1. Effect of Actuator Geometry on e31 Measurement Accuracy
Joe Evans, Naomi Montross, Spencer Smith, Bob Howard, Scott Chapman
Radiant Technologies, Inc.
International Symposium on the Application of Ferroelectrics
July 16, 2019

2. Summary
e31 cantilevers may have a built-in source of error under certain test conditions. Evaluation of cantilevers differing only by their capacitor geometries failed to demonstrate any inherent inaccuracy. The search continues.

3. Introduction Review of e31 and Cantilever Geometry
Describe Accuracy Issue
Proposed Test and Expected Results
Surprising Outcome
Re-think the Test Conditions
Final Results
Conclusion

4. Why e31?
In 1909, George Stoney derived an equation describing how a thick cantilever bends when a change in temperature causes contraction of a thin film on one surface of the cantilever.
Instead, substitute a piezoelectric film to bend the beam using voltage:
Cantilever = mechanical amplifier: The lateral stress produced by that voltage due to e31 moves the cantilever tip vertically much further than the tiny strain of the top surface of the piezoelectric film due to d33. . 0V
Vcap
𝑪𝒂𝒏𝒕𝒊𝒍𝒆𝒗𝒆𝒓 𝑻𝒊𝒑 𝑫𝒊𝒔𝒑𝒍𝒂𝒄𝒆𝒎𝒆𝒏𝒕= 𝒎𝒆𝒄𝒉𝒂𝒏𝒊𝒄𝒂𝒍 𝒂𝒎𝒑𝒍𝒊𝒇𝒊𝒄𝒂𝒕𝒊𝒐𝒏 x 𝑒 31 𝑬 3
Larger Displacement =
Cheaper Displacement Sensor =
Inexpensive Instrument!

5. E31 Fixture with Cantilever

6. The Mazzalai Equation
Radiant uses the Mazzalai Equation to calculate the e31 coefficient that produced the measured cantilever tip displacement.
v = Poisson’s ratio for the substrate
x1 = Distance from clamp point to the end of the piezoelectric capacitor
x2 = Distance from clamp point to the displacement sensor.
The equation states that e31 can be accurately calculated for any cantilever of any geometry.
Experience indicates that this assertion may not be true.
[Mazzalai, Balma, Chidambaram, Jin, & Muralt, International Symposium on Applications of Ferroelectrics – ISAF/PFM, Prague, Czech Republic; 07/2013]
𝑒 31 =− 𝑌 𝐶𝑎𝑛𝑡 𝑡 𝐶𝑎𝑛𝑡 −𝑣 𝑥 1 2 𝑥 2 − 𝑥 𝑪𝒂𝒏𝒕𝒊𝒍𝒆𝒗𝒆𝒓 𝑾𝒊𝒅𝒕𝒉 𝑪𝒂𝒑𝒂𝒄𝒊𝒕𝒐𝒓 𝑾𝒊𝒅𝒕𝒉 𝒛 𝑎𝑡 𝑥 2 𝑉 .

7. Accuracy vs Geometry?
Radiant Technologies published “A Proposal for a Universal e31 System” in 2017 that can be found on the company’s web site.
Radiant offers standardized fixtures for testing cantilevers fabricated from 4” and 2” silicon wafers in compliance with this proposal.
The shorter cantilevers have wider actuator capacitors in order to keep the cantilever tip displacement for both cantilever lengths in the 2µm to 4µm range for optimal performance of the built-in photonic sensor.
The shorter cantilevers with wider electrodes seem to always produce smaller e31 coefficients. (See Page 76.)
Radiant has heard from other researchers that they see the same inaccuracy. .

8. Short & Wide is Inaccurate?
From Page 76 of “A Proposal for Universal e31 System”:
The table below compares the measured e31 coefficients for three compositions versus the aspect ratio of the actuator width to the cantilever width.
The wider the actuator, the lower the measured e31.
The wider actuator probably bends the cantilever across its width, changing the shape of the cantilever……
Composition
2” e31
4” e31
4/20/80 PLZT
-10.0
-13.9
4/20/80 PNZT
-7.3
-12.1
20/80 PZT
-6.9
-12.5
Note: the e31 coefficient shown above for 20/80 PZT was an early value. We now know it is closer to -7 C/m2.

9. Proposed Test
It appears that the width of the actuator capacitor compared to the width of the cantilever substrate may affect the accuracy of the e31 coefficient calculation.
PROPOSAL:
Build a series of e31 cantilevers on the same wafer having a range of actuator capacitor widths for a fixed cantilever width.
Measure and compare e31 coefficients for all cantilevers.

10. Samples
Five of the 12 cantilevers are shown starting with the 4500µm width at the bottom of the image.
The top electrode widths varied from 4500µm down to 100µm in 12 steps. Otherwise the cantilevers are identical.
2µm-thick 52/48 PZT

11. Test Procedure
The following tests were executed in the same order on each cantilever from within a Vision Test Definition.
Virgin 40V bipolar hysteresis. (Each cantilever was untouched at the start.)
40V Remanent Hysteresis Task.
40V monopolar hysteresis w/ switching & non-switching half loops.
40V monopolar hysteresis with only non-switching half loops.
Single 5V monopolar hysteresis.
5V monopolar hysteresis averaged 16 times.
Other loops using ratio’d voltages or constant displacement conditions.
The averaged 5V monopolar actuator displacements were used to calculate e31. The 5V amplitude was selected so that the widest cantilever would produce a displacement within the best sensitivity range of the photonic sensor. The narrowest electrodes required averaging. .

12. Sample Response

13. Expected Results
The prediction is that a wide top electrode will not only bend the cantilever along its long axis but also across its narrow axis. This “cupping” of the cantilever makes it less willing to bend along its long axis due to the I-beam effect.
The measured data can be plotted two ways looking for this effect:
e31 vs Top Electrode Width
Absolute Displacement vs TE Width
The expected result is that e31 and tip displacement will deviate to smaller values as the top electrode gets wider. .
TE Width
e31
TE Width
Absolute Tip Displacement

14. 5V Results . No effect of TE Width on e31 is apparent in the data!
Systematic variation in e31 due to variance vertically top-to-bottom across wafer vs position of each cantilever on the wafer. .
No effect of TE Width on e31 is apparent in the data!
Might the low voltage be responsible?

15. 40V Results .
Again, no effect of TE Width on e31 is apparent at 40V.

16. Conclusion
Surprisingly, a careful study of cantilever tip displacement versus capacitor width for cantilevers all from the same wafer found no significant variance of the e31 coefficient with capacitor width.
Further research will continue to determine if the discrepancy is real and from where it originates.