Technical Report EXP17AO Rev D - ferrodevices.com · application note: “A Proposal for a...

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Radiant Technologies, Inc. 1

Technical Report

EXP17AO

Rev D

Title: Measure the e31 Constant for Super Thin Films

Date: November 5, 2017

Author: Joe Evans

Theory: At present, the thin piezoelectric film community calculates e31 using the Stoney Equation as

modified by Mazzalai.

[International Symposium on Applications of Ferroelectrics - ISAF, Prague, Czech Republic; 07/2013]

where

YCant = Young’s Modulus of the cantilever substrate

tCant = Thickness of the cantilever substrate

z(x2) = Cantilever tip displacement at x2

v = Poisson’s ratio for the cantilever material

cf = Ratio of the actuator (top electrode) width to cantilever width

x1 = Length of the actuator capacitor from the clamp point

x2 = Distance of the displacement measurement point from the clamp.

This equation rearranges into


Note that the thickness of the piezoelectric film of the actuator capacitor (tpiezoelectric) is not in

the equation describing cantilever tip displacement. Mazzalai incorporated Stoney’s assumption

that, as long as the thickness of the piezoelectric film is infinitesimal compared to the thickness of

the cantilever, the thickness of the piezoelectric film will have no affect on cantilever

performance at a fixed voltage.

The experiment described in this report tests this prediction of the Stoney Equation: Using

cantilevers that are identical in all other ways but with varying piezoelectric film thicknesses, will

cantilever tip displacement at a single fixed voltage generate the same tip displacement for all

film thicknesses?

Result: Yes.

Limitations to the Experiment: The e31 cantilever measurement is becoming popular for characterizing thin piezoelectric and

ferroelectric films. It is inexpensive to execute and produces consistent results. The piezoMEMS

community so far has been concerned with films that are one micron and thicker. Radiant

designed its E31 fixtures to work with Radiant test equipment in a way to allow films of any

thickness to be evaluated for e31, not just the thicker films. While thick films will be necessary

for piezoMEMS actuators, thin films may find usefulness in sensors in the future.

The issue restricting the testing of piezoelectric or ferroelectric films thinner than 1 µm for e31 is

the area of the actuator capacitor. Films a micron thick or thicker can yield capacitors with areas

of a square centimeter or more. Such large actuator areas will move the cantilever tips more than

a micron so that inexpensive displacement sensors can be used to measure tip motion and

calculate e31. Because defect density increases for thinner films, the maximum allowed actuator

area reduces as the films become thinner than 1 µm. Smaller actuators mean smaller tip

displacement, so much so that films at 100 nanometers or less on wafer-thickness silicon

cantilevers require laser vibrometers to resolve cantilever tip motion.

For example, films below 0.1 µm (100nm or 1000Å) cannot support capacitor areas larger than

0.0001 square centimeters (10,000 square microns). Films less than 600Å probably cannot yield

capacitor areas larger than 1,000 square microns. The area of the capacitor is represented in the

Stoney Equation by x1 times the numerator of cf . Since this product is in the numerator of the

Stoney Equation, tip displacement z(x2) will decrease linearly with decreasing actuator area.

Another very restrictive limitation is that thinner films break down at lower voltages, limiting the

maximum test voltage which in turn limits the maximum testable tip displacement

This experiment was designed to measure multiple cantilevers of different actuator thicknesses

using a fixed voltage. By definition, the maximum test voltage for all cantilevers will be set by

the maximum voltage of the cantilever with the thinnest yielding PLZT film. On the other hand,

the LDV resolution at the test frequency sets a bottom floor for tip displacement. The test

voltage and the actuator must be large enough so an LDV can see the tip move. Calculations

using the modified Stoney Equation predict that an actuator capacitor area of at least 20,000


square centimeters would be needed to cause tip displacements in the nanometer range.

According to Stoney, increasing the actuator length while sacrificing actuator width to keep the

actuator area constant increases tip displacement linearly. Consequently, an actuator length of

2000 microns with a very narrow width of 10 microns was selected for the photolithography

mask with which to fabricate the cantilevers. The 10 micron line width is the minimum that can

be fabricated in Radiant’s facilities while remaining continuous for the 2000 micron (2mm)

length.

A diagram of a Radiant E31 cantilever with a 0.2 square centimeter actuator area suitable for

1µm-thick piezoelectric films is below. This actuator typically moves the cantilever tip between

2 microns to 4 microns at 20 volts. Beneath that diagram is an image of the cantilever for this

experiment with its 2000 µm x 10 µm actuator area.

Typical Cantilever Geometry for Thick Piezoelectric Films

Electrode Geometry for Super Thin Piezoelectric Films

The perceived width of the 2000x10µm actuator electrode in the bottom image above is larger

than its true dimensions. Such a tiny electrode is not visible to the naked eye. With these two

images as perspective, that such a tiny cantilever on a 1500Å PLZT can move the tip of the

550µm-thick silicon cantilever is amazing! For more information about Radiant’s standard

cantilevers, Radiant E31 fixtures, and the mathematics of piezoelectric cantilevers, see the

application note: “A Proposal for a Universal e31 system.pdf”.

The anticipated tip displacement calculated from the modified Stoney Equation for a 2000 micron

long by 10 micron wide actuator on a 46-millimeter-long, 550-micron-thick silicon cantilever

Equation is 2.4 nanometers.

Experimental Procedure:

Twelve wafers each holding 12 cantilevers were fabricated with 4/20/80 PLZT films ranging

from 300Å up to 10,300Å (1.03 microns). Film thicknesses were adjusted by depositing different

numbers of layers using four different molarities.


WAFER LAYERS ~THICKNESS(Å) Largest TESTABLE Cap (µm)

Wafer 1 3 layers of 0.15M ~720 25x25

Wafer 2 5 layers of 0.15M 800 200x200

Wafer 3 3 layers of 0.075M ~300 None

Wafer 4 5 layers of 0.075M ~500 None

Wafer 5 4 layers of 0.25M 1,500 200x200


Wafer 7 2 layers of 0.25M ~800 50x50




Wafer 11 4 Layers of 0.6M 4,600 Not Tested

Wafer 12 9 layers of 0.6M 10,300 Not Tested

Note: wafers with “~” have estimated thicknesses.

Each wafer held square test capacitors. These capacitors were evaluated after fabrication to

predict which cantilevers would probably work in the E31 fixture. Note that the actual cantilever

actuator capacitors were not tested immediately after fabrication in order to prevent ageing before

they are tested for tip displacement. In order for the 20,000 µm2 actuators (2000µm x 10 µm) to

yield, the 40,000 µm2 (200µm x 200µm) square test capacitors had to function. Wafers 3 and 4

fabricated with the very low molarity gel produced no functional capacitors. Wafers 1 and 7 had

functional capacitors but none as large as the 200x200 level. The wafers that yielded 200x200

square capacitors were measured for film thickness using a DekTak stylus profiler. The yielding

wafers thicknesses were:

WAFER Measured THICKNESS (Å)

Wafer 2 800

Wafer 8 1,270

Wafer 5 1,500

Wafer 9 1,700

Wafer 10 2,300

Wafer 6 2,450

Wafer 11 4,600

The cantilevers were transported to Polytec Inc. in Irvine, California along with a Rev F Precision

Multiferroic Non-linear Materials tester plus a Radiant E31 Tower fixture. The equipment was

connected to a Polytec OFV-534 LDV and its OFV-5000 controller offering both velocity and

displacement output signals. The Radiant E31 Tower fixture held the cantilevers in place beneath

the LDV. The LDV and test fixture sat on an active isolation table with pneumatic legs. The first

measurements were acquired on a different table with no active noise rejection. Without active

noise isolation, the ambient mechanical noise was 0.5 microns. With active noise isolation

provided by the air table, the single pass noise level decreased to approximately ±5 nanometers.

Averaging 64 times reduced the noise level to approximately ±0.5 nanometers peak to peak.


The cantilever with the 1µm-thick PLZT actuator film was loaded first because this is a known-

good film thickness with 100% yield for 20,000 square micron capacitors. A bipolar butterfly

loop was acquired at 20 volts as a reference for all measurements. At 20 volts the tip moved

approximately 5.5 nanometers in its upper butterfly wings. A monopolar 25 volt loop was also

acquired where the tip moved approximately 7 nanometers.

The next step was to test the cantilevers with the thinnest piezoelectric films for actuator yield at

1 volt. The cantilever functionalities are listed below. All cantilever actuators with 800Å and

1,270Å thickness films were shorted. The cantilever with a 1500Å film (150nm) yielded a

functional cantilever and proved that it could handle up to 9 volts without breaking down. All

cantilevers with thicker PLZT produced at least one cantilever that could be tested. All

cantilevers marked “Good” in the table below were tested at 9 volts, 1 Hertz.

WAFER Measured THICKNESS (Å) CANTILEVER Function?

Wafer 2 800 All Shorted

Wafer 8 1,270 All Shorted

Wafer 5 1,500 Good

Wafer 9 1,700 Good

Wafer 10 2,300 Good

Wafer 6 2,450 Good

Wafer 11 4,600 Good

Wafer 12 10,300 Good

Wafer 10 at 2,300Å was not tested due to time constraints. It was selected out because its

thickness is so close to that of Wafer 6 at 2,450Å.

Noise Limitations:

The measurements were taken in open air without an enclosure even though the absolute

cantilever tip displacement at 9 volts never exceeded 5.5 nanometers (55Å). The loops were

measured at 1 Hertz, a frequency 1/300th of the resonant frequency for the 42 millimeter-long

cantilevers. The low test frequency was necessary to prevent excitation of the high-Q cantilever

resonance by the test voltages. Even at such a loop frequency so far from resonance, small

oscillations at resonance were activated by the hysteresis loops and are visible in the data. A 60

Hertz signal also appeared in the measurements but it is unclear whether this was an intrusive

EMF or an actual mechanical displacement of the cantilever tips. Despite the low signal-to-noise

of the measurements, the results were clean enough to successfully draw conclusions from the

experiment.

NOTE: The butterfly loop consisted of a full positive/negative voltage cycle conducted at 1

Hertz. Monopolar loops simulated actuator voltage profiles by executing only a half cycle. For

this experiment, monopolar loops were executed with 500ms periods to maintain a 1 Hertz full

cycle frequency.


Due to the low signal-to-noise ratio of the measurements, loops were measured 64 times each and

averaged to produce the plots that follow.

Results:

The 20 volt butterfly loop of the cantilever with 1µm of PLZT set the displacement envelope

within which the experiment was conducted. All data below come from the dataset

“Polytec Visit 10_27_2017.dst”.

A +25-volt monopolar loop is added to the next plot.

-6

-4

-2

0

2

4

6

8

10

-20 -15 -10 -5 0 5 10 15 20 25

EXP17AO e31 vs PLZT Film Thickness

Na

no

me

ters

Volts

Wafer 12 10300A 20V Butterfly: Sensor Value


The monopolar loop overlays the upper arm of the butterfly loop as expected. These two

measurements were taken on two different cantilevers. The butterfly was taken at the beginning

of the test to establish the envelope. The monopolar loop was taken at the end of the test list on a

second cantilever with 1µm of PLZT. The second cantilever was also tested for its 9-volt

monopolar loop, plotted in black below along with the 20V butterfly and 25V actuator loops.

-6

-4

-2

0

2

4

6

8

10

-20 -15 -10 -5 0 5 10 15 20 25


Na

no

me

ters

Volts

Wafer 12 10300A 20V Butterfly: Sensor Value Wafer 12 10300A 25V Mono: Sensor Value

-6

-4

-2

0

2

4

6

8

10

-20 -15 -10 -5 0 5 10 15 20 25


Na

no

me

ters

Volts

Wafer 12 10300A: Sensor Value Wafer 12 10300A 20V Butterfly: Sensor Value

Wafer 12 10300A 25V Mono: Sensor Value


The second 1µm-thick cantilever generated a little more than 2 nanometers of tip displacement at

9 volts and overlays the two previous measurements.

The 9-volt monopolar loops for all cantilevers are plotted below.

Within the noise limits of these measurements, all cantilevers produced essentially the same

cantilever tip displacement at 9 volts independent of the PLZT film thickness.

The average displacement for the six cantilevers at 9 volts is:

WAFER Measured THICKNESS (Å) Extremes z (nm)

Wafer 5 1,500 3.485-(+0.965) = 2.52

Wafer 9 1,700 2.747-(+0.152) = 2.60

Wafer 10 2,300 Not Measured

Wafer 6 2,450 2.666-(-0.055) = 2.72

Wafer 11 4,600 2.282-(-0.171) = 2.45

Wafer 12 10,300 2.496-0.344 = 2.15

Average 2.488

Note that the oscillatory noise in the data prevents using the raw data to determine absolute

displacements. Instead, the Extremes in the table above were captured from a linear fit to each

measurement where the fit crossed the 0 volt and 9 volt lines in the plot.

e31 value for this PLZT estimated for the 9 volt tests is -15.3 C/m2.

-6

-4

-2

0

2

4

6

8

10

-20 -15 -10 -5 0 5 10 15 20 25


Na

no

me

ters

Volts

Wafer 12 10300A: Sensor Value Wafer 11 4600A: Sensor Value

Wafer 6 - 2450A: Sensor Value Wafer 9 - 1700A: Sensor Value

Wafer 5 - 1500A: Sensor Value


Where

YCant = 1.69 x 1011

tCant = 550µm

z(x2) = 2.49 nm

v = 0.064

cf = 10µm/5000µm = 0.002

x1 = 2mm

x2 = 41.96 mm.

This result is consistent with measurements performed on 1µm of 4/20/80 PLZT using the

standard Radiant E31 cantilever with the 0.2cm2 actuator are shown earlier in this document. All

measurements of PLZT executed at Radiant over the last year range from -14 C/m2 to -16 C/m

2.

Analysis:

The experiment proved that the e31 coefficient can be accurately determined for very thin

piezoelectric and ferroelectric films using cantilevers. Finer analysis of the cantilever motion

requires a better Signal-to-Noise ratio (SNR) from the measurements. SNR can be improved by

increasing cantilever tip displacement and increasing the test frequency into a range better suited

for LDV resolution

Laser Doppler Vibrometers can have a displacement noise floor as low as a picometer or less but

that resolution occurs are high frequencies when the surface being tested has high velocity. At


the 1 Hertz test frequency of this experiment, the noise floor of an LDV increases to the level of a

few Ångstroms or more meaning that this experiment operated near the resolution limit of the

OFV-534. The very slow test frequency used for this experiment was necessary to avoid exciting

the resonant mechanical frequency of the cantilever at 300 Hertz. If the resonant frequency of the

cantilever can be increased, the test frequency can be increased to allow the LDV to operate with

a better SNR even if tip displacement remains constant. Increasing cantilever resonant frequency

is accomplished by reducing the length of the cantilever. The issue is that cantilever tip

displacement decreases with the square of the decrease in length. On the other hand, tip

displacement increases with the square of the decrease in thickness of the cantilever. These

cantilevers with a resonant frequency near 300 Hertz were fabricated using 100mm silicon wafers

having a thickness of 550 micrometers. By moving to 50 mm (2”) silicon wafers, the cantilevers

will reduce in length by half but decrease in thickness by roughly half to exhibit the

approximately same tip displacement. Despite having the same tip displacement, the cantilever

resonant frequency will increase and the LDV will have a lower noise floor.

Tip displacement can be increased by reducing the width of the cantilever while keeping the

width of the actuator capacitor constant. The practical limit for reducing cantilever width is that

width necessary to load the cantilever into the test fixture. Assuming a minimum of 1 millimeter

width to fit mechanically into the fixture, the tip displacement can be increased by a factor of five

from those achieved in this experiment.

Of course, reducing the defect density of the super thin films to allow for larger actuator areas

will increase tip displacement. As well, techniques such as DRIE combined with SOI silicon

wafers may allow cantilevers with extremely thin substrate thicknesses to be created so tip

displacement remain at a measureable scale while films less than 600Å might be characterized.

Conclusion:

The piezoelectric property of very thin piezoelectric or ferroelectric films can be accurately

measured using the e31 cantilever technique coupled with a very sensitive laser Doppler

vibrometer. Films as thin as 150 nanometers were successfully measured in this experiment with

tip displacements as low as 2 nanometers. By adjusting the geometry of the cantilever to increase

tip displacement and raise the cantilever resonant frequency, higher SNR is possible to provide a

finer resolution of cantilever tip trajectories. Or, with those same changes, the same SNR can be

maintained for films thinner than 150 nanometers.

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