Component Model of Ferroelectric Capacitors · PDF fileComponent Model of Ferroelectric...

Radiant Technologies, Inc.

Component Model ofFerroelectric Capacitors

Joe T. Evans,Radiant Technologies, Inc.

January 16, 2011www.ferrodevices.com

Radiant Technologies, Inc.2

An Excellent Hysteresis Loop

• This loop is nearly “perfect”. Most loops are not. Afterthis presentation, you should be able to discern thedifference upon inspection.

-7.5 -5.0 -2.5 0.0 2.5 5.0 7.5

Voltage

What is this?

• Is this loop as good as the previous loop?

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

Voltage

What is this?

• Real clean. This one is easy.

-4 -3 -2 -1 -0 1 2 3 4

Voltage

A Harder One

-4 -3 -2 -1 -0 1 2 3 4

Voltage

• Quite a few papers include loops that look like this.

Is this Ferroelectric?

-7.5 -5.0 -2.5 0.0 2.5 5.0 7.5

Voltage

What Happened Here?

-7.5 -5.0 -2.5 0.0 2.5 5.0 7.5 10.0

Voltage

• In electrical engineering, a fundamental approachto understanding a system is to break it intocomponents and model each component.

– Each component responds independently to thestimulus.

– The output of a component is either the input to anothercomponent or is summed with the outputs of othercomponents to form the response of the device.

Modeling Nonlinear Capacitance

• According to Joe:– Linear capacitance– Non-linear capacitance– Remanent polarization– Remanent and nonremanent leakage– Remanent and nonremanent small signal capacitance– Reverse bias diode electrode interfaces– Left-overs

The Components

A Mathematical ToolThe hysteresis loop is polarization responding to appliedvoltage: P(V). Its derivative with respect to voltage is

δP/δV => (δQ/δV)/Area

which equals Large Signal Capacitance per Unit Area.RT17 Normalized C(V)

-12 -9 -6 -3 0 3 6 9 12

Vdrive

RT17 Hysteresis

RT17 nC(V)

3500Å 20/80 PZT

RT17 Hysteresis

-12 -9 -6 -3 0 3 6 9 12

Vdrive

RT17 Hysteresis

3500Å 20/80 PZT

Normalized CVThe normalized CV [nCV] has the formula

δP/δV => (δQ/δV)/Area

and has the units of

µF/cm2

when the derivation is performed on the polarization units of

µC/cm2.

Integration

• Some measurements determine capacitance.– Small signal capacitance vs. Voltage

• Mathematical integration will convert the capacitance to itsequivalent polarization contribution at each voltage.

Linear Capacitance

-4 -3 -2 -1 -0 1 2 3 4

Voltage

• Q = CxV where C is a constant

-4 -3 -2 -1 -0 1 2 3 4

1nF Linear Capacitor

Voltage

Derivative of Linear Capacitance

• C is a constant slope so the derivative of linear capacitanceis simply a vertical offset on the nCV plot.

Capacitance vs Frequency• Capacitance is about separation of charge!

– Electrons are fast (light speed!).– Atoms are slow!– Domains are real slow!

Capacitance

Domains

Nuclei

Electrons Frequency

Non-linear Capacitance

• When the electric field begins to move atoms in the lattice,the lattice stretches, changing its spring constant.Capacitance goes down.

-10.0 -7.5 -5.0 -2.5 0.0 2.5 5.0 7.5 10.0

Radiant 9/65/35 PLZT[ 1700A ]

The Derivative

A non-linear capacitor has decreasing capacitance as the appliedvoltage increases.

-10.0 -7.5 -5.0 -2.5 0.0 2.5 5.0 7.5 10.0

nCV of 9/65/35 PLZTuF

Linear vs. Non-linear Capacitance

This device has both linear and non-linear capacitance. The linearcapacitance is the vertical offset of the nCV plot. The tips do nottouch zero.

-10.0 -7.5 -5.0 -2.5 0.0 2.5 5.0 7.5 10.0

nCV of 9/65/35 PLZTuF

Remanent Polarization• The PUND test is a familiar measurement:

• Any matched pair of switched and non-switched pulses may besubtracted from each other to get the remanent polarization.

Drive Voltage

PresetPulse

DelayPeriod

±Vmax

PositiveSwitched

PositiveUnswitched

NegativeSwitched

NegativeUnswitched

Remanent Hysteresis• The same measurement may be made using half-hysteresis

loops instead of pulses:

• The difference between the switching and non-switching measurementswill give the Remanent Polarization vs Voltage function.

Switching Non-switching

1/2Period

Switching and Non-switching half loops:Switching & Non-switching Loops

0 1 2 3 4 5Volts

SwitchingNon-switching

Remanent Hysteresis

• PUND: P*r - P^r = dP = Qswitched• Hysteresis: Switching - Non-switching = Remanence:

Remanent Hysteresis Calculation

0 1 2 3 4 5Volts

SwitchingDifferenceNon-Switching

RemanentHalf Loop

Remanent Hysteresis

• The test may be executed in both voltage directions and the two halvesjoined to show the switching of the remanent polarization that takesplace inside the full loop.

Remanent Hysteresis

-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6

Remanent Hysteresis[ Type AB WHITE ]

Voltage

Unswitched - Logic 0 Switched - Logic 1 Remanent

• The first stage of the experiment consisted of measuring two 4Vhysteresis loops going in opposite directions (including their gaps) and a4V remanent polarization loop.

Remanent vs. Normal Hysteresis

-4 -3 -2 -1 0 1 2 3 4

1 Second Hyst vs 1 second Rhyst[ Radiant Type AB White, 9V preset ]

Voltage

+4V 1s Rhyst: Polarization (µC/cm2) -4V 1s Hyst: Polarization (µC/cm2)

+4V 1s Hyst: Polarization (µC/cm2)

• The remanenthysteresis is in blue.

• The full loops inopposite directionsoverlay exactly.

• The Vc of theremanent loop liesoutside that of thenormal loops. Why?(Hint: the reason ispurely mathematical.)

• The Vc of theremanent loop is thetrue Vc.

The Derivative

• The nCV of the remanent polarization loop rests on the X-axis because it has no capacitance on the re-trace.

-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6

nCV of Remanent Hysteresis Loop[ Type AB ]

Voltage

Remanent

The Perfect Capacitor• A perfect capacitor combines non-linear capacitance with

remanent polarization.

-30 -25 -20 -15 -10 -5 0 5 10 15 20 25 30

Remanent Hysteresis[ 1u 4/20/80 PNZT ]

Voltage

Switched - Logic 1 Remanent

Hysteresis in Small SignalCapacitance

• The small signal capacitance versus bias voltage isdetermined by measuring the sample capacitance with alow amplitude signal at a series of bias voltages.

– Theoretically, the signal amplitude should be small enough that itdoes not disturb the state of the capacitor.

• While this is a noble effort, it cannot be ignored that theremanent polarization modulates the small signalcapacitance.

• The state of the remanent polarization must be managedduring measurements of small signal capacitance.

Radiant Technologies, Inc. 28

Small Signal vs Large Signal

• The ferroelectric hysteresis measurement is defined at Radiant as a“large signal” measurement of the polarization properties of the sample.

• “Large signal” means that the test waveform has a large enoughamplitude to switch dipoles in the ferroelectric material.

• As well, the “large signal” measurement captures and integrates allchanges the sample experiences during the test waveform, showing itsentire trajectory.

• The measurement result contains contributions from all components ofthe sample, including the remanent polarization and parasitics.

• A “large signal” measurement captures every electron that moves intoor out of the capacitor during the stimulus waveform.

Integrate theentire signal

Small Signal vs Large Signal• The “small signal” measurement is defined as one where the test

amplitude is small compared to that required to switch remanentpolarization in a ferroelectric capacitor.

• Since the response of a non-linear sample changes with the absolutevalue of the voltage applied and the remanent polarization state, the“small signal” measurement must also have a steady state voltagecomponent as well as a remanent polarization pre-set procedure to putthe sample in the appropriate state.

• Therefore, the “small signal” measurement captures and integrates onlythose changes the sample experiences during a small amplitudestimulation of the sample at a specified voltage and polarization state.

By definition, the “small signal” measurement contains no contributionfrom switching dipoles!

• In “small signal” measurements, many small measurements are taken thatcapture only the small changes associated with small stimuli.

• In a “small signal” measurement, the sequence of DC bias values is the same asthe voltage profile used for hysteresis so the two can be compared directly.

Integration periods.

Small Signal vs Large Signal• Radiant testers execute both standard “large signal” hysteresis and

“small signal” capacitance measurements.

– “large signal” hysteresis results are normally given in units ofpolarization (µC/cm2) but can be converted to capacitance usingthe CV or Normalized CV plotting functions of the HysteresisTask or the Hysteresis Filter.

– “small signal” measurements are normally given in units ofcapacitance (nF or µF/cm2) but can be converted to equivalentpolarization using the appropriate plotting function of theAdvanced CV measurement task.

Small Signal vs Large Signal• Comparison of the Hysteresis and Polarization of the Small Signal

Capacitance is shown below:

Large and Small Signal Polarization

-4 -2 0 2 4

Vdrive

Hysteresis SSAC

100ux100u900A 20/80Pt/PZT/Pt

Hysteresis = 1KHzSSAC = 4KHz

Small Signal vs Large Signal• Comparison of the Large and and Small Signal Capacitance is shown

below:

Large and Small Signal Capacitance Density

102030405060708090

-4 -3 -2 -1 0 1 2 3 4Vdrive

HysteresisSSAC

100ux100u900A 20/80Pt/PZT/Pt

Hysteresis = 1KHzSSAC = 4KHz

Stimulus

Switching

Stimulus

Non-switching

Stimulus

Non-switching

Stimulus

Switching

• 1KHz 0.2V test with 182 points

Non-switching CV for the Sampleunder Test

-4 -3 -2 -1 0 1 2 3 4

Non-switching Small Signal CV[ Radiant Type AB WHITE ]

Voltage

Switching CV for the Sampleunder Test

-4 -3 -2 -1 0 1 2 3 4

Switching Small Signal CV[ Radiant Type AB WHITE ]

Voltage

Non-switching vs Switching CV

-4 -3 -2 -1 0 1 2 3 4

1KHz SW vs nSW CV[ Radiant Type AB White, 9V preset ]

1ms 4V CV nSW: Capacitance (nF) 1ms 4V CV SW: Capacitance (nF)

Q vs V from Small SignalCapacitance

• The small signal capacitance can be multiplied bythe dV to get the dQ per test step.

• The dQs may be integrated to see the polarizationhysteresis contributed by the modulation of smallsignal capacitance by remanent polarization!

Small Signal CapacitancePolarization

• Small signal capacitance forms a hysteresis of its own.

Small Signal CapacitancePolarization

• The contribution of small signal capacitance hysteresis to the overallloop is small in this case.

Resistive Leakage in a HysteresisLoop

A common problem in ferroelectric ceramics is alinear resistance function. Usually, it appears due toleakage along grain boundaries although it can occurfrom dopants or defects in the grains themselves.

Linear resistance is easy for a triangle wave: ∆P=(Current• ∆time)/Area∴P=(Σk

n=0 n • ∆V/R • ∆t)/Area∆time=time step per point

∆V=fixed voltage step oftriangle wave

n = point number of digitized triangle wave

Result = “Football”

PLT_F DC Leakage Response

-0.004

-0.003

-0.002

-0.001

-5 -4 -3 -2 -1 0 1 2 3 4 5

Vdrive

PLT_F Leakage

1200Å 4/20/80 PNZT

The derivative of pure resistive leakage is an “X”.

-4 -3 -2 -1 -0 1 2 3 4

Hysteresis of Linear Resistor[ 2.5Mohm 4V 1ms ]

Voltage

-4 -3 -2 -1 -0 1 2 3 4

nCV of Linear Resistance[ 2.5Mohm 4V 1ms ]

Voltage

IV Test

• The IV, or Current vs Voltage, test is a series of leakage testsexecuted over the voltage profile used for the traditionalhysteresis loop.

• It is necessary because resistive leakage is not always linear!

tThe red line indicatesthe period during whichthe leakage through thesample is measuredafter a soak period.

Hysteresis in Leakage• Leakage in ferroelectric materials does not have to be linear.• Leakage can have its own hysteresis modulated by remanent

polarization.

-4 -3 -2 -1 0 1 2 3 4

Switched vs Unswitched 1s IV[ Radiant Type AB BLUE ]

4V 1s nSW IV: Current (Amps) 4V 1s SW IV: Current (Amps)

Leakage vs CV vs RemanentPolarization

Hysteresis Parameters

-6 -4 -2 0 2 4 6

SW CV*10

nSW CV*10

SW IV*2.5

nSW IV*2.5

• Are the remanent polarization, IV, and CV related? YES!

Leakage vs CV vs RemanentPolarization

nCV Parameters

-6 -4 -2 0 2 4 6Volts

SW CV*10

nSW CV*10

SW IV*2.5

nSW IV*2.5

RnCV/4

• The derivative of the remanent polarization makes therelationships clear.

• If remanent polarization saturates nicely, why is the fullloop (both switching and non-switching) open at the end?

Remanent Hysteresis Calculation

0 1 2 3 4 5Volts

SwitchingDifferenceNon-Switching

Saturated

The Gap

Not saturated

Something Left OverIf we measure the remanent polarization, small signalcapacitance, and leakage and then subtract them from the fullloop, something is left over:

This is the “gap” in the hysteresis loop!

PLT_F Left Overs!

-5 -4 -3 -2 -1 0 1 2 3 4 5

Vdrive

Measured - Mode

1200Å 4/20/80 PNZT

PLT_F Components vs Measured Hysteresis

-5 -4 -3 -2 -1 0 1 2 3 4 5

Vdrive

Full Hysteresis

1200Å 4/20/80 PNZT

Reversed Bias Diodes• A platinum electrode based capacitor has two opposing

diodes at the ferroelectric/platinum interface, one of whichis always turned off.

• In reverse bias, a diode has a constant current independent of voltage.This makes a “bow tie” on the nCV plot.

1N4002 Diode Conduction

-2.0 -1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 2.0

Anode Voltage

1N4002Reverse Bias Conduction

Forward BiasConduction

C(V) / A

Translated toC(V) plot

Reversed Bias Diodes• A pair of back to back diodes have a unique signature on a

virtual ground test system which uses a triangular drivevoltage.

Reversed Bias Diode Breakdown• The derivative of a polarization hysteresis loop clearly

shows the diode reverse-biased leakage if it is present.

• The leakage of diode reverse-biased breakdown is marked byexponentially increasing current. This produces a “trumpet flare”instead of the “X” of linear leakage.

RN101B at 14V with Vdrive = Top Electrode

-15 -12 -9 -6 -3 0 3 6 9 12 15

Vdrive

14V RN101B

Beginning of Low CurrentBreakdownDiode Leakage

The Components• Remanent polarization• Linear small signal capacitance (dielectric constant)• Nonlinear small signal capacitance (dielectric constant)• Hysteretic small signal capacitance (remanent polarization

modulation)• Linear resistive leakage• Hysteretic resistive leakage• Electrode diode reverse-biased leakage• Electrode diode reverse-biased exponential breakdown

All of these components are visible in the derivative of thepolarization hysteresis loop!

What is this?

Let’s analyze some capacitors!

• The nearly “perfect” loop. 20/80 PZT on platinum.

-7.5 -5.0 -2.5 0.0 2.5 5.0 7.5

Voltage

• The 20/80 PZT on platinum is so square that the instantaneouscapacitance increases by x250 or more during switching.

-7.5 -5.0 -2.5 0.0 2.5 5.0 7.5

20/80 PZT on Platinum[ 0.26u thick ]

Voltage

What is this?

• Is this loop as good as the previous loop? Yes! It is 4/20/80PNZT, a different composition from 20/80 PZT. So, it has adifferent shape.

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

Voltage

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

4/20/80 PNZT[ 1u thick ]

Voltage

What is this?

• This loop is good for 4/20/80 PNZT. Note the extra “diode”leakage in the tails that make the saturated tips of the loop openup.

What is this?

• A linear capacitor.

-4 -3 -2 -1 -0 1 2 3 4

Voltage

-4 -3 -2 -1 -0 1 2 3 4

1nF Linear Capacitor

Voltage

What is this?

• A linear capacitor. In this measurement, it has acapacitance density of 10µF/cm2.

A Harder One

-4 -3 -2 -1 -0 1 2 3 4

Voltage

• Quite a few papers include loops that look like this.

A Harder One

• It is only a resistor and a linear capacitor in parallel.

-4 -3 -2 -1 -0 1 2 3 4

Linear Resistor || Linear CapacitorN

Voltage

-7.5 -5.0 -2.5 0.0 2.5 5.0 7.5

Voltage

• Yes, it is! See the ferroelectric switching peaks stickingout of the resistor “X”.

-7.5 -5.0 -2.5 0.0 2.5 5.0 7.5

Ferroelectric Capacitor || Linear Resistor[ Test Period = 2 seconds ]

Voltage

What Happened Here?

-7.5 -5.0 -2.5 0.0 2.5 5.0 7.5 10.0

Voltage

What Happened Here?

• Different electrodes on each interface means a different switchingcharacteristic with direction. No linear leakage but classic back-to-back diodeleakage. Surface diode breakdown at one of the electrode/ferroelectricinterfaces.

-15-10-505

101520253035

-7.5 -5.0 -2.5 0.0 2.5 5.0 7.5 10.0

PZT on Nickel Lanthanate - 300ms Period[ EXP09BQ Rev A ]

Voltage

Gotcha!

• What is this??? Is it some kind of breakdown???

-5 -4 -3 -2 -1 0 1 2 3 4 5

Voltage

Hyst 100ms: Polarization (µC/cm2): 2

Partial Switching!

• It is a sub-saturated loop which can sometimes look like breakdown.

-5 -4 -3 -2 -1 0 1 2 3 4 5

Nested Loops[ LSCO/PNAT/LSCO ]

Voltage

Hyst 100ms: Polarization (µC/cm2): 2 Hyst 100ms: Polarization (µC/cm2): 4

Partial Switching!

• Here are the hysteresis loops.

-5 -4 -3 -2 -1 0 1 2 3 4 5

Nested Loops[ LSCO/PNZT/LSCO ]

Voltage

Hyst 100ms: Polarization (µC/cm2): 2 Hyst 100ms: Polarization (µC/cm2): 4

Triangle Wave• All of the modeling described above is dependent upon

using a triangle wave to stimulate the sample.

• ∆V/ ∆t is constant.

nCV = ∆Q/ ∆V

I = ∆Q/ ∆t ≈ ∆Q /k ∆V = k x nCV

Conclusion of Components• Geometry is everything, well almost.

• The ferroelectric hysteresis loop may be broken down intoindependent components.

• The mathematical derivative of the PE loop is a tool thatallows identification by inspection of the componentscontributing to the response of the sample.

• Practice makes perfect.

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