used for calibration and validation by comparing the result with the reported values from literature. Extracted values using the current approach match well values extracted by existing methods.

Claims

1. A method for determining piezoelectric parameters of a piezoelectric structure, comprising: applying a biasing electrical field E over the piezoelectric structure, where the biasing electrical field may have one or more selected directions in relation to a poling direction P of said piezoelectric structure; and determining the longitudinal piezoelectric voltage constant d.sub.33 and transversal piezoelectric voltage constant d.sub.31 of said piezoelectric structure in relation to a selected parameter ratio.

2. The method for determining piezoelectric parameters of a piezoelectric structure in accordance with claim 1, wherein: the predetermined parameter ratio is a capacitance ratio C.sub.r=C.sub.↓↑/C.sub.↑↑ based on the quota between a first capacitance C.sub.↓↑ of the piezoelectric structure due to an applied biasing electrical field with a direction antiparallel to said poling direction and a second capacitance C.sub.↑↑ of the piezoelectric structure due to an applied biasing electrical field with a direction parallel with said poling direction.

3. The method for determining piezoelectric parameters of a piezoelectric structure in accordance with claim 2, wherein: the relation between the piezoelectric voltage constants d.sub.33 and d.sub.31 is determined based on the relation according to the following equation (8) C.sub.r−2C.sub.rd.sub.33E=1+4d.sub.31E+2(d.sub.31E).sup.2.

4. The method for determining piezoelectric parameters of a piezoelectric structure in accordance with claim 3, wherein: the piezoelectric voltage constant d.sub.31 is determined based on the relation according to the following equation (11) d.sub.31=(−(C.sub.r+1)+√{square root over (C.sub.r.sup.2+2.5C.sub.r+0.5))}E.sup.−1.

5. The method for determining piezoelectric parameters of a piezoelectric structure in accordance with claim 4, further comprising: applying a said biasing electric field E with a selection of voltage values with negative and positive polarities over the piezoelectric structure; measuring capacitance values for said voltage values; calculating the electric field E and the capacitance ratio Cr based on said capacitance values, said voltage values and a value for the thickness T of a piezoelectric layer of the piezoelectric structure.

6. A method for determining piezoelectric parameters of a piezoelectric structure in accordance with claim 2, further comprising calibrating the relation between the piezoelectric voltage constant d.sub.31 and the capacitance ratio C.sub.r by: determining, for an unclamped piezoelectric material used in the piezoelectric structure, a first value for the piezoelectric voltage constant d.sub.31 with a first method based on the relation according to the following equation (11) d.sub.31=(−(C.sub.r+1)+√{square root over (C.sub.r.sup.2+2.5C.sub.r+0.5)})E.sup.−1 and a second value for the piezoelectric voltage constant d.sub.31 with a second method, for example based on measurement by means of a Berlincourt meter; determining a correction factor Corr for the relation between the piezoelectric voltage constant d.sub.31 and the capacitance ratio Cr based on the quota between said second value and said first value for the piezoelectric voltage constant d.sub.31; applying said correction factor such that piezoelectric voltage constant d.sub.31 is determined based on the relation according to the following equation (12B) d.sub.31=Corr(−(C.sub.r(+1)+√{square root over (C.sub.r.sup.2+2.5C.sub.r+0.5))}E.sup.−1.

7. The method for determining piezoelectric parameters of a piezoelectric structure in accordance with claim 1, wherein: the piezoelectric voltage constant d.sub.33 is determined as d.sub.33=xd.sub.31, where an approximation factor x may assume values between 1 and 3, preferably x=2 such that d.sub.33=2d.sub.31.

8. The method for determining piezoelectric parameters of a piezoelectric structure in accordance with claim 2, comprising: measuring Cr values for a selection of said applied voltage values; determining the piezoelectric voltage constants d.sub.33 and d.sub.31 based on the relation with the capacitance ratio Cr as a function of applied voltage, wherein the applied voltage is expressed in terms of electric field E, according to the following equation (13) C.sub.r=(1+4d.sub.31E+2(d.sub.31E).sup.2)(1−2d.sub.33E).sup.−1 and by fitting the measured values for Cr for specific voltage values using a fitting method, preferably quadratic fitting.

9. The method for determining piezoelectric parameters of a piezoelectric structure in accordance with claim 1, comprising: incorporating the piezoelectric structure in a resonator structure having a measurable resonance frequency fr and an effective inductor L; driving the resonator against and along polarization/poling directions; measuring the resonance frequency fr for a selection of values for the applied electric field E; determining the piezoelectric voltage constants d.sub.33 and d.sub.31 based on the relation with a normalized frequency ratio frn as a function of applied voltage, wherein the applied voltage is expressed in terms of electric field E, according to the following equation (24) f.sub.rn=1+(2d.sub.31+d.sub.33)E+(2d.sub.31d.sub.33+d.sub.31.sup.2)E.sup.2+d.sub.31.sup.2d.sub.33E.sup.3 and by fitting values for the normalized frequency ratio frn for specific values for the applied electric field E using a fitting method, preferably cubic equation fitting.

10. The method for determining piezoelectric parameters of a piezoelectric structure in accordance with claim 1, wherein: the piezoelectric structure comprises a first piezoelectric layer constituting a first outer electrode and a second piezoelectric layer constituting a second outer electrode on a metallic sheet (Shim) constituting a common electrode sandwiched between said first and second electrodes.

11. A method of manufacturing a piezoelectric structure, comprising: providing a first piezoelectric layer constituting a first outer electrode and a second piezoelectric layer constituting a second outer electrode on a metallic sheet (Shim) constituting a common electrode sandwiched between said first and second electrodes; poling said first and second piezoelectric layers such that each of said piezoelectric layers has a defined poling direction.

12. The method of manufacturing a piezoelectric structure in accordance with claim 11, further comprising: depositing the first and second piezoelectric layers by a selection of sputtering or spin coating.

13. The method of manufacturing a piezoelectric structure in accordance with claim 11, further comprising: coating the metallic sheet with noble materials PLT/Pt/Ti as a seeding layer for the deposition of the piezoelectric thin film materials layers on both faces of the metallic sheet.

14. The method of manufacturing a piezoelectric structure in accordance with claim 11, wherein the metallic sheet is a steel sheet.

15. The method of manufacturing a piezoelectric structure in accordance with claim 11, wherein the poling directions of said first and second piezoelectric layers have the same direction or opposite directions.

16. A piezoelectric structure, comprising: a first piezoelectric layer constituting a first outer electrode; a second piezoelectric layer constituting a second outer electrode; and a metallic sheet (Shim) constituting a common electrode sandwiched between said first and second electrodes.

17. The piezoelectric structure in accordance with claim 16, further comprising: a coating on the metallic sheet with noble materials PLT/Pt/Ti forming a seeding layer for the piezoelectric thin film materials layers deposited on both faces of the metallic sheet.

18. The piezoelectric structure in accordance with claim 16, wherein the metallic sheet is a steel sheet.

Description

BRIEF DESCRIPTION OF THE DRAWINGS

[0024] Embodiments disclosed herein will be further explained with reference to the accompanying drawings, in which:

[0025] FIGS. 1A, 1B, and 1C show illustrations of the response of a piezoelectric material in the shape of a disc subjected to an applied electrical field biasing the disk according to different biasing schemes, where FIG. 1A shows an unbiased disk, FIG. 1B shows a disk subjected to an anti-parallel biasing scheme, and FIG. 1C shows a disc subjected to a parallel biasing scheme.

[0026] FIG. 2 shows an illustration of double piezoelectric layers (PZE1 and PZE2) sandwiching a metallic sheet (Shim) with poling directions (↓P), in accordance with embodiments.

[0027] FIG. 3 shows XRD measurements of the fabricated PZT layer to assess the quality of embodiments, for example such embodiments as described in connection with FIG. 2.

[0028] FIG. 4 shows measured CV of a bulk sample, in accordance with embodiments.

[0029] FIG. 5 shows capacitance versus frequency measurements for unbiased (at zero volts), anti (at +3 volt) and parallel biasing (at −3 volt), in accordance with embodiments.

[0030] FIG. 6 shows simultaneous CV measurements of the two piezoelectric layers, in accordance with embodiments.

[0031] FIG. 7 shows extracted piezoelectric voltage constants in accordance with embodiments using equation (12).

[0032] FIG. 8 shows polarization curve for a piezoelectric structure with films of piezoelectric material deposited on steel sheet, in accordance with embodiments.

[0033] FIGS. 9A and 9B show extracted normalized capacitance ratio (Cr) values versus normalized applied electric field (E.sub.n) voltage along with their corresponding fitting curves, in accordance with embodiments using bulk sample used for calibration (FIG. 9A) and thin film structure under study (FIG. 9B).

DETAILED DESCRIPTION

[0034] This disclosure describes in different embodiments the use of CV characteristics to extract piezoelectric parameters such as the piezoelectric voltage constants utilizing the change in capacitance. Embodiments of a structure comprising two piezoelectric layers and analysis of this structure using embodiments of a developed method are disclosed. The following sections illustrate embodiments of an approach of characterizing the piezoelectric material or structure, the properties of the prepared sample, and of a calibration technique to optimize the characterization algorithm.

Capacitive-Voltage Approach

[0035] When a piezoelectric material that is sandwiched between two electrodes is subjected to either mechanical or electrical strains, its geometrical dimensions and dielectric constant will change according to the direction and magnitude of an applied field of mechanical or electric force. FIGS. 1A, 1B, and 1C show illustrations of the response of a piezoelectric material in the shape of a disk subjected to an applied electrical field biasing the disk according to different biasing schemes, where FIG. 1A shows an unbiased disk, FIG. 1B shows a disk subjected to an anti-parallel biasing scheme, and FIG. 1C shows a disc subjected to a parallel biasing schemes.

[0036] FIG. 1A thus illustrates a circular disk 100 of a piezoelectric material without applying any type of field exerting a force on the disk, for example neither an electrical nor a mechanical field. The circular disk 100 is shown to have a thickness T and surface area A. FIG. 1B illustrates the situation when disk 100 is driven by or subjected to an applied electrical field (not shown), with a direction opposite to the poling direction, the thickness T decreases and the area A increases simultaneously by an amount ΔT and ΔA, respectively. This results in an increase in capacitance under such condition. On the other hand, as is illustrated in FIG. 1C, if the applied electric field (not shown) is aligned with the poling direction, the thickness T is shown to increase and the area A is shown to decrease simultaneously by an amount ΔT and ΔA, respectively. This results in the decrease in capacitance under such conditions.

[0037] Mathematically, equation (1) expresses the capacitance of a parallel plate as follow:

C=εA/T  (1)

[0038] In equation (1) ε, A and T are respectively the dielectric constant, area and thickness of a piezoelectric layer sandwiched between a common and an outer electrode. The application of a DC field opposite to the poled field will result in the contraction of the layer thickness T and expansion in the area A. Hence, the capacitance is expressed as per equation (2):

C.sub.↓↑=E(A+ΔA)(T−ΔT).sup.−1  (2)

[0039] In equation (2) C.sub.↓↑ signifies the capacitance when the electric field is applied opposite to the poled field and where ΔA and ΔT are the variations in area and thickness, respectively. Meanwhile, the application of a DC field parallel to the poled field will result in the contraction of the layer area A and expansion in the thickness T. Hence, the capacitance is expressed as per equation (3):

C.sub.↑↑=ε(A−ΔA)(T+ΔT).sup.−1  (3)

[0040] In equation (3) C.sub.↑↑ signifies the capacitance when the electric field is applied parallel to the poled field. Dividing (2) over (3), yields:

C.sub.r(T−ΔT)(T+ΔT).sup.−1=(A+ΔA)(A−ΔA).sup.−1  (4)

[0041] In equation (4) C.sub.r=C.sub.↓↑/C.sub.↑↑. Equation (4) connects the change in capacitance ratio C.sub.r with the change in dimensions due to the piezoelectric effect. Equation (4) may be simplified using the Taylor approximation (1+x).sup.n=1+nx, where x<<1. Therefore, when ΔT/T<<1 and when ΔA/A<<1, applying the Taylor approximation to equation (4) yields:

C.sub.r(1−2ΔT/T)=(1+2ΔA/A)  (5)

[0042] Equation (5) correlates changes in capacitance ratio C.sub.r to both changes in thickness T and area A, assuming T or A are not zero.

[0043] In embodiments for manufacturing a piezoelectric material or structure as disclosed herein, a first layer of a piezoelectric material and a second layer of piezoelectric material are deposited on opposite sides of a substrate forming a common electrode. The deposition process is to lay down the piezoelectric material on the common electrode for example by sputtering or by spin coating. It is worth mentioning that the deposition process of both layers may end up with different thicknesses and dielectric constants, as they are deposited sequentially. To overcome such discrepancies, the variation in areas, thicknesses, and dielectric constants is expressed in terms of an applied electric field E, rather than the applied voltage. Doing this will allow the geometrical variations and change in dielectric constants to be normalized. The variation in thickness T and area A in terms of applied electric field (E) can be expressed as follow:

±ΔT/T=±d.sub.33E  (6)

±ΔA/A=±2d.sub.31E+(d.sub.31E).sup.2  (7)

[0044] In equations (6) and (7) d.sub.33 and d.sub.31 are the longitudinal and transversal piezoelectric voltage constants, respectively. Equation (6) provides that the variation in thickness exhibits a linear relationship with the applied field E. Equation (7) shows that the variation in area exhibits a quadratic relationship with the applied field E. Substituting equations (6) and (7) into (5), produces a relation expressed in the following equation (8):

C.sub.r−2C.sub.rd.sub.33E=1+4d.sub.31E+2(d.sub.31E).sup.2  (8)

[0045] In bulk piezoceramic, it is assumed that the relative change in the material dimensions is the same in both horizontal and vertical directions. Under this assumption, the relationship between the longitudinal d.sub.33 and transversal d.sub.31 piezoelectric voltage constants is approximatively d.sub.33=2d.sub.31, This is a standard approximation accepted in the field. Using this assumption and rearranging equation (8) yields:

2E.sup.2d.sub.31.sup.2+(4EC.sub.r+4E)d.sub.31+(1−C.sub.r)=0  (9)

[0046] Solving equation (9) for d.sub.31, yields:

d.sub.31=(−(C.sub.r+1)±√{square root over (C.sub.r.sup.2+2.5C.sub.r+0.5))}E.sup.−1  (10)

[0047] Equation (10) states that there are two possible solutions. However, the negative value solution has been eliminated because the final value based on the formulations should be positive (i.e., due to physical limitations). Also, it should be noted that if the materials exhibit no piezoelectric effect, then C.sub.r is equal to 1 and d.sub.31 is equal to zero. When applying these physical limitations, equation (10) is balanced only when the solution to d.sub.31 reads as following equation (11):

d.sub.31=(−(C.sub.r+1)+√{square root over (C.sub.r.sup.2+2.5C.sub.r+0.5))}E.sup.−1  (11)

[0048] The significance of equation (11) is that the equation may be used to determine d.sub.31 without requiring any knowledge and information about the change in dielectric constant or any other variations. The only needed parameter is the thickness of the sputtered thin film of the piezoelectric layer or layers. Said differently, under the above-mentioned assumptions regarding the relationship between d.sub.31 and d.sub.33, the two parameters are calculated by knowing the applied voltage (and hence the electric field) to a piezoelectric sensor based on such a piezoelectric structure and the capacitance ratio between two scenarios where the DC electric field applied over the piezoelectric structure is in the same or opposite direction to the poling direction respectively. Hence for a given piezoelectric film, after having conducted a poling process wherein the dipoles of the piezoelectric material are subjected to a constant electric field to force the dipoles to align in a poling direction, the capacitances are measured and recorded corresponding to a selection of specific voltage values with negative and positive polarities. The electric field (E) and capacitance ratio (C.sub.r) are then computed based on the measured and recorded capacitance and voltage values.

[0049] It is worth noting that the assumed approximated relation of d.sub.33=2d.sub.31, may be replaced by a more general one, specifically d.sub.33=xd.sub.31, where an approximation factor x may assume values between 1 and 3, for example to adjust the approximation to different circumstances, conditions or practical applications of embodiments. Furthermore, almost 95% of the published literature in PZT based piezoelectric materials have reported numerically values for d.sub.33 and d.sub.31 supporting the approximation d.sub.33=2d.sub.31 and the mentioned optional adjustments to this approximation. Adjustments to the approximation of the relation between d.sub.33 and d.sub.31 may be based on the fact that for PZT based materials, the domain structure of the grains in the material has a strong influence on the ratio d.sub.33/d.sub.31.

Sample Preparation

[0050] To demonstrate the current approach described in this disclosure, a non-limiting exemplary embodiment has been utilized. In the exemplary embodiment, a thin piezoelectric film is deposited on both sides of a steel sheet using the sputtering technique. The deposition conditions are listed in Table 1. The film post annealing process was done at 700° C. for one hour. The thickness of the employed steel flexible sheet is of 50 μm, and the thickness of the deposited piezoelectric layers on both steel sides was measured to be 2.41 μm.

[0051] FIG. 2 shows an illustration of a film shaped structure 200 with a stack of layers with a first and a second piezoelectric layers PZE1, PZE2 sandwiching a metallic sheet Shim 100 with poling directions (↓P). In FIG. 2 the stack of layers is composed with a thin metallic steel sheet that is coated from both sides with noble materials PLT/Pt/Ti as a seeding layer for the deposition of the piezoelectric thin film materials layers on both faces of the steel. As illustrated in FIG. 2, the shim layer 100 is a flexible steel sheet which acts as a common electrode. Furthermore, both outer surfaces of PZE1 and PZE2 are coated with Al metallization to form electrical contacts. The poling of piezoelectric material in the film structure has been conducted at room temperature with the application of 250 kV/cm applied for 20 minutes. The CV measurements are conducted between the metallic shim and the outer electrical electrodes.

TABLE-US-00001 TABLE 1 PZT thin film deposition conditions Ti Pt PLT PZT Deposition temp [° C.] 500 500 650 700 Sputtering Pressure [Pa] 0.8 0.5 0.5 0.5 RF power [W] 80 100 150 90 Ar/O.sub.2 [sccm] 20/0 20/0 19.5/0.5 19.5/0.5 Deposition time [min] 6 8 15 300

[0052] As illustrated in FIG. 2 poling directions are indicated by the arrows to the left in the drawing, with a first poling direction 102 for the first piezoelectric layer PZE1 forming a first outer electrode of the structure and a second poling direction 104 for the second piezoelectric layer PZE2 forming a second outer electrode of the structure. In this example, both layers have been poled in the same direction, hence for identically applied voltage polarities applied across them, the respective thicknesses of the piezoelectric layers PZE1 and PZE2 will either both increase or decrease simultaneously. Meanwhile, if the applied voltages polarities are opposite to each other, one layer will increase in thickness and the other layer thickness will decrease.

[0053] To assess the efficiency of the fabrication process, X-Ray Powder Diffraction (XRD) measurements have been conducted for the steel flexible sheet before PZT deposition, i.e. in the blank state, and for the flexible sheet with a PZT deposited over steel, i.e. in the coated state. FIG. 3 shows a graph representing XRD measurements of the fabricated PZT layer to assess the quality of for example such embodiments as described in connection with FIG. 2. In FIG. 3, the XRD measurements is represented in the graph such that the filled circles represent the diffraction peaks of steel flexible sheet, and the open circles represent the diffraction peaks of the PZT deposition layer on the flexible steel sheet. The existence of multiple peaks and the distribution of the peaks along the angle of orientation indicates that the PZT deposition quality is high and the sample does not have disordered materials.

Calibration Method

[0054] To further calibrate embodiment methods, CV measurements may be conducted over samples of piezoelectric material. FIG. 4 shows measured CV of a bulk sample, in accordance with embodiments. The graph in FIG. 4 represents the results using a calibration method comprising CV measurements on a piezoelectric PZT unclamped bulk ceramic with thickness 0.24 mm. The sample unclamped piezoelectric material is thus not attached to any other material, substrate or artefact. The graph depicts capacitance measurements for voltages applied over the sample, with a first direction of the biasing voltage in parallel with and a second direction antiparallel to the poling direction of the piezoelectric structure of the sample. The measured d.sub.31 using Berlincourt meter is of 250 μm per volt of this unclamped sample. The extracted d.sub.31 piezoelectric voltage constant using (11) yields 190 μm per volt. The current method is in this example corrected by multiplying (11) with a correction factor Corr that in this example is calculated and approximated to (4/3), to correct the discrepancy between measured and extracted values; hence equation (11) is modified as per equation (12). It should be noted that this correction factor depends on the structure and on the process used to make the piezoelectric. Therefore, such factor value may change if other known structures or processes known in the field are used. Equation (12) incorporates the correction factor Corr for calibration, where Corr is 4/3 in the following equation (12).

d.sub.31=(4/3)(−(C.sub.r(+1)+√{square root over (C.sub.r.sup.2+2.5C.sub.r+0.5))}E.sup.−1  (12)

FURTHER METHOD EMBODIMENTS

[0055] In order to verify the results of the calibration method embodiments comprises reference measurements. FIG. 5 shows capacitance versus frequency measurements for an unbiased sample structure, i.e., at zero volts applied voltage, anti-parallel biasing at +3 volts applied voltage and parallel biasing at −3 volt applied voltage, in accordance with embodiments The electrical measurement was taken using a Gamry 3000 reference equipment. Capacitances versus frequency measurements at zero bias were conducted to determine the frequency range and its self-resonance frequency, shown in the middle data point series in the graph of FIG. 5. The capacitance shows a smooth response over the frequency as displayed by FIG. 5. The measurements were taken along frequency variations in the range of 1-80 kHz. Further, a dc voltage was applied to both layers for capacitance versus frequency at +3 volt dc rendering an anti-parallel bias (also called anti-polarization) as shown in the lower data point series in the graph, and at −3 volt dc voltage rendering a parallel bias (i.e., parallel to polarization) as shown in the upper data point series in the graph.

[0056] FIG. 5 shows that when the applied voltage is anti-parallel to the poling direction of the PZT material, the values of the measured capacitance increases with the frequency as it reaches around 230 nF. On the other hand, applying voltage parallel to the poling direction decreases the values of the measured capacitance slightly with the frequency and reaches around 75 nF. While in the zero-biasing case, the capacitance measurements stay nearly constant around the value 135 nF along the range of the frequency. As anticipated, the applied voltage in directions either parallel or opposite to the poling direction will cause a change in the dimension which is pronounced by the capacitance measurements. In the measurement example illustrated with FIG. 5, the two layers have been subjected to biasing voltages in opposite directions with constant dc voltage over frequency variation.

[0057] FIG. 6 shows the two piezoelectric capacitance voltage responses, i.e., for each of the two piezoelectric layers, when the applied voltage has been swept from −0.3 to +0.3 volts. For layer 1, represented by data points indicated by filled circles in the graph of FIG. 6, the capacitance increase for the positive applied field as it is driven opposite to the poling field direction, meanwhile it decreases for the negative values as it is driven parallel to the to the poling field direction. On the contrary, for layer 2, represented by data points indicated by open circles in the graph of FIG. 6, the capacitance increases for negative applied field as it is driven parallel to the poling field direction, meanwhile it decreases for positive values as it is driven opposite to the to the poling field direction.

[0058] In embodiments, the piezoelectric voltage constants d.sub.31 and d.sub.33 are extracted and estimated using equation (12) and the data or similar data as presented in FIG. 5. Examples of the extracted d.sub.31 and d.sub.33 piezoelectric voltage constants have been estimated and depicted in relation to frequency in a graph in FIG. 7, with the values for constant d.sub.31 is indicated with filled circles and values for the constant d.sub.33 is indicated with open circles. In this exemplifying instance, the piezoelectric structure has been subjected to an applied electric field of 1.25 volt per μm. As observed from FIG. 7, the constants exhibit a smooth behavior over the wide frequency range. For example, at a frequency of 40 kHz, the d.sub.33 and d.sub.31 constants read 284 and 142 pm per volt, respectively. These values are comparable with values that have been determined with other, previously known, methods as reported in published related art literature for the same or similar thin film thickness.

[0059] Equation (12) along with CV measurements presented in FIG. 6 have been used to extract the piezoelectric constants of both layer 1 PZE1 and layer 2 PZE2 of the stack according to the embodiment shown in FIG. 2. The estimated mean values of values for d.sub.31 of layer 1 PZE1 and layer 2 PZE2 are in this embodiment 125 and 130 μm per volt, respectively. Furthermore, the variation in the capacitance of structures incorporating piezoelectric materials, due to the applied voltage can be generated either by determining the dielectric constant voltage dependency and/or or by determining the expansion/contraction in the dimensions of the poled material. FIG. 8 shows a graph illustrating the dielectric constant voltage dependency as a polarization curve for a piezoelectric structure with films of piezoelectric material deposited on steel sheet, in accordance with embodiments as shown in FIG. 2. The values according to the graph in FIG. 8 reveals that the change in the polarization due to the change in electric field is equal to 1.6 C/cm.sup.2. Hence the equivalent capacitance change from this change in polarization is of 0.6 pF. Therefore, 1% of the measured capacitance change is due to dielectric constant voltage dependency while 99% of this change is referred to the variations of sample dimensions due to the piezoelectric effect. Another piezoelectric constant called e31,f can be correlated with the piezoelectric constant d.sub.33 may be measured with other setups of measurement, and in the context of FIG. 8 the piezoelectric constant (e31,f) is then estimated to be −3.8 C/m.sup.2. The dielectric constant of the film is computed to be of 240 on average.

Embodiments for Direct Extraction of d.sub.33 and d.sub.31 from Cr-E

[0060] It is also possible to extract simultaneously the d.sub.31 and d.sub.33 piezoelectric constants directly from (8). It should be noted that this section does not rely on the assumption that d.sub.33=2 d.sub.31 as in above described embodiments. Equations (8) could be arranged to express the capacitance ratio (Cr) as a function of applied voltage, wherein the applied voltage is expressed in the equation in terms of electric field E, as per equation (13):

C.sub.r=(1+4d.sub.31E+2(d.sub.31E).sup.2)(1−2d.sub.33E).sup.−1  (13)

[0061] Assuming that 2d.sub.33E<<1, the Taylor approximation (1+x).sup.n=1+nx, may be applied to equation (13) to yield:

C.sub.r=(1+4d.sub.31E+2(d.sub.31E).sup.2)(1+2d.sub.33E)  (14)

[0062] Equation (14) could be further simplified as follows in equation (15):

C.sub.r=1+2(d.sub.33+2d.sub.31)E+2(4d.sub.31d.sub.33+d.sub.31d.sub.31)E.sup.2+4d.sub.31.sup.2d.sub.33E.sup.3  (15)

[0063] The last cubic term of equation (15) can be neglected, due to its very small value. As such, equation (15) may be expressed as follows in equation (16):

C.sub.r=1+2(d.sub.33+2d.sub.31)E+2(4d.sub.31d.sub.33+d.sub.31d.sub.31)E.sup.2  (16)

[0064] Equation (16) indicates that d.sub.31 and d.sub.33 can be extracted simultaneously by fitting the measured Cr values versus E using quadratic fitting. For calibration purposes, a piezoelectric bulk ceramic material of thickness 0.150 mm with d.sub.33 and d.sub.31 of 430 and 230 μm per volts, respectively, has been utilized in embodiments. Nevertheless, as both the calibration sample and sample under test have different thicknesses of more than three order of magnitudes, the normalized applied electric field may be used in embodiments to account for this difference. FIGS. 9A and 9B show graphs that display the measured capacitance ratio (Cr) values versus the normalized applied electric field (En) voltage for the bulk and thin film samples, respectively. The fitting equation of the extracted capacitance ratio versus the normalized applied electric field for the bulk sample is found to be in accordance with equation (17):

C.sub.r=1−0.04511E.sub.n−0.08492E.sub.n.sup.2  (17)

[0065] Comparing equation (17) with equation (16), the second and the third terms account for the piezoelectric effect. Hence equations (18 and (19):

2(d.sub.33+2d.sub.31)=−0.04511  (18)

2(4d.sub.31d.sub.33+d.sub.31d.sub.31)=−0.0849  (19)

[0066] Solving equations (18) and (19) simultaneously for d.sub.31 and d.sub.33 yields 0.0846 pC/N and 0.1666 pC/N, respectively. Hence for calibration the solution for equations (18) and (19) should be multiplied by a factor of 2945 to calibrate the method. This number considers the normalization of electric field and the method calibration. For the electric field normalization consideration, it is required to multiply back by the maximum applied field (V/T) max; voltage over thickness=20/(0.12e−3)=166,666.666. Multiplying (0.0846) by this maximum field yields 14,100. This value now should be compared with the Berlincourt value 250. Therefore, for the calibration of method and computation the 14,100 should be multiplied by 0.0177; hence this number, i.e., “factor number “2945”, is the result of the multiplication of (166,666.666*0.0177)=“2945”.

[0067] Therefore, the actual d.sub.33 and d.sub.31 reads 448 pC/N and 228 pC/N, respectively, i.e., d.sub.33 is equal to 1.96 times d.sub.31 (approximately d.sub.33≈2d.sub.31). For the film under study, the corresponding fitting equation is found to be in accordance with the following equation (20):

C.sub.r=0.95+0.06814E.sub.n−0.02134E.sub.n.sup.2  (20)

[0068] Solving the mathematical model generated by comparing equation (20) to equation (16) and incorporating the calibration step yields d.sub.33 and d.sub.31 of 134 pC/N and 256 pC/N, respectively. It is noted that the direct extraction method using the Cr-E approach produces a maximum error of 5%.

Embodiments for Direct Extraction of d.sub.33 and d.sub.31 from Fr-E

[0069] Incorporating a piezoelectric material in a resonator structure that has a measurable resonance frequency, with the possibility to drive this resonator against and along polarization/poling directions, the resonance frequency may be written as the following equation (17-1):

[00001]$\begin{array}{cc}f=\frac{1}{2\pi \sqrt{\mathrm{LC}}}& \left(17\text{-}1\right)\end{array}$

[0070] In equation (17-1) f is the resonance frequency, C is the capacitance and L is the effective inductor of the resonator, which will not change with driving the piezoelectric against or along the poling field.

[0071] The frequency ratio fr between the resonance frequency along the polarization over the resonance frequency measured when driving against the poling is in accordance with the following equation (18-1):

[00002]$\begin{array}{cc}{f}_r=\frac{1}{2\pi \sqrt{{\mathrm{LC}}_p}}/\frac{1}{2\pi \sqrt{{\mathrm{LC}}_a}}& \left(18\text{-}1\right)\end{array}$

[0072] The above expression may be simplified to yield an fr in accordance with the following equation (19-1):

[00003]$\begin{array}{cc}{f}_r=\sqrt{\frac{C_a}{C_p}}& \left(19\text{-}1\right)\end{array}$

[0073] However, as described above, the ratio of the capacitance when the electrical field is opposite the poling direction and is parallel to the poling direction is expressed as Cr. Therefore, the above expression may be simplified to yield the fr in accordance with the following equation (20-1):

f.sub.r=C.sub.r  (20-1)

[0074] Substituting the expression from Cr, as provided in equation (20B), using equation (49 yields:

C.sub.r(T−ΔT)(T+ΔT).sup.−1=(A+ΔA)(A−ΔA).sup.−1  (4)

C.sub.r=(A+ΔA)(A−ΔA).sup.−1/(T−ΔT)(T+ΔT).sup.−1  (4)

C.sub.r=(A+ΔA)(A+ΔA)(T+ΔT)(T+ΔT)  (4)

C.sub.r=AATT(1+ΔA/A)(1+ΔA/A)(1+ΔT/T)(1+ΔT/T)  (420B-4)

C.sub.r=AATT(1+ΔA/A)(1+ΔA/A)(1+ΔT/T)(1+ΔT/T)  (20B)

This results in fr in accordance with the following equation (21):

[00004]$\begin{array}{cc}{f}_r=\mathrm{AT}\left(1+\frac{\Delta A}{A}\right)\left(1+\frac{\Delta T}{T}\right)& \left(21\right)\end{array}$

Therefore, the absolute value of the normalized frequency ratio frn is in accordance with the following equation (22):

[00005]$\begin{array}{cc}{f}_{rn}=\left(1+\frac{\Delta A}{A}\right)\left(1+\frac{\Delta T}{T}\right)& \left(22\right)\end{array}$

[0075] It should be note that Normalization is done based on the volume of the piezoelectric layer at zero bias in order to simplify the calculation and mathematical derivations. So, equation (22) produces the following equation (23) for the normalized frequency ratio frn:

f.sub.rn=(1+2d.sub.31E+(d.sub.31E).sup.2)(1+d.sub.33E)  (23)

[0076] And therefore the normalized frequency ration frn is in accordance with the following equation (24):

f.sub.rn=1+(2d.sub.31+d.sub.33)E+(2d.sub.31d.sub.33+d.sub.31.sup.2)E.sup.2+d.sub.31.sup.2d.sub.33E.sup.3  (24)

[0077] Thus, by fitting the normalized frequency ratio frn versus applied electric field E with cubic equation, the coefficient d31 and d33 can be extracted, in accordance with embodiments.

[0078] Therefore, the characterization of piezoelectric constants relevant to a specific application will enhance their use. This disclosure describes several embodiments of methodologies and structures that may be used to determine the piezoelectric constants. The piezoelectric material should be incorporated as a capacitance dielectric material in the shape of a film. An electric applied field is then applied to drive the film parallel and anti-parallel to the poling field direction. This is usually done by sweeping the voltage from negative to positive values. The variations in geometric dimensions and the corresponding dielectric constant of the materials due to the applied field will be reflected in the measured capacitance. The developed models require only the pre-knowledge of the film thickness and automatically de-embed the change in dielectric constant due to the applied stress. The embodiment methods have been calibrated using unclamped bulk PZT ceramic and validated using conventional meters. The estimated and measured values are well corroborated with each other. The techniques in accordance with embodiments herein do not require any sample heavy preparation steps and provide a rapid response along with accurate estimations.