.-s

INTERFACIAL FORCE MICROSCOPE APPLICATION TOPOLYMER SURFACES

J. E. Houston* and R. M. Winterw

%andia National Lab$ P.O. Box 5800Albuquerque NM S7185-1421;**~e~try and Cherni@l Engineering DeparbrknG south Dxo@

School of Mines and Technology, Rapid City, SD 57701-3995

Introduction%mrring-pmbe rnicroscopies (SPM) arc presently widely used in

remarkablydiverse applicationsand, w evidenced by this symposiuw thesetechniquesarcrapidlyexpandinginto the importantareasof polymersurfacesand interfaces. The Atomic Force Microscope (MM) is presently the mostwidely used of the scanning-probetechniques. However,the AFM’s rangeofapplication suffers from an inherent mc&aoicaf instabii~tyin its deflectionforce sensor. The instabilityproblemhas been overcomeby the developmentof tie InterracialForce Microscope (IFM)l, which utilizes a fome-feedbacksensor concept. In the following, we present several examples of polymerapplicationsto illustratethe utilityof the 1PMsensorcon~pt.

ExperimentalIrrterfacialForce Microscopy (IFM) is a scanning probe technique

similar to the AFM but distinguishedby its use of a stabIeand quantitativeforce sensor. The sensor (shown schernaticaftyin Fig. 1) consists of adifferentialcapacitorinvolving a common plate (CP) suspended above twocapacitorpadsCl andCzby torsionbars. A tip is placedon one end of the CPand interracial forces between it and a sample surface rotates the CP,increasing one capacitance and decreasing the other. This displacement isdetected by au RF bridge circuit and the force bakmced by placing theappropriatevoltages on C, and CZ. Within the frequency response of thecontrollerthatestablishesthe propervoltages, the force measurementis madewithouttip movementandis, thus, stable. In addition,since the force-voltagerelationship only depends on the capacitor vahres, tie measruement isquantitative. These attributesgive rise to a degree of measurementcontrolthatpermitsthestudyof a broadspectrumof materialsproperties.

xyz PiezoController

Common Plate Sampleand Torsion Bars

Probe

C*

Figure1. A schematicdiagramof the IFMsensor.

Results and DiscussionOne of the modes of IFM operation,which has been extensively

used on metal, self-assemblingmonolayerandpolymersurfacesz~,is thatof anrmoindenter.In thk mode, a parabolichardprobe tip (such as diamond ortungsten)is used in a classic “H@zian” indentationexperiment. This allowsmeasurementsof the elastic modulus (both real and imaginary), the yieldstress at the plastic threshold and both creep and relaxation. Normafindentationexperimentstake the form of force profiles, i.e., force applied tothe tip versus tip displacement (stress-straincurves) while approachingandwithdrawingfromthe surfaceat a constantrate. Such profiles are illustratedin Fig. 2 in a direct comparison of the surfaces of polyethylene (PE) end

(a)4

.., 1 I ,

.. .. . E“. 44 MPa

3 - “. %... -t . ... Hettdcm Fit

z “. ....MX2

:$

... %.6 “..g %~1 ..*

-A.%

o -

I Patyethytena

.1 , , ,

-5 -4 -3 -2 -1 0

Relative Displacement-Ax10

m) 15, , m f

\ E’= 16.8GPa 1

-5 I (-1.5 -1 -0.5 0

Relative D@acement-hd O

Figure2. (a) shows the forceprofile for a polyethylenesample. Arrowsindicatetip motionapproachingthe sample fromthe right. Aftercontactthecurvefollows theHertziarrbehaviorindicatedby the solid grcy fine. (b) showsa similarset of datafor poly(methylmethucrylate).

poly(methylmethacrylate)(PMMA). After contact, the behavioron approachinitial appearselastic closely following the Hcrtzianelastic force/deformationrelationship, F-~RE*d3nwhere R is the tip radius, E* is the indentationmodulusandd is the deformation. F@rre2a shows the behaviorfor PE takenwith 2500 ~ W tip. The E*vafue calculatedfrom this dataturnsout to be 44MPa and the deviation from the Hertzirmcurve signals the plastic threshold,which occurs at -15 MPa of averageapplied stress. These figures comparereasonably welf with the range of values (95-260 MPa and 4-16 Mpa,respectively)given for low density PE.6 After a furtherdeformationof morethe 20C0 & the withdmwaf curve shows hysteresis, which signafs a“permanent”deformation (a dent). However, the concave nature of thewithdrawal curve indicates a considerable level of heahg as the tip isremoved leaving art indentation depti of about 1000 ~ (over time, thisindentationsignificantlyheals). Thus, the hystereticbehavior is arrelastic—energy is lost but recoveryis significant. In contrast,13g. 2b shows the forceprofile for PMMA, which indicates a much less compfiant surface and verylittle hysteresis. In adthtiou a significant level of adhesive interactionisobserved, which is largely responsible for the hysteresis (i.e., adhesivehysteresis). The modulus vafue cufculatcdfrom this data is afmost 17 Gpaiwhich is high comparedto the handbook vahreof -3 GPa.6 However, thesurfacedeformationhere is only about 70 ~ and the high E* vafue indicatesthepresenceof a thinbut stiff skin on the sample surface.

The relaxationbehavioralluded to in Fig. 2a for the PE surfacecm becharacterizedby performinga creep test. This measurementbegins with thetip “hovering”in light contactwiti the surface(- 100 oN). The hoverforce isthen suddenlyincreasedto 4 VN (above the plastic thresholdin Fig. 2a) andthe motion of the tip is recorded. The results arc shown in Fig. 3. Theincreusc in force causes the tip to suddenly push into the surfaceby afmost4000 ~ andas the samplerelaxesit to continueto push into thesurface,overaperiodof-6 see, by an additional-1500 ~. This relaxationcan be adequatelycharacterizedby two time constantswith values of aboutone and six seconds,essentiallyspfittingtherelaxationdisplacementequafly. Materialssuch as PEme usuaflynonfinear,which means thatthe creepbehaviorwill dependon thedegreeof deformation.

Polymer Preprints 2000, 41(2), xxxx

*--

0

1 , , 1 I

Polyethylene

, 1 , ,

-5 0 5 10152025

l“tme-sec

Figure3. Creepdatatakenon a polyethylenesamplewhereat timeequals zerothe force is suddenlysteppedto 4 pN. Thetip dkplaces intothe sample in orderto establishthe forceand slowly creeps furtherintothe sampleorderto maintaintheconstant-forcevrdue.

Another useful mode of IFM operation, espcially forheterogeneous materials, involves applying a small modulation of thetiphrnple separation and synchronously &tecting the signal to obtain asingfe-frequencyvafue of the ac modulus, i.e., the slope of the stress-straincurve at a given level of applied force. This procedurepermitsimages to beobtained in the normaf repulsive-force mode while also acquiring 2Dinformation on the ac modulus. Figure 4a shows an example of an acmodulus image takenon a sampleconsisting of 10 pm glass fibersembeddedin an epoxy matrix. The terracein the upperright-handcomer is on the edgeof the glass fiber and the brightercolor indicates a higher value of the acmodulus thanfor the epoxy in the lower left-handterrace. The tlamess of thetwo terracesshows that the modulus is constantin these areas. The crossingpattern in the area separatingthe terraces could be mken as part of the“interphase”region. However, the constantrepulsive-forceimage in Fig. 4bshows that it results from morphology variations. The sample polishing

(b)

500 A

Figure4. (a) A 1 pm x lpm uc modulusimage of a 10pm glass fiberembeddedin an epoxy matrix. The highermodulusglass appearsas a brightercoloron this frdsecolorimage, (b) a constantrepulsive-forceimage of thesame regionshowing thatthe epoxy has been distortednearthe interfaceby the polishingprocdm.

procedurehas left an uneven boundarybetween the fiber and matrix. Thisresultclearlyifhrstratesthe vrdueof takingsimultaneousmodulrdmorphologydatasince measuredmodulusvalues dependon the local surfacestructure.

hr earlier work on the @ass/epoxy system,4 individual forceprofiles were acquiredafong a line perpendicularto the interface. Resultswere compared for systems with varying thickness of interfaeial silanecoupling agents between the glass and epoxy. They showed considerablesystematicvariationin the measuredmodulus as a function of distance fromthe fibersurfhce. In genemfthe modulus initiallydecayed to one-hrdfthe bulkmodulus thenrose to two times the bulk modulusbeforedecaying to the valueof the modulus in the bulk as shown in llg. 5. The dktancc over which thisoccurredwas 2 to 8 microns dependingon the surface treatment. ‘l%isis inagreementwith the Wifliatnset aL7but in considerablyexcess of the thicknessof adsorbed coupling agent. Thii is a clear indication that the interfacinginteractionis playing a significant role in rdteringmatrixpropertiesover anextended range. Being able to measure these changes as a function of thechemicaf nature of the coupfirtgagent will have a significant impact oneffective fiber-reinforcedcornpos;te~esign.

0.0 5.0 10.0 15.0 20.0 25.0 30.0

Distsrtce from Fiber Surfeee (microns)

Figure5.Modulusis shown to varyas a functionof positionfromthe fibersurface.

ConclusionsIn W brief discussion, we have tried to cover a few examples of IF’M

applicationsto polymersurfacesto illustrateits unique capabilities. Its chiefstrengthsareits abiity to obtainquantitativeinformationon interracialforcesin a controllablemanner. The abifityto controlthe sensormakes it possible toperformmeasurementsto determinea wide variety of materirdspammeters.In addhion, the “rero-compliance”mtum of the sensor insuresthatno energyis stored during the act of measuring the force. Thus, for systems whichinvolve relaxationphenomena(so prevalentin polymermaterials),the sensordoes not mask individualprocesses by dumpingits storedenergyat the onsetof relaxation. These uniquecapabilities,coupled with the advantagesinherentin being able to smdies materialspropertieson a very local level, make theIFMa veryappealingtool for advancedpolymerresearch.

Acknowledgement. San&mis a multiprograrnlaboratoryoperatedbySandiaCorporation,a LcckheedMartincompany,forthe DOE underContractDE-AC04-94AL85000(JEH)and DE-FG02-98ER45733(RMW). Additionalfunding for RMW was provided by the Nationaf Science FoundationundergrantOSR-9452894(RMW).

(1)(2)(3)(4)(5)

(6)

(7)

Joyce, S. A.; Houston,J. E. Rev. Sci. Irrstrum.1991,62,710.Kiely, J. D.; HoustoL J. E. Phys. Rev B 1998,5712588.Kiely, J. D.; Houston,J. E. Lungmuir1999,15,4513.Winter,R. M.; Houston,J. E. Mat. Res. Sot. Proc. 1998.Johnson,K. L. ConkzcrMechanics 1996, CambridgeUniv. Press,NewYork,NY.Rodriguez,F. Prirwfpfeso$Polymer Sys@ms 1989, HemispherePubfishingCo., 31’Edition,pp. 598-595.Williams,J.G.;James,M.R.; Morris,W.L. 1994, Mat.Sci. Eng. A126,305.

Polymer Preprints 2000, 41(2), xxxx

DISCLAIMER

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DISCLAIMER

Portions of this document may be illegiblein electronic imageproduced from thedocument.

products. Images are

best available original

, ‘ ‘ Hannon, 11 13ykov et al., 12 and Abraham et al. 13 conclude that Mnz+ substitutes at the B-.,

site (henceforth referred to as hfnT~), inducing a captive oxygen vacancy (V. ) in its first

neighbor shell. Recent work by Laguta and coworkers14 and Laulicht et al., 15 on the other

hand, concludes that Mn2+ resides on the A-site (i.e., lfn~). This Mn displaces w 0.9A

in the (100) lattice direction, without any charge compensating defect such as K-vacancy

in the immediate neighborhood. 14 The impurity substitution pattern in the closely related

incipient ferroelectric SrTiO~ appears similarly controversial. Thus, ESR spectroscopy re-

sults for KTaO~: Mn2- have been interpreted as consistent with both A-site16 and B-site17

substitution.

Tracking ESR spectra as a function of temperature provides further information about

the impurity environment. At elevated temperature, the fine structure splitting vanishes, 12’1~

signaling that the impurity dipole has acquired sufficient thermal energy to overcome

the barrier for dipole reorientation to another symmetry-related direction. -According to

these experiments, an activation energy of 0.115 eV accompanies dipole reorientation in

KTa03:Mn2+. .An impurity dipole reorientation barrier of 0.11 eV is also deduced from

frequency-dependent dielectric relaxation measurements by Nowick et al. 18 The ESR work

of Ref. 15 gives a barrier of 0.065 eV, which seems out of line with the three other estimates

cited herein; henceforth we will assume an experimental value of w 0.11 cl;. Citing the early

work of Bykov et al, 12Nowich et al. conclude that this barrier is associated with the hopping

of an oxygen vacancy between two equivalent sites in the first neighbor shell of M residing

at the Ta-site. This interpretation does not agree with the more recent work of Ref. 14.

The possibility of multiple charge states in transition ions such as L/In brings compli-

cations when one attempts to determine the most stable defect structure(s). In view of

this, recent experiments on Ca-doped KTY19 and KTa03:Ba20 are especially illuminating.

In the KTN:Ca sample in question (2.3% Xb, 0.055’% Ca), the small amount of Ca added

significantly increases the Curie temperature. 19 This suggests that Ca substituents create

dipoles that couple strongly to the ferroelectric host. The dielectric loss spectra is consis-

tent with activated dynamics with an activation energy of 0.08 eV. The magnitude of this

3

., ‘ barrier is comparable to the one observed in KTaOq :Mn2+.

KTY samples. 21 Given these observations, a comparative

This signature is absent for pure

theoretical study of Ca and M

impurities should shed light on the structure and dynamics of dipolar defects in KTaO~, and

perhaps in perovskites in general.

In this paper, we apply first principles methods to study the energetic, structures, and

dynamics associated with Ca2+ and Mn2+ impurities in cubic KTaO~. There have been rela-

tively few first principles calculations on defects in perovskites, especially in K’T’iaO~.Exner

et U1.22have compiled a list of formation energies and favorable solid state chemical reactions

for incorporating a series of mono-, di-, and tri-valent substituents, using a shell model for

the ions. Li impurities have been examined using the full potential LMTO method.23 Li2q

and N’b25’2bsubstituted at the K and Ta sites, respectively, hay-e also been studied using a

semi-empirical Hartree-Fock method. However ~the full potential used in Ref. 23 give results

that exhibit discrepancies with all electron LAPW-LDA first principles predictions, partic-

27 More recentlv first principlesular with regard to the soft phonon modes in pure KTa03. .?

studies of Pt-impurites,b Pb-vacancies,28 and O-vacancies*g in PbTi03 have been performed.

This list, by no means exhaustive, highlights the value of performing more first principles

studies to benchmark semi-empirical calculations.

The present work on Ca-substituent energetic follows the framework of Ref. 22, but

applies density functional theory instead of empirical models. We also consider the acti-

vation energies for oxygen vacancy hopping within the first neighbor shell of the Ta-site

substituents, as well as the dynamics involving Mn and Ca motion at the K-sites. While

the system is ionic and its structural properties should be reasonably well treated within

the shell model of Ref. 22, the predictions for dynamics can be sensitive to the parameters

used in empirical models. With our test cases of KTa03:Ca and KTa03: Mn. we hope to

provide a useful first principles paradigm for dipole reorientation dynamics in ferroelectric

perovskit es.

The paper is organized as follows. Section II describes the details of the first principles

implement ation. Validation of this implementation is given in Sec. 111. where special at-

4

II. METHOD

The calculations are performed using the Vienna .Atomic Simulation Package

(V.W3P),30 which utilizes ultrasoft pseudopotentials. ‘1 The local density functional (LDA)

approximation 32 for the exchange correlation functional is used. When unpaired electrons

are present, the spin polarized local density functional approximation (SLDA)33 is applied

instead. The Generalized Gradient .4pproximation (GGA)3~ to the exchange correlation

function, which has been known to give diffusion barriers closer to experimental values in

some insulators, 35 is also applied to provide a consistency test for determining activation

tention is paid to the convergence of defect formation energies with superce~l dimensions. ~ ‘ ‘,,

Sections IV and V discuss the predictions for impurity incorporation energetic and dipole.,

reorientation dynamics, respectively, and Sec. VI concludes the paper with further discus-

sion.

energies. The pseudopotentials for K, Ca, and Mn include pseudovalent p-shell electrons.

The defect structures are computed using periodically replicated supercells of varying sizes.

The Brillouin zones of these supercells are sampled using Monkhorst-Pack grids30 equivalent

to or denser than 6 x 6 x 6 k-points within one primitive cell. .411calculations are performed

at the equilibrium LD.\ or GG.k lattice constants, as the case might be, except where in-

dicated. To compute the activation energy of dipole reorientation, the nudged elastic band

method37 is applied. This method will be discussed in more detail in Sec. V.

111, ACCURACY AND CONVERGENCE TESTS

To determine the defect structures and dynamics, accurate formation energies and barrier

heights are required. First principles predictions are far more reliable than semi-empirical

results for this purpose. However, the high cost of these calculations limits the size of

supercells that can be used to mimic isolated defects. The convergence behavior with respect

to cell size is discussed in detail in this section.

.5

A. Validation of LDA and Pseudopotentials.3

The ultrasoft pseudopotentials used herein, in conjunction with LDA, gives a lattice con-

stant of 3.957A for KTaO~, which is 0.6~0 smaller than the experimental lattice constant

and is in excellent agreement with the 3.96A obtained using L.APW.27 GGA predicts a lat-

tice constant of 4.027~, which is in excellent agreement with Ref. 38 and is 1910 larger than

the experimental value. As a further validation of the pseudopotentials, Table I lists the

~-point phonon frequencies computed at both LD-4 and experimental lattice constants. Also

shown are the LAPW results computed at experimental lattice constant,27 and experimen-

t al phonon frequenciesag extrapolated to zero temperature in Ref. 27. The close agreement

between our phonon frequencies and LAPW results demonstrates that our ultrasoft pseu-

clopotential implementation closely mimics all-electron LDA despite the omission of inner

shell electrons in the Ta pseudopotential.

The soft TO mode frequency differs from the experimental value by 60 cm-l at the

experimental lattice constant, and by as much as 94 cm–l at the LDA lattice constant.

This frequency, which is close to vanishing, is proportional to the square root of the lowest

eigenvalue of the dynamical matrix and is sensitive to small errors in lattice constants. The

disagreement with experimental force constant should not be considered excessive. However,

it does indicate that the system at its LDA lattice constant is farther away from critical-

ity than experimental KTa03. LD.A KTaO~ will therefore exhibit a lower static dielectric

constant COas well as a smaller correlation length. -Jssuming a linear dielectric effective

medium for the host KTa03, this will cause a small uncertainty in the absolute solvation

energy of charged species which, depending on the ionic radius, can be a few tenths of an

electron volt for highly charged defects. As a result, the binding energy between oppositely

charged defects may be somewhat unreliable if the species are highly charged.

In general, LDA equilibrium structures are reliable for perovskites. On the other hand,

LD.~ has been known to give diffusion barriers that are OR by 337C, while GGA has been

known to be off by 20Y0.35

6

B. Convergence with respect to Cell Size ‘ - ,,

Too small a supercell causes artificial strain fields and periodic charge fields that stem -

from the unphysical periodic boundary conditions. In this work, the effect of image charges

is controlled by adding a monopole Madelung correction. lo The LD.A high frequency dielec-

‘DA = 5.11.5, which is 10?10largertric constant is required for this correction. We assume CM

- 41 ‘The monopole and dipole interactions among super-than the experimental value of 4.6a.

cell images can in principle be completely eliminated using a reference charge distribution

technique.42’43

Figure 1 depicts the convergence behavior as a function of supercell size. The defect

formation energy Ef for several structures we will investigate in Sec. IV is plotted against the

cell dimensions (NZ, NY, N.) along the three cartesian directions. We define Ef as the change

in cohesive energy between the supercell containing the defect and an identical supercell

consisting of perfect crystal KTa03. 13f is normalized for the charge in the supercell; Sec. IV

will show that, for the purpose of determining the most favorable mode of incorporating

Ca, this normalized energy is the relevant quantity. In general, the defect structures that

do not involve oxygen vacancies are apparently converged to 0.25 eV per charge for N. = 3.

Howe\~er, when one (or more) oxygen vacancy exists in the first neighbor shell of a large

metal ion (e. ~~., in KT~-Vo )? the metal displacement can exceed 1.?. This induces large

strain and leads to slow Ef convergence. As we will show, K~,-Vo is not a low energy

defect, and therefore this does not prevent us from assigning the lowest energy structures

for Ca impurities (Sec. IV). In cases where cubic symmetry-breaking oxygen vacancies exist,

Ef converges more slowly with cell size in the direction perpendicuhzr to the direction of the

metal-V. dipole, and up to 4 x 4 sampling in the transverse directions is used. This is

despite the fact that the supercell is a well behaved insulator which exhibits no states in the

gap. We also note that our largest supercell of 240-atoms is 3 times as large as those used

in first principles LD.& studies reported in the literature. G’z8’z9

7

IV. DEFECT,STRUCTURES AND, ..,

A. Candidates

Given the highly ionic nature of the

ENERGETIC OF CA SUBSTITUENTS*

for Defect Structure

species present in KTa03 :Ca, we have focused

on nominal ionic charges when considering possible defect structures. When substituting

a Ca2- for either K+ or Ta5+, or when creating one oxygen vacancy by removing Oz–,

charged defects are formed and the supercell cannot be kept neutral. The intrinsic defect

concentration of KTa03 is of order a few hundred parts per million,4~ and are not sufficient

to accept excess electrons from (or donate them to) the extrinsic Ca-containing defects which

want to be positively (negatively) charged. Therefore we assume that, whenever a positively

charged defect is created by a impurity, a compensating negatively charged defect must exist

to preserve overall charge neutrality.

The choices of possible low energy defect structures are guided by a previous shell-

model study.yl The isolated positively charged defects examined are Ca~, (V~-Vo)+, (Ca~,-

(vo)~)+, and V&. The candidates for isolated negatively charged defects are (CaT.-Vo)-,

Vi, (K~,-Vo)z-, and Ca&. Figures 2 and 3 depict some of these structures. Relatively

small lattice relaxations accompany Ca~. This defect does not have Ca residing off-center

and therefore forms no dipoles. On the other hand, in (CaT~-Vo)+, Ca displaces N 0.1.1

towards the oxygen vacancy in its first neighbor shell, forming a large dipole. In Ca~; ,

the large size of Ca2+ relative to the Ta5+ normally residing there forces the oxygen ions

surrounding the cramped Ta-site to displace outwards by 0.17~. (The nearest neighbor K-O

distance in bulk KTaO~ is 2.8 ~, compared to 2 .?i for Ta-O.) The K+ ion is even larger than

Ca2+; as alluded to before, it displaces more than 1.1 towards the oxygen vacancy that is

bound to it at the Ta-site.

To compute the solvation energy of Ca assocated with each defect, we need to add or

subtract the chemical potentials of each species added to or removed from the supercell,

respectively. In the svnthesis,ll’lg Tkaq05 and excess KqC03 are heated to 1000 K in air.

8

KZC03 is replaced by K20 in our calculations since COZ is readily eliminated from the’ “ ‘.

carbonate. The chemical potentials p~, pt., and ~~, are determined from the cohesive

energies of K20, CaO, and KTa03. 45 Ta205 is irrelevant here because it is the reactant

assumed to be completely exhausted during synthesis. So, for example, ~&h~~ive= PCa + #0.

p. can in principle be calculated from the partial pressure in air and the cohesive energy of

OQ molecules.~5 Since mass is conserved and all ions retain their nominal charges, Y. can

actually be eliminated in the final expression when defects pairs with equal but opposite

charges are combined. Therefore PO can be left unspecified. The solvation energies are

referenced to constant PK, /%Cal PTal and

E$’ =

PO. For instance,

E;aK – ~ca - flK.

Lsing these solvation energies, we search for the most stable pair of defects with equal

and opposite charges. The possible structures are first screened using under-converged (i.e..

iVZ = ;Vv = N. = 2) supercells; no attempt is made to further converge Ef for structures

with very unfavorable energies. Table II confirms that some defects under consideration are

evidently so costly they can be ignored. The most favorable structures are converged to

~ ().~.~e~l per charge, From Table II, we predict the most favorable solid state reaction to

be

2Ca0 + KTa03 4 Ca~ + (CaT, – Vo)- (1)

The non-dipolar defect Ca~~ is only slightly higher in energy (W 0.3eV) per Ca impurity:

4Ca0 + 3KTa03 -+ 3Ca~ + Ca& + K20 (2)

The stability of these two structures relative to other possible defects is in qualitative agree-

ment with Ref. 22. Table II shows that potassium vacancies, which might naively be expected

to accompany Ca~, are highly unfavorable. The trends in Table II can be rationalized in

terms of ionic charges and sizes. Ca2~ substituting for Ta54 creates a large localized charge

in addition to forcing a larger ion into a congested Ta site. These Ta-site disadvantages can

9

./

~ , }Jealleviate~by binding an oxygen vacancy to CaT.. The K+ ionissimply toolarge to fit,- .

into the Ta site even with an oxygen vacancy in its first neighbor shell. In the more stable

(Ca~, – Vo)- defect, a dipole is formed, consistent with the experimental observation that

Ca stablizes the ferroelectric phase in KTN. 19

Our theoretical treatment differs from experiments in that 2.3 YOPJb actually exists in

19 This amount of l-b is sufficient to cause a paraelectric-to-the KThT:Ca sample in question .

ferroelectric phase transition. 9’10Our first principles implementation, which uses the local

density functional approximation (LDA) ,32 overestimates the hardness of the soft optical

mode (see Sec. HI A and Ref. 27). It is difficult to treat the phase transition and the

~~companying lattice distortions induced by such a small amount of Fib, because a large

supercell is needed to account for the proper composition, and because the small error in

LDA lattice constant may preclude an extremely accurate prediction of the phase boundary.

Furthermore, the breaking of cubic symmetry that accompanies phase transitions in KTN

will gi~:e rise to a much larger number of possible defect structures separated by small

energies, compared to the case in high symmetry KTa03. To restrict ourselves to studying

generic features of defects! we have chosen cubic KTaOS as the host lattice.

B. Defect Clustering

Since the defects are charged, clustering is favored electrostatically and is a distinct

possibility y. ]T’e have investigated two possible charge-compensated complexes, (CaK)s-CaT,

and CaK-CaT~-Vo, which are depicted in Fig. 4. Within the precision of the calculation, no

binding energy is observed between the bound pair of Ca~ and (CaT~-vo) –. On the other

hand, because CaTa carries a large –3 charge, a binding energy of 3.0 eV obtains for (GaK)s-

CaTa. This translates into a 0.75 eV decrease in energy per Ca substituent in Eq. 2, which

will put this complex 0.35 ev lower in energy per Ca than the isolated CaK and CaT~-Vo

pair (Eq. 1).

The (CaK)s-CaT, cluster carries a dipole that cannot readily reorient at low tempera-

10

ture. It is unlikely to be responsible for the temperature-dependent dielectric 16ss signature. ‘

Experimentally, the synthesis is carried out at T = 1000 K, and the configurational en-

tropy cost of assembling 4 impurity atoms has to be considered when determining whether

Ca clustering can occur. During crystal growth, CaO is spontaneously incorporated into

KTaOq ,24 which means that the solid state reaction which incorporates Ca has a negative

free energy. Our calculations always give positive incorporation energy at zero temperature.

Barring large LD.A errors in cohesive energies, entropy must play a large role in determining

the defect structure. .% we have not systematically varied the cluster size, shape, and ori-

entation, the possible effect of clustering on dipole formation and dynamics will be deferred

to future studies.

V. DIPOLE REORIENTATION DYNAMICS

The calculations of dipole orientation barriers are performed from first principles using

multidimensional transition state theory. 46 Here the relevant barrier height is the lowest

saddle point separating the initial and final configurations in the potential energy landscape

comprising all atomic degrees of’freedom in the supercell. In

figurations are two of the possible orientations of (say) the

algorithm for finding this saddle point is the nudged elastic

our case, the initial/final con-

CaT,-Vo dipole. .\n efficient

band method.37 This method

creates a series of replicas along the reorientation (or “reaction,” to borrow a physical chem-

istry terminology) path that interpolates between the initial (reactant) and final (product)

configurations. .An energy function is introduced to penalize large changes of atomic coordi-

nates between nearest neighbors in replica space. The reaction path is found by minimizing

a suitable action that governs all degrees of freedom in the replicas, subject to the constraint

that the path passes through a saddle point. 37 This general, robust method has been imple-

mented within the VASP packageso and successfully applied to predict non-trivial reaction

pathway-s. It will be used as the standard in activation energy calculations, against which

simpler approaches will be compared.

11

—

t Al Oxygen Vacancy Hopping around CaTa and MXIT.---

Our previous results suggest that Ca resides in both K- and Ta-sites. First we consider

the activation

neighbor shell

energy for dipole reorientation via oxygen vacancy hopping within the first

of CaT~. Figure 5 depicts the lowest energy V. hopping pathway in a CaK-

CaTa-Vo complex. A NZ = ;V = NZ = 2 supercell with 2x2x 2 k-grid sampling is used. This

doubly substituted supercell has been chosen as a paradigm because this cell is uncharged

and does not suffer from artificial interactions between periodic image charges. We find

that the hopping pathway is symmetric about the x = y plane, and the oxygen atom which

undergoes hopping stays within 0.2~ of the ‘Ta-containing lattice plane. The x = y plane

appears to be a dividing surface between reactant and product. To confirm this, we impose

reflection symmetry about the x = y plane in this supercell in panel (6) and allow all other

degrees of freedom to relax. The energy difference between this constrained configuration and

the stable defect structure is 1.988 eV (Table III). This energy difference is within 0.08 eV

of the activation energy obtained via a spline fit of the nudged elastic band replica energies.

The small difference can be attributed to a slight underconvergence when applying the

nudged elastic band method. We conclude that relaxing the atom positions while preserving

reflection symmetry about the z = y plane gives the correct dipole reorientation activation

energy. The Ca atom at the K-site also undergoes displacive motion: at the saddle point

[panel (6)], it displaces 0.25 ~ off the z-y plane which contains the .I-site.

The qualitative features discussed above hold in all cases of oxygen vacancy hopping

we have studied. Since the constrained relaxation technique agrees with the nudged elastic

band method and is much less computationally expensive, it will be used for all other

defect structures in the remainder of this section. 47 Table III gives the dipole reorientation

activation energies for various defect complexes. Increasing the cell size to NZ = NY = NZ =

3 increases the barrier by 0.2-0.45 eV. We have also compared one LDA result against the

GG.A prediction computed at the GGA lattice constant of KTaO~. The difference is 0.14

eV, which is already smaller than the error due to using small supercells. We further note

that the defect with the largest metal ion displacement at the Ta site, iiamely) KT~-Vo (see ‘ ‘

Fig. 3), also exhibits the largest hopping barrier.

It is noteworthy that we predict a N 1 eV barrier for intrinsic V. hopping in undoped

KTaO~, in agreement with experiments. 49 On the other hand, this value is in substantial

disagreement with the 0.07 eIJ activation energy in pure KTa03qxedicted by the shell model

of Exner et al.22 L7sing the nudged elastic band method with up to 8 replicas, we do not find.

the transition state parallel to the ~ = g plane that Ref. 22 reports. This suggests that their

shell model may not be as useful for dynamics as it apparently is for structural properties.~8

.kll activation energies for V. hopping within the first neighbor shell of CaTa are predicted

to exceed 2 eV. This is corroborated by the results of preliminary, constant temperature ab

initio molecular dynamics simulations. Even at ?’ = 3000K, which corresponds to a thermal

energy of N 0.25 e~”, the oxygen vacancy stays in the same lattice site after 2 ps of simulation.

Therefore the oxygen vacancy hopping barrier must be significantly larger than 0.1 eV. We

conclude that w 0.1 eV activation energy deduced from dielectric loss measurements in

KTN:Ca is most likely not associated with oxygen vacancy hopping. Note that this finding

does not imply that CaTa-Vo cannot exist; we have merely shown that CaT~-vo should not

contribute to the dynamics at low temperature.

To gain further insight into the dynamics, we turn our attention to Mn2~ impurities,

which are more amenable to experimental studies because of unpaired electrons. Since Mn

readily takes on numerous charge states, searching for the most favorable defect pairs will

be even more difficult than for Ca. Instead of attempting to predict the favorable struc-

ture (s), we explicitly consider the dynamics of isolated hln~+ and Mn~~-Vo and determine

which defect is consistent with the activation energy observed in ESR and dielectric loss

spectroscopes. .4s the supercell contains an odd number of electrons, SLD.A is used in the

computation. The V. hopping barrier within the first neighbor shell of lh!tn~~ is computed

by imposing the aforementioned symmetry constraint. -% in all previous cases that in-

volve oxygen vacancy hopping, the activation energy exceeds 1 eV (Table III). This finding

seems to explicitly cent radict the oft-cited assertion of Ref. 12~namely, that oxygen vacancy

13

B. Mn hopping motion at the K-site

,-,hopping around Mn~~ exhibits a 0.11 ev barrier.

In this section, we investigate the possibility that Mn~+ is responsible for the 0.11 eV

activation energy, as suggested in Refs. 14 and 15. Figure 6 shows that Nln spontaneously

displaces off center by 0.81~ along the (100) direction, which compares favorably with the

0.9~ displacement deduced from an enharmonic model analysis of ESR results. 14 The off-

center motion of Mn2+ is in contrast to Ca2~, which exhibits no such displacement. One

difference between the two cases is the change in ionic radius, between 0.99 .~ for Ca2+

and 0.80 ~ for Mnz+. Smaller ions are more likely to sit off center at the .4-site; however,

the stronger covalent bonding between the transition metal ion Mn2+ and oxygen atoms

should also play a role. The use of SLDA is crucial here. With LDA, the atoms relax to a

configuration where Mn is displaced along (100) by 1.4.3. This alternative structure turns

out to be metastable when SLD.A is applied. On the other hand, the structure predicted by

SLDA is unstable when LD.4 is used.

\“ext, the hopping of off-center Mn between two of the six equivalent directions is in-

vestigated using both the nudged elastic band method and by imposing symmetry about

the x = y plane. The qualitative picture is very similar to Fig. 5. ‘The hopping path is

confirmed to be symmetric with respect to reflection about the ~ = y plane; the barrier

lies on the x = y dividing surface; the hopping species (Mn in this case) moves in the x-y

plane that contains the A-site. However, the barrier is now predicted to be 0.182 eV when a

NZ = A?g= NZ = 3 supercell is used. This dipole reorientation activation energy is within a

factor of two of known experimental values, compared to the factor of twenty discrepancy in

the case of V. hopping. The large difference in Mn and V. hopping barriers in KTa03 :Mn

can be attributed to the fact that the K-site is much less congested than the Ta-site, as

mentioned above. Hence ?vfnKmotion around the K-site should be much less hindered than

oxygen motion around the Ta-site. Indeed, comparing the hopping pathways in Figs. .5and 6

shows that oxygen vacancy hopping is accompanied by significant motion of several other , ‘

atoms, whereas the Mn motion barely perturbs its nearest neighbors. Our prediction of a

small barrier at the K-site is reminiscent of the 55 meV gained in LiK off-center displace-

ment computed for an isolated Li impurity using full potential LMTO calculations, which

necessarily implies that the hopping barrier for LiK cannot exceed 55 meV. 23’50Even though

the convergence of the activation energy with supercell size is slow (Fig. 6), we can conclude

that the experiment ally observed N 0.1 etr activation energy in experiments is consistent

with the hopping motion of .Mn~.

C. Conjecture on Hopping Dynamics in KTN:Ca

Unlike MnK, CaK does not displace off-center and cannot be directly associated with the

temperature dependence observed in dielectric loss spectroscopy. Our conjecture is that an

off-center Ca2~ dipole obtains when Nb is also present at the Ta-site. Even though Ca2~

exhibits a unistable potential energy well centered at the K site, the potential surface is

extremely flat. Figure 7 shows that an off-center displacement of 0.2.3 in the (100) direction

costs only 7 meV, provided all other atoms are relaxed to accommodate this motion. Recall

that the 2.3$Z0Nb concentration used in Ref. 19 is sufficient to cause a transition to the

ferroelectric phase and a breaking of the cubic lattice symmetry. E~en well above the Curie

temperature, .Nb substituents in KTa03 displace off centersl and cause lattice distortion

around them. They create strain fields which may perturb the flat energy surface for GaK

sufficiently to create bistable potential wells, causing off-center motion. Since it is diflicult

to treat 2.370 Nb doping in a supercell (see the discussion in Sec. IV .4), we elect to apply

a tensile strain along the (100) direction at fixed supercell volume to test this hypothesis.

The calculations are performed using NZ = NY = NZ = 2 supercells with 2 x 2 x 2 k-point

sampling.

KPJb03 has an orthorhomic crystal structure that exhibits a maximum tetragonal dis-

tortion of 1.’FZC,52while the maximum distortion for KTN: at small Nb concentration is

‘ estimated td be less than 0.770. 53 We find that upon applying a 2% strain, Ca does become7

unstable at the K-site. It displaces off-center in the (100) direction by N 0.26~, forming a

dipole and gaining ~ 43 meV in the process (Fig. 7). This implies that a barrier of 43 meV

now exists between dipoles in the (100) and (iOO) directions. .40.79’0 uniaxial strain already

causes off-center Ca displacement, albeit an extremely small one. Due to the small LD.A

error in soft mode frequency and lattice constant, the LDA prediction of strain-induced Ca

off-center displacement may not be extremely accurate.s4 Furthermore, convergence with cell

size has not been pursued systematically. l$7ithin these uncertainties, we have nevertheless

given a plausibility argument that off-center Ca displacement can be due to a sufficiently

large uniaxial strain.

KTa03:Ba does not exhibit the activated dielectric loss signature observed in KTX:Ca.20

This is consistent with our argument that Pib is needed to create impurity dipoles, although

it must be pointed out that the Ba2~ ion is much larger than Ca2~ and that difference will

also affect the defect structure and dynamics. Pure KT?i also lacks the activated signature

seen in KTX:Ca. Comparing Ca at the .4-site with K in pure KTa03, we find that K+ sees

a much steeper potential well. This difference is most likely due to the much larger ionic

radius of K–, which hinders

:nent ioned above, we do not

The presence of Xb will

off-center displacement. L-sing the 27G artificially strained cell

find any off-center IS- displacement (Fig. 7).

also change the V. hopping barrier around a CaT~ impurity.

However, it is extremely unlikely that Xb will lower the barrier to the N 0.1 el~ observed

in experiments. Our various conjectures will be put to test in future experimental and

theoretical works. Finally, while the conjecture about strain-induced off-center Ca dipoles

appears reasonable, it should be pointed out that, in cases of more than one substituent

such as in KTN:Ca, interactions among substituent ions may lead to unexpected results.

~nfort~mately, systematic first principles treatments of these complex systems are prohibitive

at this time.

16

VI. CONCLUSIONS

Pure KTa03 is an ideal material for investigating defect structures and dynamics in

perovskites because it retains its cubic symmetry down to zero temperature. We have

considered Ca2~ and .l!In2~ substituents in KTa03 from first principles. We find that defect

formation energies converge slowly with supercell size, especially in cases where oxygen

vacancies are present and large relaxations of large ions at the Ta-site (the “B” site in

perovskites) entail. For Ca impurities, Ef generally converges to N 0.25 eV using 135- and

240-atom cells in the absence/presence of oxygen vacancy, respectively. For the KTa-Vo

complex, on the other hand, Ef per defect has barely converged to the 1 eV level in a

240-atom supercell.

Ca2- substituting for K+ or Tas~ necessarily creates charged defects, which must be

compensated with oppositely charged extrinsic defects to preserve overall charge neutrality.

Because of this, Ca goes into both K and ‘l?a sites. For oppositely charged species far

removed from each other, Ca at the Ta-site is accompanied by an oxygen vacancy in its

nearest neighbor shell. “The large electrostatic energy between highly charged defects

clustering, which is not systemically investigated herein; it should be addressed in

studies that deal with impurities in perovskites.ss

favors

future

It is shown that the N 0.1 eV activation energy deduced from ESR and dielectric loss

measurements in KTa03: Mn2~ is consistent with dipole reorientation motion of off-center

~lnK, in agreement with recent experiments. 14’15That is, the LD.A-predicted hopping barrier

is within 0.07 eV of the measured value. The experimental barrier is not consistent with

V. hopping within the first neighbor shell of a MnT. defect. KTX:Ca also exhibits a 0.08

eV activation energy in its dielectric loss spectra. 19 lVe argue that this dynamical signature

is due to reorientation of off-center CaK dipoles created by hTb-induced symmetry breaking.

In general. it appears that impurity ion hopping at the less congested -i-site exhibits a

much smaller activation energy than oxygen vacancy hopping at the B-site in KTa03. This

conjecture will be tested for other impurities and for other perovskite hosts in future works.

17

> ACKNOWLEDGMENT.,.

We thank George Samara, Dwight Jennison, Man Wright, Peter Feibelman, Tim Dunbar,

and Nicola Hill for useful discussions. This work was supported by the Department of Energy

under Contract DE-.AC04-94AL85 OOO. Sandia is a multiprogram laboratory operated by

Sandia Corporation, a Lockheed hIartin Company, for the U.S. Department of Energy.

18

I

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‘~ ~’ote that this symmetrv constraint is viable in cubic KTaO~ but will be inapplicable for“

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48 A recent shell model/semi-empirical Hartree Fock study predicts an oxygen vacancy hop-

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‘4 Because Ca off-center displacement can be expected to be sensitive to the soft-mode

frequency, which LDA does not reproduce quantitatively, it cannot be completely ruled

out (although it is highly unlikely) that Ca may in reality displace off-center in unstrained

KTa03, even in the absence of Nb.

‘5 Clustering has been investigated in Ref. 26 for uncharged defects.

FIGURES 2

FIG. 1. Convergence of defect formation energy Ef for cat, (CaT’a-Vo)-, C~~, and

(KTa-VO)2-. -Ef is normalized with respect to the charge in the supercell. The lines and cir-

cles are guides to the eye. A convergence of N 0.25 eV is attained for the Ca-impurities, while

K substituting at the Ta site, bound to an oxygen vacancy, converges poorly because of the large

relaxation of the K atom. Fortunately, this complex proves too energetically costly to be a viable

defect structure.

FIG. 2. Structures of cat and (CaTa-VO)-, depicted as cross-sections in the z-y plane. The

filled or shaded circles are in the Ta (B-site) layer while the open circles are coplanar with K

(A-site). The solid line circles represent Ca while circles with dashed lines are K atoms. These are

shaded only if t hey are in the Ta-layer. Ta are depicted as small, filled, black circles, while O atoms

in the Ta- and K-layers are represented as filled grey circles and open circles with dotted lines,

respectively. The Ca~ supercell has a cubic symmetry. Note that the Ta-O bond length for pure

KTaO~ is 1.98A according to LD.4. (CaT.-Vo) - exhibits CqVsymmetry, and Ca2~ is displaced by

-0. 1A towards the oxygen vacancy.

FIG. 3. Structures of Ca$~ and (KT.-VO )2-. For a description of the symbols, see the caption

of Fig. 2.

FIG. ~. Structures of two defect clusters, CaK-CaT.-vo and (CaK)3-CaTa (see text). For a

description of the symbols, see the caption of Fig. 2.

FIG. 5. Lowest energy oxygen vacancy hopping pathway between symmetry-related, equivalent

minima in (CaT~-Vo )– computed using LDA. The symbols are described in Fig. 2. In the inset,

the activation energy profile is plotted as a function of the reaction coordinate (i.e., replica number

within the nudge elastic band technique). Diamonds are the total energies of the respective replicas

referenced to the formation energy oft he stable defect (panels Oand 5). The dashed line is a spline

fit. The plus and cross indicate the barrier height computed by imposing symmetry about the z = g

plane (panel 6) for supercells of size N. = NV = NZ = 2 and N. = iVY= NZ = 3, respectively.

\ rFIG. 6. Lowest &nergy Mn2+ hopping pathway between symmetry-related Mnl{ minima com-.’

puted using LDA. The symbols are as in Fig. 5 except that Mn now replaces Ca. The purpose of

showing the under-converged barrier height (plus) is to demonstratee that the symmetry constraint

prediction agrees well with the nudged elastic band method applied at identical cut-offs.

FIG. 7. Effect of uniaxial strain on Ca and K off-center displacement in the (100) direction at

the A-site. Dot-dashed line: Ca, 0% strain; solid line: Ca, 0.7Yo; dashed line: Ca, 2Yo; dotted line:

K, 2%. These lines are spline fits to LDA results. The supercell volume is kept constant as the

strain varies.

TABLES > J

,,

!

mode VASP (expt. ao) VASP (LDA aO) LAPW27 experiment3gr

T02 85.8 cm-l

TOS 193.9 cm–l

TOl ‘ 276.0 cm-l

T05 555.4 cm-l

117.8 cm-l 80 cm–l 24 cm-l

204.8 cm-l 172 cm–l 197 cm-l

273.3 cm-l 264 cm–l 274 cm-l

582.1 cm-l 529 cm– 1 546 cm– 1I I II

TABLE I. Comparison of predicted phonon frequencies with LAPW results computed at exper-

imental lattice constant (Ref. 27) and experimental results3g extrapolated to zero temperature.27

I I I II./+ I *(Ca~)+l (vK-vO)+ ‘(~aT~-(v(J)’2)+ v;+ \

“(Ca~.-Vo)- I 1.4 eV

*(Ca~a)3- 1.7 eV

7.7 eV 2.5 eV 4.4 eV I

21.4 eV 3.4 eV 11.7 eV

*(vK)- 3.4 eV NA 5.7 eV X.4 ~1

TABLE II. Solvation energy per Ca in various defect pairs. The defects marked by an asterisk

and a dagger have Ef converged to 0.25 eV and 0.5 eV per charge, respectively. The other entries

are computed using ilrz = NV = NZ = 2 supercells and 2 x 2 x 2 Monkhorst-Pack k-point sampling.

.,

,- ,

defect

Ca~-Ca~a-Vo

CaK-CaT~-Vo

&T.-vo

CaTa-Vo

CaT~-Vo (GGA

i\ V.

KTa-Vo

‘i---hhIT.-vo

Mn~

il ~nK

TABLE III. 1

—

Nzx Ngx Nz activation energy (eV)

2x2x2

3x3x3

2x2x2

3X3X3

2x2x2I

2x2x2

2x2x2

2x2x2

1.988

2.204

2.749

3.198

2.611

1.006

3.965

2.159

2X2X2

3x3x3

0.Iz5

0.182

about the z = y plane.

energies are computed

, ,

?ole reorientation act ivat ion energy computed by imposing reflection symmetry

2 x z x 2 ~lonkhorst-pack ~-point grids are used throughout. AH activation

using LDA except where noted.

, I

a) -5.50Ur

-6.00

9.0

8.5

ui-

7.5

7.0

* \

.

.

“o NX=NY=NZ

NX=NY=2

(%)’-],,’

K’”e●+

I

NX=NY=3

NX=NY=4

2 3

N,

4

NX=NY=2

--’’--+=”’=@ NX+4

(CaK-Vo)-

I I

w“’’” “=..

~ NX=NY=2 i

4

w%-%)’-]/.2

● NX+JY=3

0 NX=NY=4

2 3 4

Nz

5

25.0

24.0

17.5

17.0

16.5

16.0

-m

Leung et al., Fig. 1

(CaK)+

(CaT~-Vo)-

11 I I I

I I I ;/..1..y4%-** (,” _,,. ,

‘,+=’ . . . . .,

I II

I I I I

V. K o

Leung et al., Fig. 2

(CaTa)3-

(KTa-vo)2-

,

.,----- ~..--.,

“e ~..——. —-’

“--,+a

I I I II I I

-d

I \ o

Leung et al., Fig. 3

(CaK)3-CaTa

CaK-CaT,-Vo

I 1 ,

1

I

I I I I

o———Ca

—l———

Leung et al., Fig. 4

4

Cak,

–l_–—–_

?-

+._ */?’ :;:J1 I

o ‘ Ca—,———— — _ L.

o

–;–– , ‘ _I xi),. ,,........

!

!

I

Q-- !—— —— —-

1I 1

_,__––_,. ‘ _I

42I ‘. ,,. . . . . . ..

— — — —

I

I

L-

1

~ 6-1 ,

,-—,— —— —— ,–l–––_:,”- , a-

1m,, ,

—i____ <---- _.’1.......] ,. ,, I,...-...! ........

II

,;&-~&,., , .3D* 1e.

% ‘ :3’ 1

— — — — —- -

!

3 .,.”----— — — — — —

q.,

I

,,

I

1I

c) ——————-6

1

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hopping coordinate

Leung et al., Fig. 5

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hopping coordinate

Leung et al., Fig. 6

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0.00 0.10 0.20 0.30 0.40 0.50

impurity displacement (A)

Leung et al. Fig 7