Crystal Structure and the Paraelectric-to-Ferroelectric Phase Transition of Nanoscale BaTiO3
Crystal Structure and the Paraelectric-to-Ferroelectric Phase
Transition of Nanoscale BaTiO3
Millicent B.Smith,?Katharine Page,?Theo Siegrist,§Peter L.Redmond,?
Erich C.Walter,?Ram Seshadri,?Louis E.Brus,?and Michael L.Steigerwald*,?
Department of Chemistry,Columbia Uni V ersity,3000Broadway,New York,New York10027,
Materials Department and Materials Research Laboratory,Uni V ersity of California,
Santa Barbara,California93106,and Bell Laboratories,600Mountain A V enue,
Murray Hill,New Jersey07974
Received August3,2007;E-mail:mls2064@https://www.sodocs.net/doc/5615217267.html,
Abstract:We have investigated the paraelectric-to-ferroelectric phase transition of various sizes of
nanocrystalline barium titanate(BaTiO3)by using temperature-dependent Raman spectroscopy and powder
X-ray diffraction(XRD).Synchrotron X-ray scattering has been used to elucidate the room temperature
structures of particles of different sizes by using both Rietveld re?nement and pair distribution function
(PDF)analysis.We observe the ferroelectric tetragonal phase even for the smallest particles at26nm.By
using temperature-dependent Raman spectroscopy and XRD,we?nd that the phase transition is diffuse
in temperature for the smaller particles,in contrast to the sharp transition that is found for the bulk sample.
However,the actual transition temperature is almost unchanged.Rietveld and PDF analyses suggest
increased distortions with decreasing particle size,albeit in conjunction with a tendency to a cubic average
structure.These results suggest that although structural distortions are robust to changes in particle size,
what is affected is the coherency of the distortions,which is decreased in the smaller particles.
Introduction
Barium titanate(BaTiO3)is a ferroelectric oxide that under-
goes a transition from a ferroelectric tetragonal phase to a
paraelectric cubic phase upon heating above130°C.In cubic
perovskite BaTiO3,the structure of which is displayed in Figure
1a,titanium atoms are octahedrally coordinated by six oxygen
atoms.Ferroelectricity in tetragonal BaTiO3is due to an average
relative displacement along the c-axis of titanium from its
centrosymmetric position in the unit cell and consequently the
creation of a permanent electric dipole.The tetragonal unit cell
is shown in Figure1b.The elongation of the unit cell along the
c-axis and consequently the deviation of the c/a ratio from unity
are used as an indication of the presence of the ferroelectric
phase.1–3
Ferroelectric properties and a high dielectric constant make BaTiO3useful in an array of applications such as multilayer ceramic capacitors,4,5gate dielectrics,6waveguide modulators,7,8IR detectors,9and holographic memory.10The dielectric and ferroelectric properties of BaTiO3are known to correlate with size,and the technological trend toward decreasing dimensions makes it of interest to examine this correlation when sizes are at the nanoscale.11–16
?Columbia University.
?University of California.
§Bell Laboratories.
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Figure1.Unit cell of BaTiO3in both the(a)cubic Pm-3m structure and (b)tetragonal P4mm structure.In the tetragonal unit cell,atoms are displaced in the z-direction,and the cell is elongated along the c-axis.Atom positions: Ba at(0,0,0);Ti at(1/2,1/2,z);O1at(1/2,1/2,z);and O2at(1/2,0,z). Displacements have been exaggerated for
clarity.
Published on Web05/08/2008
10.1021/ja0758436CCC:$40.75 2008American Chemical Society J.AM.CHEM.SOC.2008,130,6955–696396955
Many experimental and theoretical17–25studies have indicated that the phase-transition temperature of BaTiO3is size-depend-ent,with the ferroelectric phase becoming unstable at room temperature when particle diameter decreases below a critical size.However,both theoretical and experimental reports of this critical size encompass a broad range of sizes.The experimental discrepancies may arise because of intrinsic differences between ferroelectric samples,because the transition is sensitive to conditions such as compositional variation,26lattice defects,12 strain,27or surface charges.20Furthermore,the differences in cell parameters between the two phases are small compared to other sources of broadening in diffraction data,likely leading to an overestimation of the critical size.Recent work by Fong et al.on perovskite(PbTiO3)thin?lms indicates that ferroelec-tric behavior persists down to a thickness of only three unit cells,25a value signi?cantly less than that suggested by previous experimental studies.
Several theoretical studies have been particularly useful in furthering the understanding of the observed behavior of ferroelectrics at small sizes.17However,ferroelectrics are particularly sensitive to surface effects,making modeling increasingly complicated as dimensions are reduced.Many models based on Landau theory18overestimate critical sizes;it has been suggested that this overestimation has resulted from the use of material parameters in the free-energy expression that were derived from the bulk material.19Spanier et al.have found by theoretical modeling that certain surface termination of thin ?lms can stabilize polarization down to a thickness of only several unit cells.20Their calculations,which take into account experimentally determined nanoscale material parameters,es-timate the critical size for a BaTiO3sphere to be4.2nm.Other theoretical treatments,such as effective Hamiltonian and ab initio calculations,have predicted the presence of ferroelectricity in perovskite?lms as thin as three unit cells.23,24
Various experimental probes of the structure of BaTiO3have revealed a complex and sometimes controversial picture.In the study of bulk material,structural transformations have been explained by averaging domains that are locally rhombo-hedral.28,29For the tetragonal phase,the titanium atoms are distorted in the?111?directions and oriented with a net displacement in the c-direction.A number of studies have reported evidence of disorder within BaTiO3above the transition temperature,supporting the existence of distortions within the cubic phase.30–32X-ray diffraction(XRD)studies produce data that are consistent with an increasingly cubic structure at smaller particle sizes,not distinguishing between average and local structure.12,33In contrast,Raman results have supported the existence of tetragonal symmetry at small dimensions,even though it was not discernible by XRD.34The disagreement between Raman and diffraction studies suggests that the phase transition in bulk BaTiO3is complex,with order-disorder as well as displacive character.12,35,36
Extended X-ray absorption?ne structure(EXAFS)and X-ray absorption near-edge structure(XANES)studies of bulk BaTiO3 have supported a dominant order-disorder component to the structural phase transitions.29In EXAFS and XANES analysis of10,35,and70nm BaTiO3particles,37Frenkel et al.?nd titanium displacements for all samples studied,in contrast to their cubic macroscopic crystal structures from laboratory XRD. Petkov et al.38have recently demonstrated the use of the pair distribution function(PDF)to understand local structure distor-tions and polar behavior in Ba x Sr1-x TiO3(x)1,0.5,0) nanocrystals.They found that locally,re?ning over the?rst15?,the tetragonal model was the best?t to the experimental PDF;however,over longer distances(15-28?),the cubic model was the best?t.Their conclusion was that5nm BaTiO3 is on average cubic,but that tetragonal-type distortions in the Ti-O distances are present within the cubic structure.They did not,however,?nd the distortions to be inherent to small particles because they were not present in the perovskite SrTiO3. Several preparation strategies have been reported in recent years for high-quality,well-de?ned BaTiO3nanocrystalline samples.Hydrothermal or solvothermal methods have been systematically used to make nanocrystalline BaTiO3.39–42O’Brien et al.43and Urban et al.21,44have produced BaTiO3particles and rods,respectively,from the reaction of a bimetallic alkoxide precursor with hydrogen peroxide.Niederberger et al.report a solvothermal preparation of5nm particles of BaTiO3and
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6956J.AM.CHEM.SOC.9VOL.130,NO.22,2008
A R T I C L E S Smith et al.
SrTiO3from titanium isopropoxide and metallic barium or strontium in benzyl alcohol.45
Here,we describe the use of a bimetallic alkoxide precursor in conjunction with solvothermal techniques to produce high-quality nanoparticles of BaTiO3with controllable sizes.We have studied particles with average sizes of26,45,and70nm by temperature-dependent Raman spectroscopy and XRD and with room temperature Rietveld and atomic PDF analysis of high-energy,high momentum-transfer synchrotron X-ray diffraction data.The sample particles are unstrained,because they are not thin-?lm samples and are compositionally homogeneous with, in particular,no discernible OH impurities that are known to plague many low-temperature solution preparations of ferro-electric oxides.12,33,36
The complementary structural methods we employ provide information on different time and length scales.Raman spectra re?ect the local symmetry around the scattering sites and are averaged over different parts of the sample.The X-ray techniques both allow an average depiction of the structure (through pattern matching and Rietveld analysis)and provide information on the near-neighbor length scale through PDF. The outcomes of the current study are consistent between the different techniques and are somewhat surprising.Raman spectroscopy indicates that the small particles undergo a more diffuse phase transition than in the bulk,although the T C remains nearly unchanged.Careful temperature-dependent XRD studies show that all sizes of particles are tetragonal until close to the bulk T C,and yet the smaller particles seem more cubic by using the c/a ratio as the metric.Average(Rietveld)and local(PDF) structure analyses of X-ray synchrotron data show that as the particle size is reduced,there is a clear and surprising trend toward increasing structural distortion.The increase in the off-centering of the titanium cation as particle size decreases in conjunction with the decrease in the c/a ratios is consistent with diminished structural coherence in smaller particles. Experimental Section
Preparation of BaTiO3Nanoparticles.Anhydrous benzene, isopropanol,dendritic barium(99.99%),and titanium isopropoxide (99.999%)were obtained from Aldrich Chemical Co.and used as received.Sintered pieces of BaTiO3were also purchased from Aldrich for use as a bulk standard.The bimetallic precursor BaTi[OC3H7]6was prepared according to Urban et al.44Parr acid digestion bombs with23mL Te?on liners were used for the solvothermal reaction.In a typical synthesis,10mmol(5.4g)of the precursor,BaTi[OC3H7]6,was added to the Te?on liner of a digestion bomb under an inert atmosphere.A total of10mL of solvent was added to the precursor under?owing argon according to the water and isopropanol ratios in Table1.In none of the solvents used did the precursor dissolve,but rather it formed a thick white suspension.The Te?on liner was tightly sealed inside the acid digestion bomb,and the mixture was heated in an oven at 220°C for18h.The resulting white precipitate was collected by centrifugation,washed with ethanol,and allowed to dry under
ambient conditions.A white powder suitable for powder XRD and
Raman measurements was produced with a typical yield of1.93g.
Transmission electron microscope(TEM)images were taken on a
JEOL100CX instrument by using an accelerating voltage of100
kV.
Raman Spectroscopy.Raman spectroscopy was performed in air by using a backscattering micro-Raman spectrometer with
helium-neon laser(633nm)excitation.A home-built thermoelec-
tric heating stage was used for temperature-dependent measure-
ments.Spectra were taken at temperatures ranging from room
temperature to above150°C.The300cm-1peak35was?t to a
Lorentzian line shape on a sloping baseline,and from this?t,the
scaled peak area and linewidth were determined.
Differential Scanning Calorimetry.Differential scanning cal-orimetry(DSC)was performed on a Perkin-Elmer Pyris1DSC.
For each scan,3-4mg of sample was used.The heating pro?le
consisted of two cycles of heating from0to150°C at a rate of10
°C/min and then cooling from150to0°C at that same rate. Thermodiffraction.X-ray diffraction data were obtained by using a Rigaku rotating anode together with a custom-built four-
circle diffractometer.Graphite monochromated Cu K radiation
(1.39217?),together with a matched graphite analyzer,was used
in Bragg-Brentano geometry.In this way,a well-de?ned powder
diffraction pro?le was obtained for all re?ections,allowing a
detailed analysis of the pro?le changes associated with the
paraelectric-to-ferroelectric phase transition.The intensities were
normalized to the incident beam to eliminate drift over the data
acquisition time.A home-built heating stage was used to reach
temperatures up to150°C.X-ray patterns above143°C were
collected to obtain a cubic reference for the expected increase in
the peak widths with2θ.Full pattern re?nements were executed
in the program Winprep46by using the pro?le parameters obtained
from the cubic phase above143°C.
Synchrotron X-ray Diffraction.Synchrotron powder diffrac-tion data were collected in transmission mode at beamline11-ID-B of the Advanced Photon Source,Argonne National Laboratory,by utilizing high-energy X-rays(～90kV)at room temperature.The use of high-energy X-rays enables measure-ments at longer wavevectors,Q)4πsin(θ/λ),which is important for the application of the PDF technique.Samples were loaded in Kapton tubes,and scattering data were collected on an image plate system(amorphous silicon detector from General Electric Healthcare)with sample-to-detector distances of660 mm for Rietveld re?nement data and150mm for PDF data. The raw data sets were processed to one-dimensional X-ray diffraction data by using the program FIT2D.47A bulk internal standard was used to calibrate the processed data,to supply an effective wavelength ofλ)0.13648?for re?nements.Rietveld re?nement of the synchrotron data was carried out in the XND program.48Lattice parameters,atomic positions,and atomic displacement parameters were re?ned.The PDF,G(r))4πr[F(r)
-F
],was extracted from the processed scattering data as
described by Chupas et al.49with a maximum momentum transfer
of Q)24?-1by using the program PDFGETX2.50In this
equation,F(r)is the local atomic number density,F0is the
average atomic number density,and r is the radial distance.Full
structure pro?le re?nements were carried out in the programs
PDF?t2and PDFgui.51The scale factor,lattice parameters,
(45)Niederberger,M.;Garnweitner,G.;Pinna,N.;Antonietti,M.J.Am.
Chem.Soc.2004,126,9120–9126.(46)Stahl,K.Winprep;Lyngby,Denmark.
(47)Hammersley,A.P.;Svensson,S.O.;Han?and,M.;Fitch,A.N.;
Hausermann,D.High Pressure Res.1996,14,235–248.
(48)Bèrar,J.F.;Garnier,P.NIST Spec.Publ.1992,846,212.
(49)Chupas,P.J.;Qui,X.;Hanson,J.C.;Lee,P.L.;Grey,C.P.;Billinge,
S.J.L.J.Appl.Crystallogr.2003,36,1342–1347.
(50)Qiu,Y.;Wu,C.Q.;Nasu,K.Phys.Re V.B2005,72,224105-1–
224105-7.
(51)Farrow,C.L.;Thompson,J.W.;Billinge,S.J.L.J.Appl.Crystallogr.
2004,37,678.
Table1.Particle Size Dependence on Solvent Composition
water:isopropanol(v:v)particle size(nm)
1:070(10
40:6060(10
30:7045(9
20:8026(5
0:1～10
J.AM.CHEM.SOC.9VOL.130,NO.22,20086957 Paraelectric-to-Ferroelectric Phase Transition of Nanoscale BaTiO3A R T I C L E S
atomic displacement parameters,and atomic positions as well as broadening from the sample and the instrument resolution were re?ned.
Results and Discussion
Preparation of BaTiO 3Nanoparticles.We explored the effects
of reaction conditions such as temperature,precursor concentra-tion,solvent composition,and addition of surfactants in the preparation of BaTiO 3nanoparticles.We found that the composition of the solvent played a critical role in determining the size of the particles,pure water producing the largest sizes and pure isopropanol producing the smallest.A TEM was used to determine the particle size and morphology,and typical images are shown in Figure 2,with histograms of the particle-size distributions displayed as insets.The particles were nearly spherical in shape with average sizes of 70,45,and 26nm.Table 1gives the average particle size obtained with each solvent mixture as determined by TEM;the given error is plus
or minus one standard deviation.Scherrer analysis 52of the laboratory XRD (111)peak at room temperature gave X-ray coherence lengths (grain sizes)of 33,29,and 21nm for the 70,45and 26nm particles,respectively.The instrumental line width limits the determination of particle size to a maximum of 35nm,preventing any conclusions about the single crystal-linity (grain size)of the 70nm particles.However,for the two smaller sizes,the individual particles are likely single crystals.The ?nal size of the particles is determined by the balance between particle nucleation and growth.In order to form BaTiO 3from the alkoxide precursor,M -O -M bonds must be formed from M -OR species (M )Ti,Ba;R )-OC 3H 7).In the mixed solvent system,it is likely that several mechanisms are in competition with one another,determining the reaction pathway.In pure water,the pH of the solvent -precursor solution was 13,suggesting the partial hydrolysis of the precursor to Ba(OH)2.This M -OH species can react with a second M -OH or with an M -OR to form the M -O -M bonds and water or isopro-panol,respectively.M -O -M bonds might also form through a -hydride elimination and the reaction of the metal hydride with an M -OR.An additional effect of the solvent composition is that the isopropyl group is a better capping group than the hydroxide because -OC 3H 7is less reactive than -OH.Isopro-poxy moieties on the surface of a particle passivate the surface,inhibiting particle growth and leading to smaller particle sizes.Raman Spectroscopy.Tetragonal BaTiO 3has 10Raman-active modes.When splitting of transverse and longitudinal optical modes,as well as splitting due to differing polarizability in each unit cell direction is considered,18Raman-active phonons result.53Symmetry demands that cubic BaTiO 3should be completely Raman-inactive.However,broad peaks centered at 260and 530cm -1are still observed above the cubic-to-tetragonal phase-transition temperature.34The Raman activity of the cubic phase has been generally attributed in the literature to disorder of titanium in the nominally cubic phase.53
Figure 3shows the Raman spectrum of (a)bulk,(b)70nm,(c)45nm,and (d)26nm BaTiO 3over a range of temperatures between 25and 150°C.The assignments given to the Raman modes at the top of Figure 3are those reported in the literature.34Below 200cm -1,we ?nd some weak scattering in the nanoparticle samples due to a BaCO 3impurity.As seen by others,the BaTiO 3Raman spectra have the broad features characteristic of titanium disorder in the unit cell at all temperatures and at all sizes.In the bulk BaTiO 3spectra in Figure 3a,the intensities of the E (LO +TO),B 1peaks at ～300cm -1and E (LO),A 1(LO)peaks at ～715cm -1decrease rapidly as the temperature increases through the bulk T C ,an observation consistent with prior reports.35We interpret the disappearance of the 300cm -1peak as an indicator of the tetragonal phase and use two characteristics as an indication of the phase transition.The ?rst is an increase in peak width at the phase-transition temperature similar to that reported by Hoshina et al.,15and the second is the loss of peak intensity with increasing temperature.These values are given in Figure 4a -d.
For all samples,the linewidth for the E (LO +TO),B 1peak increases both with increasing temperature and with decreasing particle size.The much larger linewidths of the Raman peaks of the nanoparticles suggest that the tetragonality present is accompanied by a signi?cantly decreased structural coherence.
(52)Cullity,B.D.;Stock,S.R.Elements of X-ray Diffraction ,3rd ed.;
Prentice Hall:Upper Saddle River,NJ,2001.
(53)DiDomenico,M.;Wemple,S.H.;Porto,S.P.S.Phys.Re V .1968,
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Figure 2.TEM images of BaTiO 3nanoparticles.Histograms of individual
particle sizes,shown as insets,correspond to (a)70(10nm,(b)45(9nm,and (c)26(5nm.The 200nm scale bar is common to all three micrographs.
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It is interesting to note that bulk BaTiO 3near the cubic-to-tetragonal phase transition displays a Raman linewidth that is similar to the line width displayed by the 26nm particles at all temperatures.
The linewidth analysis is complemented by the analysis of scaled peak area.Figure 4shows that near the expected phase-transition temperature of 130°C,there is a sharp drop in the Raman intensity of the 300cm -1peak for the bulk sample but a more gradual decrease in intensity over the entire temperature range for the 70and 45nm particles.In contrast,the peak area of the 26nm particles in Figure 4d is nearly constant over the entire temperature range.These results indicate a phase transition that becomes increasingly diffuse in temperature as the particle size decreases.
The lack of a sharply de?ned phase transition in nanosized samples is also observed by using DSC.For bulk BaTiO 3,the DSC trace exhibits a peak near 130°C,indicative of the phase transition.Similar features are not observed in the DSC of nanoparticle samples.Together with the Raman results,these ?ndings support the idea that the phase transition is distributed over a wide range of temperatures in the nanoparticles,although it is sharply de?ned in the bulk material.
Thermodiffraction.The splitting of the X-ray diffraction peaks is well de?ned in terms of symmetry,allowing analysis of systematic changes for different (hkl )indices.Figure 5shows diffraction data for 70nm BaTiO 3at room temperature and at 148°C over a small 2θrange.In the high-symmetry cubic phase,
no re?ections are split.In the tetragonal phase,(222)remains a single peak whereas the (400)re?ection is divided into (400/040)and (004)peaks with an intensity ratio of 2:1.Because the c /a ratio is larger than 1,the (004)re?ection shifts to a lower 2θvalue,and the (400/040)re?ection correspondingly shifts to a higher 2θvalue.In spite of changes in symmetry,the cubic-to-tetragonal phase transition is usually not well resolved in diffraction studies of nanosized BaTiO 3because of inherent line broadening due to small particle size.
In our study,the phase evolution of BaTiO 3particles was determined by pattern matching to the laboratory X-ray
dif-Figure 3.Raman spectra at different temperatures for (a)bulk BaTiO 3,(b)70nm particles,(c)45nm particles,and (d)26nm particles.Temperatures increase from top to bottom in each panel.Temperatures are speci?ed to be within a range of up to (3°C.The locations of Raman modes are indicated at the top of the ?gure.The features below 200cm -1are due to a trace BaCO 3impurity,and these are not found in the bulk
sample.
Figure 4.Results from ?ts to the Raman data.Filled circles show variation
of the linewidth of the 300cm -1Raman signal as a function of temperature.Open squares are intensities of the 300cm -1Raman signal normalized to the intensity at 280cm -1.Displayed for (a)bulk powder,(b)70nm particles,(c)45nm particles,and (d)26nm
particles.
Figure 5.70nm BaTiO 3particle laboratory XRD data shown over a small 2θrange.(a)Recorded at room temperature.(b)Recorded at 148°C.Re?ections have been labeled for the cubic phase in panel b.The (222)peak does not split in the tetragonal phase,and consequently,the peak width is constant with temperature.Peaks which are degenerate in the cubic phase but not in the tetragonal phase,for example cubic (400),widen and lose intensity upon cooling.
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fraction data.Patterns taken above T C were matched to the cubic phase to determine the intrinsic peak pro?le resulting from particle size and instrumental effects.To investigate structural changes with temperature,all data sets were pattern-matched with a tetragonal unit cell,even above the phase transition,by using this ?xed pro?le function.
Figure 6a -d shows the re?ned values of the length of the a and c cell parameters at each temperature for bulk,70,45,and 26nm particles.The pseudocubic cell parameter (the cube root of the unit cell volume)is also shown.In Figure 6a,we see that for the bulk particles,there is at ?rst a gradual change in a and c cell parameters as temperature increases.The rate of change becomes the greatest at the transition temperature,at which point the c /a ratio drops to its cubic-phase value of unity.This behavior is well known for the bulk material and has been explained as a second-order phase transition followed by a ?rst-order phase transition.3In contrast to the behavior of the bulk material,the three sizes of particles studied here undergo a more gradual change in cell parameters,without a dramatic increase in slope at the phase transition temperature (Figure 6b -d).At room temperature,the c /a ratio for the bulk sample was calculated to be 1.010,close to the value of 1.011reported in the literature.28The ratio diminishes for decreasing particle size,with values of 1.0058(1),1.0055(1),and 1.0040(2)for particle sizes of 70,45,and 26nm,respectively.For 26nm particles,the deviation of c /a from unity is about 40%of that for the bulk tetragonal phase.Depressed c /a values have been modeled as a result of decreased polarization near the particle surface.22The picture of an increasingly broadened phase-transition behavior with decreasing particle size is consistent across the analysis of Raman spectra,DSC measurements,and XRD.
Synchrotron Rietveld and PDF Analysis.Rietveld re?nement
results of high-energy,long wavevector (Q )synchrotron X-ray diffraction for the three sizes of nanoparticles in this study are given in Table 2,and the corresponding ?ts are shown in Figure 7.The data have been ?t with the P 4mm tetragonal model,as established from the pattern matching analysis.An orthorhombic BaCO 3impurity phase was included in the re?nements.The right panels in Figure 7show the respective phase contributions to the ?ts from BaTiO 3and BaCO 3.Table 2includes results of a quantitative phase analysis from the re?nement.Phase impurities of 11,6,and 3mol%for particle sizes of 26,45,and 70nm are determined,respectively,in agreement with the relative increase of BaCO 3scattering in the Raman data of the smaller particles.We do not expect any in?uence of this separate phase on Raman and X-ray results.
Synchrotron Rietveld re?nement results suggest several trends as particle sizes are reduced and are displayed graphically
in
Figure 6.Change in a and c cell parameters and pseudocubic cell parameter
for (a)bulk,(b)70nm particles,(c)45nm particles,and (d)26nm particles from pattern matching to laboratory diffraction data.Uncertainty in temperature for all data points is (2°C.
Table 2.Results of Rietveld Re?nement of the Synchrotron X-ray Diffraction Data Collected at a Wavelength of λ)0.13648?a
70nm
45nm
26nm
4.0003(2) 4.0044(3) 4.0125(5)c (?) 4.0265(4) 4.0254(6) 4.030(1)c /a
1.0065(10) 1.0054(3) 1.0044(4)vol (?3)64.43(1)64.55(1)64.88(2)z (Ti)0.518(1)0.524(1)0.534(1)z (O1)0.008(8)0.004(10)0.003(9)z (O2)
0.490(5)0.506(5)0.508(5)BaCO 3mol%3611R w (%)
1.92
2.00 2.68
a
A second BaCO 3phase
was also re?ned,and the mole fractions of the second phase are presented.
Figure 7.Rietveld ?ts of the synchrotron X-ray data for (a)70nm particles,(b)45nm particles,and (c)26nm particles.Data are shown as circles,and the solid orange lines are ?ts with two phases:BaTiO 3and BaCO 3.Phase contributions from BaTiO 3and from the BaCO 3impurity are displayed across a small 2θregion at the right of each panel.In panels a and b,the most intense peak has been cut off in order to show all data sets at the same scale.
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Figure 8.For reference,previous neutron studies of the bulk material give a cell volume of 64.271(14),a c /a ratio of 1.011,and a z (Ti)position of 0.5224(6).28Lattice parameters are shown in Figure 8a,indicating larger a and c values for smaller particle sizes.In Figure 8b,the c /a ratio is shown.The c /a values decrease for smaller particle sizes:1.0065(10),1.0054(3),and 1.0044(4)for 70,45,and 26nm particles,respectively.Cell volume is presented in Figure 8c,increasing for the same series as 64.43(1),64.55(1),and 64.88(2)?3.These results are consistent with pattern matching results.The increase in unit cell volume is well established in the literature for many metal oxides 54–57and has been observed in studies of small particles of BaTiO 3.57This is in contrast to metals such as gold,the cell parameters of which shrink as their size is diminished.58The most consistent explanation for such a volume expansion in small oxide particles is the effect of truncation on the attractive Madelung potential that holds the oxide lattice together.59
The z (Ti)positions from the Rietveld re?nements are plotted in Figure 8d.Surprisingly,the titanium displacements increase with decreasing particle size,corresponding to 0.0725(1),0.0966(2),and 0.1370(3)?displacements in the unit cells of 70,45,and 26nm particles,respectively.This is surprising given that the system becomes more cubic with reduced size.This average structure result motivates us to look at the local structure.
Local structure analysis was carried out in the P 4mm space group,with only the z (Ti)position re?ned.Our PDF analysis is based entirely on metal -metal distances (metal positions are reliable for X-ray scattering)and was not changed by including the re?nement of O1and O2oxygen positions.We performed
tetragonal model ?ts from 8to 28?of real space,in 4?supplements.Figure 9displays the ?rst 10of 20??ts to the experimental nanoparticle PDFs,G (r ),in real space,and Table 3presents the PDF results of the 20?re?nements.The ?rst atom -atom distance manifested in the BaTiO 3PDF corresponds to Ti -O distances near 2?.Ba -O distances come next around 2.8?,followed by Ba -Ti distances at around 3.5?and the ?rst Ba -Ba distances around 4?.Qualitatively,the intensity of atom -atom peaks decrease,and the widths increase with decreasing particle size.
In the P 4mm space group,Ba -Ba distances are manifested in a and c cell parameters,and Ba -Ti distances depend on both the cell parameters and titanium off-centering.Consequently,cell parameters,the atom z positions,and the c /a ratio could all be used as metrics of departure from a centrosymmetric structure.Because c /a and z (Ti)are likely to be correlated in an analysis,this is perhaps best captured through examination of atom -atom distances.The off-centering of titanium creates four long and four short Ba -Ti distances within each unit cell.
(54)Ayyub,P.;Palkar,V.R.;Chattopadhyay,S.;Multani,M.S.Phys.
Re V .B 1995,51,6135–6138.
(55)Thapa,D.;Palkar,V.R.;Kurup,M.B.;Malik,S.K.Mater.Lett.
2004,58,2692–2694.
(56)Li,G.;Boerio-Goates,J.;Wood?eld,B.F.;Li,L.Appl.Phys.Lett.
2004,85,2059–2061.
(57)Ishikawa,K.;Uemori,T.Phys.Re V .B 1999,60,11841–11845.(58)Mays,C.W.;Vermaak,J.S.;Kuhlmann-Wilsdorf,D.Surf.Sci.1968,
12,134–140.
(59)Perebienos,V.;Chan,S.-W.;Zhang,F.Solid State Commun.2002,
2002,295–297
.
Figure 8.Rietveld re?nement results for synchrotron data for the different
particle sizes.(a)Cell parameters (c is the larger value),(b)c /a ratio,(c)cell volume,and (d)z position of titanium in the tetragonal P 4mm
phase.
Figure 9.PDF ?ts of the total X-ray scattering for the different particles.
Circles correspond to the experimental PDFs,and the ?ts are gray lines through the data.The difference curves are displayed in each panel and have been offset for clarity.Table 3.Results of Real Space PDF Re?nements over a 20?
Range a
26nm
45nm
70nm
a (?) 3.9972(5) 3.9961(5) 3.9926(4)c (?) 4.041(1) 4.029(1) 4.0294(8)c/a
1.0109(4) 1.0082(4) 1.0092(3)vol (?3)64.56(6)64.34(4)64.23(4)z (Ti)0.518(2)0.516(2)0.514(1)z (O1)000z (O2)0.5
0.5
0.5
U iso (Ba)0.00415(6)0.00365(5)0.00325(4)U iso (Ti)0.0090(4)0.0078(3)0.0072(2)U iso (O1,O2)0.0227(3)0.0227(3)0.0224(2)R w (%)
16.014.819.1
a
A tetragonal P 4mm model was employed,with ?xed O1and O2positions.Re?ned parameters are given with error.
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We have parameterized the off-centering by using a distortion parameter,de?ned here as
(Ba -Ti)long -(Ba -Ti)short (Ba -Ti)long +(Ba -Ti)short
The sensitivity of the PDF to short-range structure and local bond distance,rather than to long-range periodic structure,is the particular strength of the technique.This is traditionally most apparent when re?nements are carried out over short r ranges.Figure 10a shows the extracted Ba -Ti distances,and Figure 10b shows the distortion parameter for the various r ranges used in this study and an unexpected result.The distortion is the largest in the high r range (relatively constant for r g 16?,the distance across about four unit cells)and is essentially absent in the low r region.The dependence on the r range is similar for all three particle sizes,suggesting that this is related to the type of modeling that we have used or the disorder present in the system and not to any size effect.This result perhaps supports the notion of decreased sensitivity of the local structure to the order -disorder picture ?rst envisioned by Comes and Lambert.31,32Previous PDF studies of bulk BaTiO 3phases above and below phase transitions have demonstrated small or negligible effects on the experimental G (r ).60,61A possible explanation is that the PDF is better poised to probe changes in displacive disorder than changes to the orientation of displace-ments,especially over low r ranges.Thus,the behavior of our ?ts at low r may suggest the length scale at which correlated order -disorder distortions in the particles may be captured.It should be emphasized that the low r behavior in this system is not completely understood and perhaps calls for higher-quality data (such as time-of-?ight neutron PDF)or improved modeling.Figure 11shows the long and short Ba -Ti distances in panel a and the distortion parameter in panel b for the particle sizes,
as extracted from Rietveld,12?PDF,and 20?PDF analyses.It is again displayed that measured distortion increases with the length scale of the probe used for analysis.This is a counter-intuitive result,as we expect atomic distortions to be manifested most strongly in the low r region of the PDF and become less and less apparent at high r .
Despite these questions,both Rietveld and PDF suggest that as particle size decreases,the unit cell becomes metrically more cubic,but the displacement of titanium is actually enhanced.These results are reconciled if we consider that for the smaller particles,the distortions from one unit cell to another become less correlated and lose their coherence,in much the same manner as what is seen when bulk samples are heated to near the phase-transition temperature.Two physical explanations for an increasing distortion at smaller particle sizes suggest themselves.The ?rst supposes that the increase in cell volume in smaller particle sizes allows more space within the unit cell for titanium off-centering.The second presumes that the reduction in periodicity in the lattice of smaller particles diminishes the restoring Coulombic force on movable atoms.
Conclusion
We observe by using DSC and Raman spectroscopy that nanoparticulate BaTiO 3undergoes a cubic-to-tetragonal phase transition over a wide temperature range,in contrast to the sharp transition found in the bulk material.Our XRD data show that the tetragonal metric is reduced from the bulk value with decreasing size,but that particles as small as 26nm remain in the tetragonal phase until near the bulk transition temperature.By using a combination of Rietveld and PDF analysis of synchrotron X-ray diffraction data,we establish a trend of increasing distortion in Ba -Ti distances and titanium off-centering with decreasing particle size.We conclude that although the smaller particles have a greater distortion,a loss of coherence relative to the bulk material is responsible for decreased c /a values.Our results signi?cantly contrast the accepted wisdom that BaTiO 3becomes less distorted for smaller particle sizes.
(60)Kwei,G.H.;Billinge,S.J.L.;Cheong,S.-W.;Saxton,J.G.
Ferroelectrics 1995,164,57.
(61)Egami,T.;Billinge,S.J.L.Underneath the Bragg Peaks:Structural
Analysis of Complex Materials ;Pergamon Press Elsevier:Oxford,England,
2003.
Figure 10.PDF re?nement results with varying r range.(a)Long and
short Ba -Ti distances and (b)distortion parameter based on Ba -Ti distances as de?ned in the
text.
Figure 11.(a)Long and short Ba -Ti distances and (b)distortion parameter,de?ned as [(Ba -Ti)long -(Ba -Ti)short ]/[(Ba -Ti)long +(Ba -Ti)short ].Results from synchrotron Rietveld re?nements as well as the PDF analysis for different r ranges are displayed.
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Acknowledgment.The work at Columbia has been supported by the National Science Foundation through the MRSEC program (DMR-0213574),by the Nanoscale Science and Engineering Initiative of the National Science Foundation(CHE-0117752),and by the New York State Of?ce of Science,Technology,and Academic Research(NYSTAR).K.P.is supported by the National Science Foundation through a Graduate Student Fellowship,and R.S.is supported through a Career Award(DMR04-49354).Data collection at Argonne National Laboratory and the Advanced Photon Source was supported by the DOE Of?ce of Basic Energy Sciences under contract W-31-109-Eng.-38.The authors thank Peter Chupas and Karena Chapman for their help with synchrotron data collection at the11-ID-B beamline at the Advanced Photon Source.
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