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7

Ferroelectric Negative Capacitance Domain Dynamics

Michael Hoffmann,1, 2, ∗ Asif Islam Khan,3 Claudy Serrao,1 Zhongyuan Lu,1 Sayeef

Salahuddin,1, 4 Milan Pesic,2 Stefan Slesazeck,2 Uwe Schroeder,2 and Thomas Mikolajick2, 5

1Department of Electrical Engineering and Computer Sciences,University of California, Berkeley, CA 94720, USA

2NaMLab gGmbH, Noethnitzer Str. 64, D-01187, Dresden, Germany3School of Electrical and Computer Engineering,

Georgia Institute of Technology, Atlanta, GA 30332, USA4Material Science Division, Lawrence Berkeley National Laboratory, Berkeley, CA 94720, USA

5Chair of Nanoelectronic Materials, TU Dresden, D-01062 Dresden, Germany

Transient negative capacitance effects in epitaxial ferroelectric Pb(Zr0.2Ti0.8)O3 capacitors areinvestigated with a focus on the dynamical switching behavior governed by domain nucleationand growth. Voltage pulses are applied to a series connection of the ferroelectric capacitor anda resistor to directly measure the ferroelectric negative capacitance during switching. A time-dependent Ginzburg-Landau approach is used to investigate the underlying domain dynamics. Thetransient negative capacitance is shown to originate from reverse domain nucleation and unrestricteddomain growth. However, with the onset of domain coalescence, the capacitance becomes positiveagain. The persistence of the negative capacitance state is therefore limited by the speed of domainwall motion. By changing the applied electric field, capacitor area or external resistance, thisdomain wall velocity can be varied predictably over several orders of magnitude. Additionally,detailed insights into the intrinsic material properties of the ferroelectric are obtainable throughthese measurements. A new method for reliable extraction of the average negative capacitance ofthe ferroelectric is presented. Furthermore, a simple analytical model is developed, which accuratelydescribes the negative capacitance transient time as a function of the material properties and theexperimental boundary conditions.

I. INTRODUCTION

To further continue the remarkable trend of miniatur-ization of nanoelectronic devices in integrated circuits,fundamental power density constraints have to be over-come in the near future.1 These physical limits stem fromthe thermal broadening of Fermi-Dirac statistics in anydevice using the injection of electrons over an energy bar-rier as the operation principle. Therefore, the ultimatelower limit of how much voltage is needed to change thecurrent in such devices by one order of magnitude is givenby ln(10)kBT/q, where kB is the Boltzmann constant, Tis the temperature and q is the elementary charge.2

In 2008, Salahuddin and Datta proposed to use nega-tive capacitance (NC) in a ferroelectric material to over-come this sometimes called ”Boltzmann tyranny”,2 byusing the ferroelectric as an insulating layer betweenthe controlling gate electrode and the semiconductorchannel.3 This new physical property of NC was pre-dicted from Landau-Devonshire theory of ferroelectricphase transitions and was already indirectly confirmed inseveral different ferroelectric-dielectric heterostructuresand superlattices.4–8 Recently, first direct measurementsof NC in ferroelectric capacitors in series with a resistorhave been reported,9,10 opening up new ways to charac-terize and investigate this novel physical phenomenon,which is to some degree still controversially discussedin the literature.11–13 This cautious attitude towardsthe topic is comprehensible, since there is a consider-able lack of understanding of the physics of NC in gen-eral. This is especially troubling, considering first NC

devices overcoming the Boltzmann limit have been re-ported recently.14–17 To further improve these promisinginitial results, a more detailed understanding of the un-derlying physics and more reliable modeling approacheswill be crucial.

While almost all modeling efforts of NC up to now haveapplied homogeneous single-domain Landau-Khalatnikovtheory, the more subtle but critical influence of domainformation has only been discussed in a very limited man-ner so far.7,11,18 However, first reports on domain dy-namics in ferroelectric capacitors during switching arebeginning to emerge.10,19,20 Understanding the influenceof ferroelectric domain formation and dynamics on NCeffects will be critical for the utilization in actual devices.NC voltage transients were measured on high quality epi-taxially grown ferroelectric lead zirconate titanate (PZT)capacitors connected in series to a resistor as shown inthe schematic in Fig. 1. It will be elucidated how ferro-electric material properties like negative capacitance perarea, activation field and internal loss can be extractedfrom such measurements using a newly developed ana-lytical model.

II. EXPERIMENTAL SETUP

Ferroelectric thin films of Pb(Zr0.2Ti0.8)O3 (PZT) weregrown epitaxially using pulsed laser deposition on metal-lic SrRuO3 (SRO) buffered SrTiO3 (001) substrates. The100 nm PZT and the 20 nm SRO films were depositedat 720 C and 630 C, respectively. During the growth,

2

Pulse

generatorOscilloscope

Exte

rnal

resis

tor

R

Channe

l 1 (

VF)

Channe

l 2 (

VS)

SRO bottom electrodeFerroelectric PZT

STO substrate

Au top electrode

FIG. 1. Schematic sample structure and measurement cir-cuit applied to directly measure negative capacitance voltagetransients via an oscilloscope.

the oxygen partial pressure was kept at 100 mTorr andafterwards the heterostructure was slowly cooled down at1 atm of oxygen partial pressure and a rate of -5 C/minto room temperature. Laser pulses of 100 mJ of energyand ∼4 mm2 of spot size were used to ablate the targets.Ti/Au top electrodes were ex-situ deposited by e-beamevaporation and then patterned using standard litho-graphic techniques into square dots an area of (10 µm)2

to (50 µm)2. The SRO layer was used as the bottomelectrode.The schematic measurement setup is shown in Fig. 1.

An external resistor (R = 200 Ω to 200 kΩ) is connectedto the top electrode of the ferroelectric capacitor. An Ag-ilent 81150A pulse function generator was used to applysquare voltage pulses (with rise and fall times of 50 ns)to the series connection of the ferroelectric capacitor andthe resistor. The voltages of the pulse generator (VS) andacross the ferroelectric (VF ) were directly probed with adigital oscilloscope (GW Instek GDS-3354). The para-sitic parallel capacitance Cp = 35 pF of the measure-ment setup was determined by measuring the chargingtime constant τ = RCp, with the ferroelectric capaci-tor disconnected. All measurements were carried out atroom temperature.

III. MODELING OF FERROELECTRIC

DOMAIN DYNAMICS

A. Time-dependent Ginzburg-Landau Theory of

Ferroelectric Phase Transitions

According to the Landau-Ginzburg-Devonshire theoryof ferroelectric phase transitions,21 the Gibbs free energyper volume u of a ferroelectric material can be expandedin terms of an order parameter, which is the electric po-larization P , as

u = αP 2 + βP 4 + γP 6 − EP + k(∇P )2, (1)

where α, β and γ are the ferroelectric anisotropy con-stants, E is the electric field and k is the domain couplingconstant, which is related to spatial non-uniformity of P .While the −EP term in Eq. (1) describes the electro-static energy, the Ginzburg term k(∇P )2 accounts forthe energy penalty for spatial variations of P and there-fore also includes domain wall energy contributions. Inferroelectrics, α is always negative below the Curie tem-perature TC , which is the origin of the NC behavior inthis class of materials. For second-order phase transi-tions, β > 0 and γ = 0. On the other hand, for the morecommon case of first-order phase transitions, β < 0 andγ > 0.When the dynamical evolution of P is of interest, the

most straight forward approach is to apply the Landau-Khalatnikov equation22

−∂u

∂P= ρ

∂P

∂t, (2)

where ρ is a damping parameter of unit [Ωm], whichis a measure of the loss in the material. It should benoted that Eq. (2) does not describe the intrinsic dynam-ics during ferroelectric switching, but is only a diffusivelimit approximation.23 In this limit, the dynamics of thesystem is described by the domain wall velocity, whichdepends on the parameter ρ. For investigations beyondthe diffusive limit approximation of Eq. (2), moleculardynamics simulations would have to be carried out. How-ever, this is beyond the scope of this work.Eq. (1) in conjunction with Eq. (2) is es-

sentially identical to the time-dependent Ginzburg-Landau formulation which is known from the theory ofsuperconductivity24 and is commonly used in ferroelec-tric phase field modeling.25,26 In the following, the nu-merical implementation of the time-dependent Ginzburg-Landau model used for simulation of ferroelectric NCvoltage transients is described.

B. Numerical Implementation

In order to keep the model as simple as possible with-out losing its descriptive capabilities, a few assumptionsare made based on the experimental boundary condi-tions. Since ferroelectric capacitors of 100 nm thicknesswith metallic electrodes are investigated, depolarizationfields due to imperfect screening at the electrode inter-faces should play no significant role and are therefore ne-glected here. Furthermore, only the out-of-plane (z-axis)contributions to the polarization of the ferroelectric areconsidered, which is reasonable since the measured PZTfilms are c-axis oriented and switched through the appli-cation of rather high electric fields (E ≥ 0.4 MV/cm).

3

P can vary in the x-y plane of the capacitor. Therefore,only 180 domain walls can be modeled. While ferroe-lastic 90 domain wall are known to exist in PZT thinfilms,25,27 their effect on transient NC seems to be neg-ligible in our experiments, since films of 100 nm thick-ness and below show almost no 90 domain walls due tostrain imposed by the substrate.28 Other effects relatedto strain, e.g. polarization-strain coupling are not con-sidered in our model, see Eq. (1).Spatial derivatives of P are calculated by a finite dif-

ference method on a uniform square grid in the x-y plane.The spatial difference between adjacent points in x- andy-direction are identical, i.e. ∆x = ∆y. Each point ion this grid has a local energy density ui, polarizationPi and internal loss ρi. In accordance with Eq. (1), thelocal energy density at point i of the ferroelectric is thengiven by

ui = αiP2i +βiP

4i +γiP

6i −EPi+k

∑

j

(

Pi − Pj

∆x

)2

, (3)

where αi, βi and γi are the local anisotropy constants,∆x is the grid spacing and the index j denotes all cellsdirectly adjacent to cell i. While cells in the middle ofthe grid have four nearest neighbors (left, right, aboveand below), cells at the edge or at the corner of the gridhave three and two nearest neighbors, respectively, cor-responding to open boundary conditions at the capacitoredge. To calculate the total charge on the ferroelectriccapacitor QF , the individual contributions of each cellhave to be summed up as

QF = A

(

ǫ0E +1

N

N∑

i=1

Pi

)

≈A

N

N∑

i=1

Pi. (4)

Here, A is the total area of the ferroelectric capacitor,ǫ0 is the vacuum permittivity and N is the total numberof cells. The approximation in Eq. (4) is reasonable,because P >> ǫ0E in almost all ferroelectrics. Since∆x = ∆y, the area of each cell is given by A/N = (∆x)2

and therefore ∆x =√

A/N .The effective field Eeff in the ferroelectric is given by

Eeff =VF

tF− Ebias, (5)

where VF is the voltage across the ferroelectric capaci-tor, tF is the thickness of the ferroelectric and Ebias is aninternal bias field, which might be induced by electrodeswith different work functions or inhomogeneous distribu-tion of charges along the z-axis in the ferroelectric.29–31

The connection to the external circuit as seen in Fig. 1is now easily defined by Kirchhoff’s voltage and currentlaws. The voltage defined by the pulse generator VS canalso be written as

VS = iRR+ VF , (6)

where iR is the current flowing through the externalresistor R. Since there is also a parasitic capacitanceCp related to the measurement setup in parallel to theferroelectric, iR can be expressed as

iR = iF + iCp =∂QF

∂t+ Cp

∂VF

∂t. (7)

Here, iF is the total ferroelectric current and iCp is thecurrent through the parasitic capacitor Cp. The com-plete dynamic behavior of the circuit in the diffusivelimit approximation of the ferroelectric is described byEqs. (2-7). Leakage currents through the ferroelectricare neglected here, because their effect on NC voltagetransients was shown to be minimal as long as they aremuch lower compared to iR.

10

To account for any non-uniformity in the ferroelectriclayer, spatial distributions of the parameters α, β, γ, ρand Ebias can be defined. This approach can emulatedifferent nucleation times for reverse domain formationas well as general defect driven property variations in theferroelectric.

IV. RESULTS AND DISCUSSION

In this section, first, typical measured NC voltage tran-sients are presented to define characteristic NC param-eters directly obtainable from the experiment. Time-dependent Ginzburg-Landau simulations will then be fit-ted to this experimental data, to gain insights into the do-main switching dynamics during the NC transient. Sub-sequently, the effect of experimental measurement condi-tions on the NC transients will be investigated. Initially,the influence of applied and internal electric fields in theferroelectric will be discussed. Afterwards, the effect ofthe capacitor area and the external resistor on the NCtransients are examined. Furthermore, it will be shownthat a simple analytical model can be developed to ex-tract intrinsic material parameters from these measure-ments, namely the average NC per area, the activationfield and the average internal loss.

A. Characterizing Ferroelectric Negative

Capacitance Voltage Transients

An exemplary NC voltage transient of a PZT capacitorwith A = (50 µm)2, an external resistance R = 3360 Ωand a maximum voltage amplitude VS,max = 5 V isshown in Fig. 2. Initially, before the pulse VS shownin Fig. 2(a) is applied, the ferroelectric capacitor is com-pletely poled into a state where the spontaneous polar-ization vector points from the bottom towards the top

4

0 10 20 30 40 50 60-6

-4

-2

0

2

4

6

5 6 7 8 9 10

3.2

3.4

3.6

3.8

4.0

30.5 31.0 31.5 32.0 32.5-2.6

-2.4

-2.2

-2.0

(c)(b)

(c)

Volta

ge (V

)

Time (µs)

VF

VS

(b)

(a)

V F (

V) tpmin

Vpmin

Time (µs)

Vpmax

Vp =

Vpm

ax -

Vpm

in

Tp = tpmin - tpmax

tpmax

V F (V)

tnmax

tnmin

Vnmin

Vnmax

Vn =

Vnm

ax -

Vnm

in

Tn = tnmin - tnmax

Time (µs)

FIG. 2. (a) Voltage transients measured across the pulse generator VS and the ferroelectric VF for a capacitor with A = (50 µm)2

and R = 3360 Ω. Magnifications of the time frames where the ferroelectric capacitance is negative are shown in (b) for positiveVS and in (c) negative VS.

-0.4 -0.2 0.0 0.2 0.4

-75

-50

-25

0

25

50

75

P (µ

C/c

m2 )

E (MV/cm)

Negativecapacitance regions

FIG. 3. Polarization-electric field hysteresis for a capacitorwith A = (50 µm)2 and R = 3360 Ω calculated from negativecapacitance voltage transients shown in Fig. 2.

electrode. Therefore, when applying a positive voltagepulse VS , the voltage VF initially increases, until at acertain voltage VF = Vpmax the polarization will be-gin to switch. A magnification of this time frame isshown in Fig. 2(b). During switching, for a certain timetpmax < t < tpmin, the voltage across the ferroelectric de-creases. As was pointed out by Khan et al., this behaviorcan be interpreted as the ferroelectric possessing a neg-ative differential capacitance, since VF decreases whileQF increases and the ferroelectric capacitance, definedas CF = dQF /dVF , is negative in this case.9

At t = tpmin, the voltage VF shows a minimum Vpmin

after which VF again increases with time and thereforeCF > 0. After the switching and charging of the ferro-

electric is completed, no current iR is flowing anymoreand VS = VF . However, when the applied voltage VS isreversed to -5 V, a similar NC voltage transient can be ob-served during the back-switching process. A zoom-in onthis second time frame tnmax < t < tnmin where CF < 0is shown in Fig. 2(c). Analogous to the previous case, theNC transient begins at VF = Vnmax and terminates atVF = Vnmin. Such transient behavior during switchingin BaTiO3 was actually already experimentally reportedby Merz in 1954.32 However, it was not interpreted as atransient NC at that time. Furthermore, if VS(t), VF (t),R and Cp are known, the charge of the ferroelectric canbe calculated as10

QF = QF0 +1

R

t∫

0

[VS(t′)− VF (t

′)] dt′ − VFCp. (8)

Here, QF0 = QF (t = 0) is the initial charge of the fer-roelectric. From Eq. (8), we can now calculate the po-larization P ≈ QF /A and plot the P -E hysteresis of theferroelectric, which is shown in Fig. 3. The loop is notcompletely closed due to the presence of leakage currentsfor positive applied fields. The NC regions during switch-ing can be directly seen from the gray shaded areas in Fig.3, where dP/dE < 0. These regions are predicted by theS-shaped P -E relationship which is implicitly defined byLandau-Devonshire theory and was already predicted inseveral studies over a decade ago.33,34 However, it is clearthat there is still hysteresis, because the NC regions arenot stabilized in this isolated capacitor configuration andthus the NC behavior is only transient in nature. To sta-bilize NC, a ferroelectric-dielectric bilayer structure asreported before is necessary,4,7 otherwise this effect willalways be hysteretic because of spinodal decompositioninto a multi-domain state.35 The relationship between

5

4 6 8 10 12 143.0

3.5

4.0

4.5

5.0

Sim. Meas.

VF

VS

x (µm)

(f)

(e)(d)(c)

Vol

tage

(V)

Time (µs)

(b)

0 10 20 30 40 500

10

20

30

40

50

x (µm)

y (µ

m) -0.7000

-0.6000-0.5000-0.4000-0.3000-0.2000-0.10000.0000.10000.20000.30000.40000.50000.60000.70000.80000.9000

P (C/m2)

0 10 20 30 40 500

10

20

30

40

50

x (µm)

y (µ

m)

0 10 20 30 40 500

10

20

30

40

50

x (µm)

y (µ

m)

0 10 20 30 40 500

10

20

30

40

50

x (µm)

y (µ

m)

0 10 20 30 40 500

10

20

30

40

50

(a)

(f)(e)(d)

(c)(b)(a)

y (µ

m)

FIG. 4. (a) Simulated voltage transients across the ferroelectric VF in response to the applied voltage VS. (b)-(f) Spatialpolarization distribution in the ferroelectric during switching for t = 6 µs, 7 µs, 9 µs, 11 µs and 13 µs, respectively.

domain formation and hysteresis will become more ap-parent through the simulation results in the followingsection.

B. Simulating Negative Capacitance Domain

Dynamics

To understand the domain dynamics during theNC voltage transients in more detail, time-dependentGinzburg-Landau simulations were carried out and sim-ulation parameters (α, β, γ, ρ, Ebias and k) were fittedto the experimental data shown in Fig. 2. A good agree-ment was achieved by using the parameter distributionslisted in Table I. For α, β, γ, and Ebias, Gaussian distri-butions with a standard deviation σ of 7.5 % with respectto the mean values µ were used. The internal loss wasassumed to be uniformly distributed with the minimumbeing ρmin = 0.4 Ωm and a maximum of ρmax = 13 Ωm.The domain wall coupling constant k was fitted to bek = 2.5× 10−7 m3/F.

The resulting simulated NC voltage transient for VS >0 is shown in Fig. 4(a). It can be seen that a very similartransient NC behavior can be observed when comparedto the experimental data from Fig. 2(b). Only during theinitial charging and switching (t < 7 µs) of the capacitor,

TABLE I. Parameters fitted from the experimental data,which were used for time-dependent Ginzburg-Landau sim-ulations. α, β, γ and Ebias are normally distributed, ρ isuniformly distributed.

µ σ unit

α −4.4 × 107−3.3 × 106 m/F

β −9.1 × 106−6.8 × 105 m5/(C2F)

γ 7.4 × 107 5.6 × 106 m9/(C4F)Ebias 7.2 × 106 5.4 × 105 V/m

value unit

k 2.5 × 10−7 m3/Fρmin 0.4 Ωmρmax 13 Ωm

there are some deviations between simulation and mea-surement: The change of VF with time is slightly steeperand the maximum of VF appears earlier in the measure-ment compared to the simulation, which shows that theinitial switching actually happens slightly faster than ourmodel was capable of reproducing. The reason for this isthe diffusive limit approximation of Eq. (2), which can-not quantitatively predict the intrinsic dynamics duringinitial switching.

For t > 7 µs on the other hand, excellent agreementbetween data and theory was achieved, since here the

6

switching dynamics can be accurately described by aneffective domain wall velocity. Furthermore, in Fig. 4(b)-(f) the spatial polarization distribution across the capac-itor area is shown as contour plots for different points intime during switching. At the beginning of the switch-ing process (t = 6 µs), Fig. 4(b) shows that first reversedomains nucleate randomly distributed across the capac-itor area. Going further in time and looking at Fig. 4(c),the effective capacitance of the ferroelectric is still neg-ative and some indication of the origin can be extractedfrom the spatial polarization distribution: While somenew reverse nucleation sites are still forming, already nu-cleated domains can grow sideways almost unimpeded.However, the negative slope of dVF /dt is beginning toflatten out. This can be understood from the shrinkingarea that is available for reverse domain nucleation andunrestricted growth which is known from Kolmogorov-Avrami-Ishibashi (KAI) switching dynamics in epitaxialPZT thin films and single crystals.36 Therefore, at a cer-tain point in time (t = 9 µs), VF will stop to decreaseand start increasing with time again. From the contourplot in Fig. 4(d) it can be seen that at this point in time,about half of the capacitor area is already switched andunrestricted domain wall motion is mostly replaced bydomain coalescence. In Fig. 4(e) and (f), more domainscoalesce while less area of the ferroelectric capacitor isswitching which results in an increase of VF with time,which is expected for charging of a positive capacitor.

This shows, that in case of A = (50 µm)2 and R =3360 Ω, the NC transient times ∆Tp,n in these epitaxialPZT films are mostly limited by the domain wall veloc-ity and not by the initial domain nucleation. However,most of the change of VF during the initial NC tran-sient seems to happen during this initial reverse nucle-ation process. Very similar reverse domain nucleationand growth processes in Pb(Zr0.2Ti0.8)O3 were shown toalso behave according to Merz’s law by atomic force mi-croscopy experiments.37 From these results and also vis-ible in Fig. 4, it is apparent that such transient NCeffects will always be hysteretic due to the underlyingirreversible switching processes. For example, when theferroelectric is in the NC state, see Fig. 4(b)-(d), ap-plying a large negative voltage will not continue the NCeffect in the opposite direction, because there is alreadycoalescence of the -Ps (blue) domains happening in thisstate, thus the capacitance will be positive. Therefore,the whole ferroelectric has to be switched to the oppo-site polarization state before another NC transient can bemeasured. This will always result in a large P -E hystere-sis in such capacitors. This also implies, that transientNC cannot be used for small-signal voltage amplification.Therefore, transient NC effects in ferroelectric capacitorshave rather different underlying physics compared to sta-bilized NC in a ferroelectric-dielectric bilayer or super-lattice, where the stabilization of a small-signal NC wasshown to be possible.7 However, also this stabilized NCseems to stem from a complex domain structure result-ing in a negative contribution to the permittivity due to

domain wall motion.In a next step, the influence of the electric field in the

ferroelectric on the NC switching times ∆Tp and ∆Tn,will be investigated.

C. Influence of the Applied Voltage

Since the NC effect in epitaxial PZT seems to be aresult of domain wall motion, the effect of different ap-plied voltages VS was investigated. Therefore, capacitorswith A = (35 µm)2 were connected to a series resistorR = 10 kΩ and the amplitude of VS was changed, whilethe duration of the voltage pulses was kept constant. Itis intuitively expected, that by applying a larger voltagepulse VS to the series connection, the NC transient shouldget shorter, because then more charge can be supplied tothe capacitor in the same amount of time.In Fig. 5, the characteristic NC transient times ∆Tp,n

are plotted against 1/VS,max, which is the reciprocal ofthe maximum applied voltage VS . The reason for this isthe very linear relationship between ∆Tp,n and 1/VS,max

in this semi-logarithmic representation in Fig. 5. Thisapparent linearity is related to the electric field depen-dence of the domain wall velocity known empirically asMerz’s law:32

1

vDW

∝ τsw = τ0exp

(

Ea

E

)m

. (9)

Here, vDW is the domain wall velocity, τsw is the char-acteristic switching time, τ0 is the intrinsic switching timeconstant, Ea is the activation field andm is the exponent.In the initial work of Miller and Weinreich, this depen-dence was attributed to the nucleation of triangular stepsat the domain wall.38 However, this approach yielded ac-tivation fields an order of magnitude larger than exper-imentally observed.39 In the case where m = 1, the do-main wall motion is a creep process,40 which is ascribedto random field defects.41 From the activation field Ea

in Eq. (9), the domain wall energy density can be cal-culated when neglecting the depolarization energy of thereverse domain nucleus:42

Ea =cσ2

DW

PskBT. (10)

Here, c is the domain wall width, which can be ap-proximated as the in plane lattice constant, σDW is thedomain wall energy density and Ps is the spontaneouspolarization. Once we have obtained Ea from the exper-imental data, we will estimate σDW from Eq. (10).However, when looking at Fig. 5, we can see that

there are two different trends for positive and negativeNC transients, ∆Tp and ∆Tn, respectively. Since it wasalready established that our PZT capacitors exhibit aninternal bias field (tFEbias = 0.72 V), it makes sense to

7

0.05 0.10 0.15 0.20 0.25 0.30

10-6

10-5

Tp

Tn

T fit of T

T (p,n

) (s)

V-1 (V-1)

T = 0exp(tFEa/VS,max)Ea = 1.063 MV/cm

0 = 322.8 nsR2 = 0.9826

S,max

FIG. 5. Negative capacitance transient times with (∆T ) andwithout correction (∆Tp,n) for internal bias fields as a func-tion of VS,max with R = 10 kΩ and A = (35 µm)2. Solid lineshows analytical fit of ∆T .

correct this offset by plotting ∆Tp against 1/(VS,max −tFEbias) and ∆Tp against 1/(VS,max + tFEbias) as alsoshown in Fig. 5. As can be seen, after correcting ∆Tp

and ∆Tn for Ebias, all data points now lie on the sametrend line and are therefore labeled only ∆T .

Now, by fitting Eq. (9) to the experimental data (∆Tvs. V −1

S,max) shown in Fig. 5, we can estimate Ea andtherefore also σDW . In this case, we use the duration ofthe NC transient ∆T as the characteristic switching timeτsw with m = 1 and get τ0 = 322.8 ns and Ea = 1.063MV/cm. This value for the activation field Ea is invery good agreement with the results reported before(Ea ≈ 1 MV/cm).40,43 Using Eq. (10) with c = 3.95A44 and Ps = 0.7 C/m2, we can then calculate the do-main wall energy density as σDW = 27.93 mJ/m2. Thisvalue is rather low compared to ab initio calculations forPbTiO3 (σDW = 132 mJ/m2)45 and experimental valuesfor Pb(Zr0.2Ti0.8)O3 (σDW = 120 mJ/m2).46 The rea-son for this discrepancy might be a domain wall width c,which is larger than the lattice constant or an underes-timation due to the neglect of the depolarization energyin the derivation of Eq. (10).42 Additionally, ab initio

calculations were carried out only for T = 0 K.45

Nevertheless, the field independent switching time τ0 =322.8 ns, fitted to the data in Fig. 5, might also give usan insight into the intrinsic material parameters of thesample, like the average loss ρ and the average NC perarea CFE . For this, we will need an analytical modeland vary the circuit parameters like capacitor area A andexternal resistance R. Since the NC transient duration∆T is clearly limited by a domain wall creep process,we will investigate how the domain wall velocity can beprecisely controlled by limiting the charge supply through

the external circuit.

D. An Analytical Model for Transient Negative

Capacitance

From Eq. (9) we already know the field dependenceof ∆T and also the temperature dependence from Ea =f(T ) in Eq. (10). While this field and temperature de-pendence comes from the domain wall dynamics of theferroelectric itself, the external circuit will also stronglyinfluence the transient NC effect as reported before.9,10

Therefore, a simplified electrical circuit as shown in Fig.1 is considered, where the ferroelectric can be modeledaccording to the Landau-Khalatnikov equation (2) as anon-linear capacitor in series with a resistance ρtF /A,which describes the loss in ferroelectric. Since we arenow only interested in the influence of the external cir-cuitry, we neglect the Ginzburg term k(∇P )2 as well asany parasitic capacitance or resistances. Then we canwrite Kirchhoff’s voltage law

VS ≈dQF

dt(R + tF ρ/A) +

(

2αP + 4βP 3 + 6γP 5)

tF ,

(11)where dQF/dt is the current flowing through R and

ρtF /A and QF ≈ PA is the charge on the ferroelectriccapacitor. Therefore, we can write

VS ≈dP

dt(RA+ tF ρ)+

(

2αP + 4βP 3 + 6γP 5)

tF . (12)

Since we are interested in the change of P with time,we can transform Eq. (12) and get

dP

dt≈

VS −(

2αP + 4βP 3 + 6γP 5)

tF

RA+ tF ρ. (13)

From Eq. (13) it is now apparent that the time dtnecessary to change the polarization by dP is propor-tional to (RA + tF ρ). This shows that there is an in-trinsic term tF ρ and an extrinsic term RA limiting theswitching speed of the ferroelectric. The limited chargesupply through the external resistor might also help toexplain the strong dependence of the domain wall veloc-ity on the experimental measurement time, reported byother authors.39 This also shows that there are two dif-ferent NC switching regimes: First, if RA << tF ρ, thepolarization reversal is limited by the internal loss in theferroelectric material, and changing either R or A willnot significantly increase the switching time. This canbe seen in the case of NC transient measurements onferroelectric doped HfO2 capacitors for different R.10 Inthe other case, RA >> tF ρ and the internal loss of theferroelectric can be neglected compared to the limitingexternal circuit. This is the case for our simulation andmeasurement results shown in Fig. 2 and Fig. 5. In this

8

regime, the NC transient time should be proportional toR and A. Therefore, we can now propose an analyticalmodel for the NC transient time ∆T based on Eq. (9)and Eq. (13):

∆T = |CFE |(RA+ tF ρ)exp

(

Ea

E

)

. (14)

There are three free parameters in this model: the av-erage NC per area CFE , the average internal loss ρ andthe activation field Ea, which we have already determinedfrom field-dependent NC transient measurements. To de-termine CFE and ρ, further measurements on differentlysized capacitors (changing A) or different external resis-tors R are suitable. Therefore, in the following section,we investigate the effect of the capacitor area on the NCvoltage transients.

E. Influence of the Capacitor Area

PZT capacitors with different areas from (10 µm)2 to(50 µm)2 were fabricated and characterized in the sameway as was shown for A = (50 µm)2 in Fig. 2. The ex-ternal resistance was kept constant at R = 10 kΩ andthe maximum applied voltage was VS,max = 5 V.With the knowledge of the internal bias and activation

field Ea from the field-dependent measurement results,we can also correct the area dependent data for this in-ternal bias of tFEbias = 0.72 V. Therefore, we can usethe equation

∆T = ∆Tp,nexp

[

Ea

(

1

E−

1

E ± Ebias

)]

(15)

to calculate the actual NC transient time ∆T from∆Tp

and ∆Tn which are different due to the broken symmetrybecause of the internal bias field. In Eq. (15), E −Ebias

and E + Ebias is used when calculating ∆Tn and ∆Tp,respectively. The resulting dependence of ∆T and forcomparison ∆Tp,n on A is shown in Fig. 6. As expected,after correcting ∆Tp,n for Ebias, only a single linear trendfor ∆T vs. A is observed. A linear fit (∆T = cAA +τA) to the data is also presented in Fig. 6. With thesefitted values cA and τA, the parameters CFE and ρ of ouranalytic model in Eq. (14) can be determined from

cA = R|CFE |exp

(

Ea

E

)

,

τA = tF ρ|CFE |exp

(

Ea

E

)

.

(16)

From Eq. (16) we can now calculate the average NCper unit area as

|CFE | =cAR

exp

(

−Ea

E

)

. (17)

0 500 1000 1500 2000 25000

2

4

6

8 T = cAA + A

cA = 2306.58 s/m2

A = 290.3 nsR2 = 0.9895

Tp

Tn

T fit of T

T (p,n

) (µs

)

A (µm2)

FIG. 6. Negative capacitance transient times with (∆T ) andwithout correction (∆Tp,n) for internal bias fields as a func-tion of A with R = 10 kΩ and VS,max = 5 V. Solid line showslinear fit of ∆T .

By using Eq. (17) with cA and Ea fitted from the ex-perimental data in Fig. 5 and Fig. 6, respectively, we candetermine CFE = −0.023 F/m2. It should be noted here,that this value is about 5 times lower than the theoreticalvalue CFE ≈ 1/(2αtF ) which can be estimated based onhomogeneous Landau theory (in our case -0.114 F/m2),which is almost always used in the literature when cal-culating the stability condition of NC in series with apositive capacitor. Looking at our analytical model inEq. (14), we can see that CFE is actually just the NCin the limit where E → ∞ and the NC that is actuallymeasured can be expressed as CFE,m = CFEexp(Ea/E).Therefore, CFE,m seems to be proportional to the domainwall velocity vDW , while CFE is an intrinsic material pa-rameter which is independent of E. This is actually incontrast to a rather voltage-independent NC which wasextracted in a previous study.9 However, the determina-tion of CFE shown here, seems to be much more reliable.We now can also make a first estimate of the average

of ρ using Eq. (16), with the newly calculated CFE andτA from Fig. 6 and we get ρ ≈ 15 Ωm. However, thisresults should be treated with caution, since the fit in Fig.6 might not be very accurate for A → 0. Therefore, amore reliable approach is to vary R, while keeping A andVS constant, which will be elucidated in the following.

F. Influence of the Series Resistance

In this section, the effect of a change of R on the NCvoltage transients will be investigated for a constant ca-pacitor area of 2500 µm2 and VS,max = 5 V. As expectedand also shown in previous studies,9,10 a larger R willslow down the switching as can be seen from Eq. (13).

9

100 1000 10000 10000010-7

10-6

10-5

10-4

T = |CFE|(RA+ tF)exp(Ea/E)CFE = -0.023 F/m2

= 4 m (bottom fit) = 13 m (top fit)

Tp

Tn

T fit of T

T (p,n

) (s)

R ( )

FIG. 7. Negative capacitance transient times with (∆T ) andwithout correction (∆Tp,n) for internal bias fields as a func-tion of R with A = (50 µm)2 and VS,max = 5 V. Solid linesshow linear fit of ∆T .

When R increases, dP/dt is reduced and thus also theNC transient time increases as shown in the analyticalEq. (14).

To validate our newly developed analytical model forthe NC transient time ∆T in Eq. (14) and to determinethe average value of ρ more precisely than our previousestimation of 15 Ωm, first, ∆T has to be calculated fromEq. (15) to correct ∆Tp,n for the internal bias field. Thedependence of ∆T on R as well as ∆Tp,n for compari-son is shown in Fig. 7. While, Ea = 1.063 MV/cm andCFE = −0.023 F/m2 where used from the fitted exper-imental data in Fig. 5 and Fig. 6, respectively, ρ waschosen to fit the R dependent data in Fig. 7. From thisfigure it is apparent that ρ is different for the positive andnegative switching transients even after internal bias cor-rection for small values of R: The NC transient times forVS < 0 are slightly shorter compared to the ones VS > 0,which means that ρ is larger in the latter case. FromFig. 7 we can see an excellent agreement of our analyt-ical model to the experiment, when using ρ = 4 Ωm forthe VS < 0 transient and ρ = 13 Ωm for the VS > 0transient. Interestingly, these values are very similar tothe range of randomly distributed ρ values fitted for thetime-dependent Ginzburg-Landau simulations, see TableI.Finally, we can again validate our analytical model by

calculating CFE and ρ from the fit to the field-dependentdata in Fig. 5 by taking either one of the two from theprevious fit in Fig. 7. Using CFE = −0.023 F/m2 wethen get ρ = (τ0/|CFE | − RA)/tF = 17.8 Ωm. On theother hand, when using ρ = 4 Ωm to 13 Ωm, we cancalculate CFE = −τ0/(RA + tF ρ) = −0.026 F/m2 andCFE = −0.024 F/m2, respectively. These results arevery similar to the ones estimated before, showing the

consistency of our proposed model described by Eq. (14).

V. SUMMARY AND OUTLOOK

We have investigated NC voltage transients duringpolarization switching in high quality epitaxial capaci-tors of ferroelectric Pb(Zr0.2Ti0.8)O3 connected in serieswith an external resistor. A time-dependent Ginzburg-Landau approach was used to gain insight into the tran-sient switching behavior and domain dynamics using nu-merical simulations. Three main experimental parame-ter variations are investigated: A change of the appliedelectric field E, the capacitor area A and the external re-sistance R. In particular, the duration of the NC voltagetransients during switching from positive to negative po-larization and vice versa were analyzed. The simulationresults indicate, that in the regime where the NC tran-sient time is limited by transverse domain wall motionand coalescence, the NC effect is mostly caused by reversedomain nucleation and unrestricted growth. With theonset of domain coalescence, the differential capacitanceof the ferroelectric capacitor becomes positive again.

By investigating the electric field dependence of the NCtransient time, it was shown that the transverse domainwall velocity is the decisive factor limiting the switchingspeed. This was corroborated by the observation of atypical domain wall creep behavior in accordance withMerz’s law. Furthermore, it was shown that the differ-ent NC durations when applying a positive or negativevoltage pulse can be attributed to internal bias fields.By correcting for this internal field, an activation fieldof ∼1 MV/cm was determined which in turn allowed foran estimation of the domain wall energy density as ∼28mJ/m2. An analytical model was developed to relate theNC transient time ∆T to the experimental parametersR,A and E which allows to reliably determine the averageintrinsic NC per area CFE and the average internal lossρ from measurement data. While ∆T is proportional toboth R and A, the field dependence is exponential. For100 nm thin films we obtained CFE = −0.023 F/m2 andρ = 4 Ωm and 13 Ωm for negative and positive appliedfields, respectively.

This study shows that NC voltage transients in epitax-ial ferroelectric capacitors can be explained by reservedomain nucleation and growth dynamics which can beaccurately modeled by time-dependent Ginzburg-Landautheory. Since these processes are known to be irreversibleand dissipative, there will always be hysteresis effectsassociated with NC in isolated ferroelectric capacitors.This has to be considered when considering device appli-cations like low power logic transistors, where hysteresisshould be avoided. For this case, integration of the fer-roelectric directly into the gate stack seems mandatory.

10

ACKNOWLEDGMENTS

M.H., M.P., U.S., S. Slesazeck, and T.M. gratefullyacknowledge the support by the European Fund for Re-

gional Development and the Free State of Saxony. M.H.acknowledges a visiting scholar fellowship from the Uni-versity of California, Berkeley. M.H. also acknowledgesfruitful discussions with Samuel Smith at UC Berkeley.

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