Chemical Solution-Deposited BaTiO3 Thin Films on Ni Foils:Microstructure and Interfaces

Tanawadee Dechakupt, Gaiying Yang, Clive A. Randall, and Susan Trolier-McKinstryw

Materials Science and Engineering Department and Materials Research Institute, The Pennsylvania State University,University Park, PA 16802

Ian M. Reaney

University of Sheffield, Sheffield, U.K.

The microstructure and interface quality of chemical solution-deposited BaTiO3 films on Ni foil were investigated by trans-mission electron microscopy. The microstructures were found toconsist of equiaxed and uniform grains, with average grain sizesfor rapid thermal-annealed films of 12 nm (7001C) and 18 nm(7501C), respectively. Films furnace annealed at 10001C after arapid thermal anneal at 7001C showed a grain size of 42 nm. It isbelieved that the final grain size is limited by the highly reducingatmosphere and also by the existence of well-developed crystal-lites resulting from the rapid thermal annealing step. Spatiallyresolved electron energy loss spectroscopy identified the exis-tence of residual carbon and variations in the oxygen content inBaTiO3 films. High-resolution transmission electron microscopyrevealed an interfacial layer of Ni–Ba alloy (5–10 nm thick)between the BaTiO3 and Ni foil.

I. Introduction

MULTILAYER ceramic capacitors (MLCCs) have been widelyused in electronic circuits because of their small size and

high volumetric efficiency. The most commonly used dielectricsfor MLCC are formulated barium titanate ceramics as theyprovide a high dielectric constant over a wide temperaturerange.1,2 BaTiO3-based thin films are potentially interesting foreither future generation single or multilayer capacitors.3�7 It hasbeen demonstrated that by using metal foils as the substrates,the films can be processed at higher temperature and thus higherpermittivity can be achieved. Dawley et al.7 and Nagata et al.6

showed high permittivities (erB1500) with low loss tangents forbarium strontium titanate and pure BaTiO3 films on Ni foil,respectively. A higher permittivity (erB3000) was reported forpure BaTiO3 films on Cu foil.8,9

When BaTiO3 films are prepared on base metal foils, firing in ahighly reducing atmosphere is required to prevent oxidation of theelectrodes.10,11 While the oxygen vacancies introduced in the di-electric layers can be partially ameliorated by post-oxidation, theuse of low oxygen partial pressures during firing can potentiallyaffect a wide variety of factors, including the efficiency of carbonremoval, grain size, and breakdown strength, among others.

There are a number of reports on the microstructural conse-quences of low pO2 firing on BaTiO3 ceramics and the BaTiO3/Ni interface. Oxygen vacancies in low pO2-processed bariumtitanate dielectrics result in the appearance of dislocation loops

observable in a transmission electron microscope.10,12,13

Yang et al.10 showed that the oxygen content in co-firedbarium titanate is not uniform and that a higher oxygen vacan-cy concentration is observed near the Ni electrode. The localvariation of barium titanate reduction was attributed to theeffect of residual carbon. However, the uniformity of the oxygencontent in barium titanate is significantly improved after thereoxidation step. Yang et al.14 also revealed the existence of adiscrete layer (4–15 nm thick) of Ni–Ti–Ba metallic alloy be-tween the Ni and barium titanate in many commercial MLCCs.The alloy layer is formed during high temperature co-firingand is stable through the reoxidation step. Other authors havereported the effect of low pO2 on reducing the grain size inBaTiO3.

15�17

Although barium titanate thin films on Ni foil have shownpromising electrical properties, there is little information aboutthe microstructure and interface quality of chemical solution-deposited films processed at low oxygen partial pressures. In thiswork, the microstructure–processing–property relations in thefilms were explored using transmission electron microscopy(TEM). Electron energy loss spectroscopy (EELS) was also em-ployed for chemical analysis for the films.

II. Experimental Procedure

In this work, BaTiO3 thin films were prepared on high-purity Nifoil (25 mm thick, 99.99%, Alfa Aesar, Ward Hill, MA) bychemical solution deposition (CSD) from methanol-based sol-vents. Before deposition the foil was preannealed at 9001C inapproximately 1� 10�16 atm pO2, using a reducing furnace toremove oxide and contamination. The deposition was carriedout by spin coating the solution on the preannealed Ni foil at3000 rpm for 30 s. After each deposition, the films were dried at1801C for 3 min and then at 2801C for 3 min on hotplates in air.After every two spinning/drying steps, the films were heated to3501C for 1 min for further organic removal and then at 7001 or7501C for 1 min under a N2 gas flow (23 SLPM) in a rapidthermal annealer (RTA 600, Modular Process TechnologyCorp., San Jose, CA). Spinning, drying, and RTA annealingwere repeated until 8 layers were obtained.

The films were then heat-treated at 10001C for 1 h in5� 10�17 atm pO2 (setting value) in a reducing furnace andthen reoxidized at 6001C for 30 min in 1� 10�10 atm pO2 (set-ting value). The heating rate of both steps in the reducing fur-nace was 51C/min. A schematic showing the oxygen pressuresand temperatures employed for some of the steps is shown alongwith the Ellingham diagram in Fig. 1. It is clear that the con-ditions used for reoxidation could lead to oxidation of theNi foil, if the BaTiO3 layer does not provide a sufficient kineticbarrier against oxygen diffusion.

The crystallinity of the BaTiO3 films on Ni foils was moni-tored using a Scintag V X-ray diffractometer (Scintag Inc.,

Q. X. Jia—contributing editor

Article was invited by guest editors from the Electronics Division.This work was financially supported by the Royal Thai Government, Intel, and Kemet.wAuthor to whom correspondence should be addressed. e-mail:STMcKinstry@psu.edu

Manuscript No. 23865. Received October 15, 2007; approved December 22, 2007.

Journal

J. Am. Ceram. Soc., 91 [6] 1845–1850 (2008)

DOI: 10.1111/j.1551-2916.2008.02407.x

r 2008 The American Ceramic Society

1845

Cupertino, CA) with CuKa radiation. Data were collected from201 to 601 2y with a step size of 0.021 and a 0.5 s acquisition timeper step. A slow scan with a count time of 5 s was also usedwhen a small amount of some phase, i.e., NiO, was expected.

The microstructure and interface quality of BaTiO3 films onNi foils before and after a 10001C furnace anneal were observedusing a transmission electron microscope (Philips 420, Hills-boro, OR) with a 120 keV tungsten source. Cross-section TEMsamples were prepared by either conventional polishing and ionmilling or a focused ion beam (FIB) technique. It was found thatconventional polishing and ion milling are useful for BaTiO3/Nifoil samples after furnace annealing. To reduce the milling dam-age for the EELS specimens, the final thinning was performedon a cooling stage (using liquid N2) in a EA Fishione Model3000 ion mill (Export, PA). However, RTA-annealed filmsreadily detached from the Ni foil during polishing. Therefore,FIB milling was used for such samples as it enables preparationof cross-sectional TEM samples without a mechanical polishingstep.

A JEOL (2010F, Tokyo, Japan) transmission electron micro-scope with a field emission electron source combined with aGatan Enfina parallel electron energy-loss spectrometer (EELS)was used for chemical analysis of BaTiO3 films on Ni foils. Themicroscope was operated at an accelerating voltage of 200 kVand the EELS spectra were acquired using diffraction mode.The resolution of the EELS spectra was 1.1 eV at the zero-losspeak, and the electron collection angle was 14 mrad.

III. Results and Discussion

(1) Phase and Structure

X-ray diffraction patterns of 7001C RTA-annealed, 10001Cfurnace-annealed and reoxidized BaTiO3 films are shown inFig. 2. The BaTiO3 films on Ni foil have a pseudocubic perovs-kite structure with some degree of /100S orientation. The highintensity and narrow peaks suggest that the furnace-annealedfilms exhibit a larger grain size, compared with those of RTA-annealed films. NiO was not observed in normal scan X-raydiffractions of all films. However, a NiO 200 peak was revealedin slow scan X-ray diffraction measurements of the reoxidizedfilm only, suggesting that Ni oxidation occurs after reoxidation.

(2) Microstructure

(A) RTA-Annealed BaTiO3 Films: Some degree of sur-face damage was observed in FIB-milled samples, as shown inFig. 3. Discrete crystallization layers are apparent in the BaTiO3

films annealed at 7001 and 7501C, confirming that crystallization

occurred during each RTA annealing. There is no significantdifference in grain size across the film thickness, as clearly shownin the undamaged area in Fig. 3a. The result implied that rep-etition of the RTA annealing induced crystallization of newlydeposited layers, but did not cause considerable grain growth ofthe previously crystallized BaTiO3. The high-magnificationbright-field TEMmicrographs in Figs. 3b and 3c show the micro-structure of the BaTiO3 films on Ni foil after RTA annealing at7501 and 7001C, respectively. The equiaxed, polycrystallinestructure of the BaTiO3 films suggests that nucleation occursthroughout the films. A BaTiO3 thickness of approximately200 nm was achieved with eight coatings and four RTA an-neals, yielding 25 nm for each layer of deposition. On increasingthe RTA temperature, the average grain size slightly increasesfrom 12 nm (7001C) to 18 nm (7501C). Both films contain afew nano-size (o10 nm) pores. No cracking was observed. TheBaTiO3/Ni foil interfaces in samples annealed at 7001 and 7501Cshow no evidence of any interfacial layer such as NiO, consistentwith the X-ray diffraction result.

The grain size of the CSD-based BaTiO3 films in this work iscompared to previous reports in Table I. The morphology ofmost of the films was reported to be equiaxed; it is noted if theauthors stated otherwise. The grain size range of 5–30 nm of theRTA-annealed films at 7001 and 7501C in this experiment wassmall. Neglecting the difference in precursor chemistry and sub-strate materials, this small grain size is reasonable, comparedwith the reported grain size of 15–20 nm of (Ba,Sr)TiO3 films onPt/Si substrates prepared using similar conditions.18 A largergrain size (20–60 nm) was reported for films that were annealedat 7001 or 7501C for longer times (0.5–1 h).19�21

The absence of a clear interfacial layer between the BaTiO3

films annealed at 7001 or 7501C and the Ni foil indicated thatcovering the Ni foil with at leastB50 nm of amorphous BaTiO3

film (first 2 layers) and using N2 flow during both film pyrolysisat 3501C and RTA annealing at 7001 or 7501C prohibited ox-idation of the underlying Ni foil. Finally, it is noted that theobserved cross-sectional BaTiO3 films are on a single grain ofNi. Therefore, no comment can be made about the presence ofan oxide layer (i.e., of contaminants in the Ni foil or as a resultof NiO formation) at the junction of the BaTiO3 film and thegrain boundary in the Ni substrate.

(B) Furnace-Annealed BaTiO3 Films: Figure 3d shows aTEM cross-section of a furnace-annealed (crystallized andreoxidized) BaTiO3 film that was RTA-annealed at 7001C.The film exhibits polycrystalline, equiaxed grains. The filmsurface is rough due to grain growth at high temperature. Theaverage grain size of the furnace-annealed film is 42 nm. Thefilm contains five to six grains across a film thickness of B200nm. There are a few isolated voids (with a size on the order ofthe grain size) distributed in the film. The voids may be fromeither porosity or damage due to (Ar1) ion milling. There is no

600 800 1000 1200 1400

−60

−50

−40

−30

−20

−10

0

10Lo

g pO

2 (a

tm)

Temperature (°C)

Ni/NiO

Ti/TiO2

Ti/Ti2O3

Ti/TiO

Re-oxidation

Ni pre-annealingCrystallization

Ti2O3 /TiO2

400

Fig. 1. Stability of Ni and various oxidation states of titanium (circles,J and �, show typical setting values and measured values for the pO2

for each heat treatment) (Adapted from Yang et al.14)

Fig. 2. X-ray diffraction of 7001C RTA-annealed, crystallized, andreoxidized BaTiO3 films on Ni foil and their slow scans.

1846 Journal of the American Ceramic Society—Dechakupt et al. Vol. 91, No. 6

interfacial layer between the BaTiO3 film and the Ni foil, indi-cating that neither the high-temperature heat treatment nor thereoxidation caused extensive Ni oxidation. It is important tonote the contrast to the X-ray diffraction results, where someNiO was observed after reoxidation. The result suggests that theNiOmust be spatially distributed, so that it is difficult to observein the small sample area typically characterized in the transmis-sion electron microscope. It is possible that the NiO might beconcentrated at grain boundaries in the Ni foil.

The change in microstructure of the furnace-annealed BaTiO3

film was expected to be dominated by the 10001C step becausethe reoxidation was performed at a much lower temperature(6001C), where the cation mobility is low.22,23 However, consid-ering the high pO2 used at 6001C, the reoxidation step can affectthe interface quality and oxygen content in the BaTiO3 lattice,which will be discussed in Section III (3).

There are a few reports of preparing BaTiO3 films by theCSD method at temperatures higher than 8001C, as shown in

Fig. 3. Transmission elcetron bright-field micrograph of microstructure and interface of BaTiO3 films on Ni foils (a) RTA annealed at 7501C (lowmagnification), (b) RTA annealed at 7501C, (c) RTA annealed at 7001C, and (d) furnace annealed at 10001C and reoxidized at 6001C.

Table I. Reported Grain Size and Average Grain Size (in parentheses) of BaTiO3–Based Films Prepared by Various Groups Under aVariety of Different Conditions

Film/substrate

Annealing condition

Grain size and

(avg grain size) (nm)

Total thickness

(nm) ReferenceT(1C)/Time Atmosphere

This work: BaTiO3/Ni foil 7001C, 1 minw N2 5–20 (12) 2007501C, 1 minw N2 10–30 (18) 200 —10001C, 1 h 10�16 atm pO2 24–75 (42) 200

BaTiO3/Pt-Si 7001C, 30 min O2 20–60 (40) 250 Gust et al.19

(Ba,Sr)TiO3/Pt-Si 700, 1 h Air 50 400 Tahan et al.21

Ce-(Ba,Sr)TiO3/Pt-Si 7501C, 2 minw NA (15–20) 200 Sedlar et al.18

BaTiO3/Si 7501C, 1 h O2 20–60 300 Gust et al.20

(Ba,Sr)TiO3/Ni foil 9001C, 1 h 10�18 atm pO2 30–35z 400 Dawley and Clem7

9001C, 1 hw 10�18 atm pO2 40–50z 400BaTiO3/Cu foil 9001C, 30 min 10�9 atm pO2 100–300 (200) 600 Ihlefeld and others8,9

wMultiple annealing. zNo evidence of grain morphology reported.

June 2008 Chemical Solution-Deposited BaTiO3 Thin Films on Ni Foils 1847

Table I. The grain size of the furnace-annealed BaTiO3 film inthis experiment is comparable to that reported in Dawley et al.’swork, where barium strontium titanate films on Ni foil wereheat-treated at 9001C in a reducing atmosphere.7 In contrast, thegrain size of the furnace-annealed film here is much smaller thanthat of BaTiO3 films on Cu foil, which consists of two equiaxedgrains of 300 nm size across the film thickness, as shown inIhlefeld et al.’s work.8,9 Because the solution chemistry of thetwo films was similar, the differences may be due to the anneal-ing conditions. In this experiment, the BaTiO3 film was precrys-tallized in an RTA at 7001C before being subjected to thefurnace anneal (10001C), while in Ihlefeld’s work, the BaTiO3

was crystallized once at 9001C. Because the RTA-annealedBaTiO3 film contains a high concentration of small grains, grainimpingement during growth may limit the final grain size duringfurnace annealing. In addition, the pO2 used for annealingBaTiO3 films on Ni foils (10�16 atm) is much lower than thatof Cu foil (10�9 atm). It has been reported that prior H2 heattreatments of BaTiO3 ceramics can freeze abnormal graingrowth in subsequent air sintering.15,16 Also, Polotai et al.have shown that the average grain size of a BaTiO3 ceramicstrongly decreased from 296 to 137 nm as the pO2 during sinte-ring decreased from 10�1 to 10�19 atm.17 Thus, it is speculatedthat the difference in grain size for films on Cu and Ni can beattributed to a combination of the number of crystallized grainsbefore the high-temperature furnace annealing and the effect ofthe pO2 atmosphere.

Although a columnar structure has been reported to improvethe dielectric properties of BaTiO3-based films,24�26 equiaxedgrains are necessary to limit oxygen vacancy mobility in basemetal electrode MLCCs. Many studies have shown that the dcinsulation resistance degradation originates from electromigra-tion of oxygen vacancies in the perovskite lattice.10,11,27�29 Inequiaxed structures, the grain boundaries parallel to the inter-face provide barriers for migration of the oxygen vacancies cre-ated by the reducing atmosphere processing. Thus, it isencouraging that there are multiple grains across the thicknessof quarter-micrometer-thick CSD-derived layers.

(3) Chemical Analysis and Oxygen Nonstoichiometry

Oxygen nonstoichiometry is very important to the electricalproperties and reliability of BaTiO3 capacitors. In an attemptto study this parameter, furnace-annealed (reoxidized) BaTiO3

films on Ni foils were investigated by electron energy loss spec-troscopy attached to a transmission electron microscope. Theenergy loss peaks for Ba, Ti, and O were observed using the BaM4,5 (781 eV), Ti L2,3 (455.5 eV), and O K edges (532 eV), re-spectively. Because the films were processed at low pO2, it is alsoof interest whether there is any residual C in the film. The C Kedge is at 283.8 eV. In this experiment, all EELS spectra werecalibrated (to shift the peaks to the correct position) by using theonset of the O K edge.

Figure 4 shows a cross-section of a bright-field image ofBaTiO3 on Ni foil. EELS analysis with an electron beam sizeof B10 nm was randomly performed on four to five grains ofthe furnace-annealed BaTiO3 film on Ni foil. All grains showsimilar EELS spectra, which consist of a strong C K edge peakassociated with the existence of BaM4,5, Ti L2,3, and O K edges,as shown in Fig. 5. EELS spectra of the Ni foil show only the NiL2,3 edge (854 eV) without the C K edge. The result suggestedthat C really existed in the BaTiO3 film and was not solelysurface contamination.

Oxygen nonstoichiometry is usually determined using theTi2,3 and O K edges. A reference EELS spectrum of an air-sintered, precious metal electrode (PME) MLCC (Fig. 6b)shows the fine structure in the Ti L2,3 and O K edges for oxygen-stoichiometric dielectrics. The splitting of the Ti L3 edge (lowerenergy) into eg and t2g features and the high intensity of the O Kedge peak series are characteristic of a low concentration ofoxygen vacancies in the PME dielectrics. Of interest for thisdiscussion is the clarity of the splitting in the Ti L2,3 edge.

EELS analysis on a furnace-annealed BaTiO3 film on Ni foilwas performed at three to four areas as a function of distancefrom the Ni electrode. Fig. 4 is marked with four spots where theEELS spectra were acquired. EELS spectra for the C K, Ti L2,3,O K, and BaM4,5 edges were first collected from each spot usingan energy dispersion of 0.5 eV/channel to cover the broad rangeof C, Ti, O, and Ba edge positions. Finer resolution of the Ti L2,3

and O K edges was obtained by remeasuring each spot with anenergy dispersion of 0.1 eV/channel.

Figure 6a shows broad-range EELS spectra from the fourspots in Fig. 4. It was found that, in this area of the BaTiO3 film,every spot contains at least some C. Lower C concentrationswere often observed near either the Ni electrode or BaTiO3 grainboundaries (spots 1 and 3). The EELS spectrum near the Ni foil(Point 1) shows no splitting of eg and t2g peaks and a lower in-tensity of the O K edge, implying a partial reduction of the Tioxidation state from 41 to 31, which is usually associated withlow oxygen content. The spectrum at Point 4 shows almost fullyoxidized Ti, similar to the PME dielectric. The fact that near-stoichiometric BaTiO3 was observed in some regions of the

Fig. 4. Cross-sectional bright-field image of BaTiO3 film on Ni foil(positions for electron energy loss spectroscopy analysis are marked.)

200 300 400 500 600 700 800 900

920880 960

Cou

nts

Energy-loss (eV)

840

Ni L2,3

C K

Ti L2,3

O K

Ba M4,5

Ni

BaTiO3 grain

Fig. 5. Electron energy loss spectroscopy spectra of BaTiO3 grain andNi foil, showing the existence of C in BaTiO3.

1848 Journal of the American Ceramic Society—Dechakupt et al. Vol. 91, No. 6

sample suggests that the reduced regions observed elsewherewere not simply the product of ion milling. It is important tonote that other regions of the sample did not show any system-atic trend in the oxygen content as a function of the distancefrom the Ni foil.

High-resolution transmission electron microscopy (HRTEM)was performed to investigate the BaTiO3/Ni interface, as shownin Fig. 7. The clear atomic arrangement in the BaTiO3 confirmsthat there is no contamination on the sample surface or damagedue to sample preparation. The Moire fringes in the Ni regionare not fully understood. It is noted that NiO was not observedin HRTEM images.

The Ni foil surface shows some image contrast at the topB5–10 nm. Figure 8 shows representative EELS spectra obtainedfrom deeper in the Ni region and from this interfacial layer. TheNi L2,3 edge confirms that in both cases the Ni is in a metallicstate, not an oxide.30 Thus, the associated O K edge is due to Oabsorbed on the Ni surface. The interfacial layer shows primar-ily metallic Ni L2,3 with a small Ba M4,5 edge and no O K edge.The result suggested that the interfacial layer is a Ni–Ba alloy.EELS spectra of the interfacial layer and Ni region were con-firmed at three to five spots and the same results were obtained.

Yang et al.10,11,14 have previously reported the presence of aNi–Ti–Ba alloy layer (4–15 nm thick) between the BaTiO3 andthe Ni in commercial MLCCs.14 It was found that the metallic

alloy layer develops during the co-firing step (13001C, 10�10 atmpO2) and survives the reoxidation step (8001C, 10�8 atm pO2).Carbon residue was proved to be the cause of the metallic alloyinterface layer and the local variation of oxygen content in theco-fired BaTiO3. Belated decomposition of the organics, espe-cially C, from the Ni paste may rob oxygen from the BaTiO3

lattice, resulting in oxygen vacancies near the electrode.10

It is interesting that the alloy composition seen in the thinfilms does not apparently show any Ti. This could be a result ofthe lower process temperatures used for thin film firing than forbulk ceramics. It will be important in the future to determine thealloy composition more quantitatively and to estimate its impacton the Schottky barrier layer at the film/foil interface.

EELS analysis of the films from this work shows spatialdistribution in both the C and the oxygen content in reoxidizedsamples. The existence of C is probably a consequence of itsincomplete removal from the organic precursors in the film. Thisis reasonable given the low pO2 conditions used. Because BaO isvery stable at the annealing conditions, it is proposed that BaOwas reduced by a low local pO2. C. Yang et al. suggested thatthe process is also catalyzed by the Ni electrode.14 The observedspatial nonuniformity in the C content may be linked to the lo-cal variations in the Ti31/Ti41 ratio.

300 400 500 600 700 800 900

Cou

nts

Energy-loss (eV)

Ti L2,3

Point 1

Point 2

Point 3

Point 4

C K Ba M4,5

O K

(a)

440 460 480 500 520 540 560

PME BaTiO3

Point 4

Point 3

Point 2

Point 1

eg Ti L2,3

Cou

nts

Energy-loss (eV)

tgO K

(b)

Fig. 6. Electron energy loss spectroscopy spectra collected fromBaTiO3 film as a function of distance from Ni foil (a) broad-range spec-tra, (b) resolved spectra of the fine structure of Ti L2,3 and O K edgesincluding that of precious metal electrode BaTiO3.

Fig. 7. High-resolution transmission electron microscopy image of fur-nace-annealed BaTiO3 film/Ni foil interface.

O K

Cou

nts

Energy-loss (eV)

Ni L2,3

Ba M4,5

Ni foil

Interfacial layer

400 500 600 700 800 900

Fig. 8. Electron energy loss spectroscopy spectra collected from Niregion and interfacial layer between BaTiO3 and Ni.

June 2008 Chemical Solution-Deposited BaTiO3 Thin Films on Ni Foils 1849

IV. Conclusions

The BaTiO3/Ni interface quality as well as the dielectric micro-structure was studied for BaTiO3 films on Ni foil. The films werepolycrystalline, which was considered an advantage for reducingthe migration of oxygen vacancies. The small grain size of thefilm is believed to be due to the highly reducing atmosphere andthe existence of well-developed crystallites produced during theRTA step. NiO is revealed by slow-scan X-ray diffraction butnot observed in cross-sectional TEM samples. Therefore, it isexpected that the NiO is spatially distributed in the film andpossibly concentrated at the grain boundaries of Ni. EELS analy-sis showed that C is retained in the furnace-annealed BaTiO3

films, which is probably due to the incomplete organic removalin reducing atmosphere. Both reduced and nearly stoichiometricBaTiO3 were observed. In addition, an interfacial layer of Ni–Baalloy was revealed between Ni and BaTiO3.

Acknowledgments

We also gratefully acknowledge the technical contributions of Trevor Clark, aswell as useful discussions with Mike Randall.

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