Equilibrium Shape of Copper Crystals Grown on Sapphire

INTERFACE SCIENCE 12, 7–18, 2004c© 2004 Kluwer Academic Publishers. Manufactured in The Netherlands.

Equilibrium Shape of Copper Crystals Grown on Sapphire

DOMINIQUE CHATAIN∗

Centre de Recherche sur les Mecanismes de la Croissance Cristalline†, CNRS, Campus de Luminy,Case 913, 13288 Marseille, France

[email protected]

VERONIQUE GHETTALaboratoire de Thermodynamique et Physico-Chimie Metallurgiques, ENSEEG-INPG, BP 75,

38402 Saint Martin d’Heres, France

PAUL WYNBLATTDepartment of Materials Science and Engineering, Carnegie Mellon University, Pittsburgh, PA 15213, USA

Abstract. The equilibrium crystal shape (ECS) of copper has been studied by scanning electron microscopy onµm-sized copper crystallites supported on single-crystals of α-alumina. In addition, the orientation relationshipsbetween copper crystals and the sapphire substrate were investigated by X-ray techniques. A detailed discussion ofthe kinetic factors that can inhibit equilibration is provided, and it is shown that only crystals ranging in radius from3 to 4.5 µm can achieve equilibrium shapes under the conditions of the experiment. The maximum anisotropy ofsurface energy was found to be about 1.02, which is significantly lower than that of the other two fcc metals (leadand gold) for which reliable data are available. Another distinction between copper and those other fcc metals is thatits ECS displays {110} facets, and possibly {311} facets, in addition to the commonly observed {111} and {100}facets, at temperatures where equilibration is possible. The observed facets connect tangentially to the curved partsof the ECS, so that all possible surface orientations are present on the copper ECS.

Keywords: surface energy, equilibrium shape, anisotropy, copper

1. Introduction

The anisotropy of surface energy of crystalline mate-rials has most frequently been determined from stud-ies of the equilibrium shape of isolated crystals [1–10], or of cavities in single crystals [11, 12]. Manysurface properties of materials are related to surfaceenergy anisotropy, including the equilibrium crystalshape (ECS), the stability of certain surface orien-tations, the phenomenon of faceting, the energy ofmono-atomic steps on surfaces, as well as other sur-face characteristics.

∗To whom all correspondence should be addressed.†Laboratoire associe aux Universites d’Aix-Marseille 2 et 3.

The focus of the present work was a determination ofsurface energy anisotropy of pure copper from an anal-ysis of the ECS of micron-sized crystals supported onsingle crystals of α-alumina. Because of the slow rateof the mass transport processes which allow solids toreach their ECS, shape equilibration of solids can onlybe achieved in reasonable times at temperatures closeto the melting point. In addition, equilibration must beconducted under conditions which maintain the purityof the surface. For such reactive materials as lead [4,8, 10], indium [5], gold [3, 9] and silicon [7], stud-ies of the ECS have therefore been performed underultrahigh vacuum (UHV) conditions. In contrast, ma-terials such as high melting temperature oxides can beinvestigated in air [11, 12]. Copper cannot be studied

8 Chatain, Ghetta and Wynblatt

in UHV because of its high vapor pressure at the tem-peratures needed for equilibration. Thus, the presentstudy has followed the approach used in previous stud-ies of the copper ECS [13, 14], by equilibrating at atmo-spheric pressure in a carefully controlled gas mixture[15] aimed at suppressing evaporation while maintain-ing a surface free of adsorbed impurities. This is par-ticularly important, since the surface energy of copper,as well as its anisotropy, are modified by oxygen ad-sorption [13, 14]. Experiments were also performed tostudy the effects of adsorbed oxygen on the ECS ofcopper, but these will be reported elsewhere.

The surface energy anisotropy of copper has beeninvestigated previously. The first study was reported inthe pioneering work of Sundquist [2] who attempted tomeasure the orientation dependence of surface energyfor several face centered cubic (fcc) metals by study-ing their ECS. Unfortunately, at the time that work wasperformed, methods for ensuring that surfaces werefree of adsorbed impurities had not yet been developed.Robertson and Shewmon [16] measured the grooves as-sociated with the intersection of both twins and grainboundaries with the surfaces of polycrystalline coppersamples, and obtained some limited information on sur-face energy anisotropy. McLean [13], and McLean andHondros [14] also studied grain boundary grooves, ei-ther in foils or wires with bamboo grain structures, andreported the surface energy anisotropy for a large num-ber of surface orientations within the standard stereo-graphic triangle. McLean and Hondros [14] performedtheir studies as a function of oxygen partial pressure,and included results at sufficiently low oxygen pres-sures that no measurable influence of adsorbed oxygenon the surface energy could be detected. However, thesurface orientations they studied were randomly dis-tributed within the stereographic triangle, so that the in-formation density along the principal crystallographiczones was insufficient for a precise identification offaceted orientations on the ECS. The present study hasfocused on those principal zones, and has also providedthe direct observation of facets.

2. Experimental

Samples were prepared as follows. Copper of 99.999%purity was deposited in ultrahigh vacuum by physicalvapor deposition onto α-alumina single-crystal sub-strates of 99.993% purity. The resulting copper filmswere about 400 nm thick. The optically polished sub-

strates had a {1120} oriented surface or a surface planetilted by a few degrees from the {1120} orientationabout the 〈0001〉 axis. They were annealed at 1500 Kfor six hours under a 50%He-50%O2 gas mixture at atotal pressure of 105 Pa (1 atmosphere) prior to copperdeposition.

After deposition, the samples were transferredthrough air to a furnace operating under a He-H2 atmo-sphere at a total pressure of 105 Pa [15]. The fractionof hydrogen in the atmosphere was varied from 5 to30% in different experiments. Samples were heated to1373 K, just above the melting point of copper, to breakup the metal films into discrete drops (ranging in ra-dius from about 250 nm to 5 µm). After holding at thistemperature for 10 min, the samples were cooled downto 973 K so as to crystallize the droplets (and over-come any possible undercooling). The copper dropletscrystallize mostly into single crystals, although someparticles contain low angle grain boundaries. In orderto equilibrate the shape of the copper crystals, a sec-ond annealing treatment was performed at 1240 K forvarious times ranging from 4 to 105 hours.

The oxygen partial pressure, p(O2), was measuredin the gas flowing downstream from the sample, usingan external electrochemical oxygen sensor operating at1023 K. The active gases which fix the oxygen partialpressure in the gas mixture are hydrogen and traces ofwater vapor. The oxygen partial pressure in the furnaceat 1240 K is obtained from the p(O2) value measuredby the sensor at 1023 K, together with the standardfree energy change for the H2 +1/202 = H2O reactionat equilibrium [15]. Under an atmosphere containingmainly hydrogen, the oxygen partial pressure can beextremely low (e.g., for p(H2O)/p(H2) = 1.1 × 10−5

at 1240 K, p(O2) = 10−20 Pa). It has been shown [17]that if the p(H2O)/p(H2) ratio is less than 10−3, thegaseous mixture is not buffered and behaves like purehydrogen. The sensor cannot measure an equilibriumoxygen pressure but gives a miminum value for the ac-tual p(O2) in the furnace atmosphere. Thus, our work-ing atmosphere has an oxygen partial pressure lowerthan ∼10−15 Pa. The lowest values of p(O2) reportedby Hondros and Mclean [14] in their experiments arequestionable. In fact their values were calculated fromthe standard free energy change for the H2 + 1/202 =H2O reaction, from a p(H2O) measured upstream fromthe samples with a dew point sensor. As in our appara-tus, their p(H2O)/p(H2) ratio is too low to permit thep(O2) to be fixed. The actual atmospheres in their fur-nace were probably buffered only for values of p(O2)

Equilibrium Shape of Copper Crystals Grown on Sapphire 9

Figure 1. Typical secondary electron image of a sample showingthe range of crystallites size.

higher than 10−16 Pa (at 1200 K, where their experi-ments were performed).

After heat treatment, samples were quenched toroom temperature by pulling them into the cold zoneof the furnace. This procedure led to an initial cool-ing rate of several hundred degrees per minute, fromwhich we conclude that the high temperature structureis retained without significant modification. This con-clusion is supported by the relatively slow equilibrationkinetics discussed below.

The ECS of the copper crystallites was investigatedby examining samples in a field emission gun scan-ning electron microscope (SEM, Jeol 6320F). Sampleswere carbon coated prior to examination in order to pre-vent charging. Shape measurements were performed onSEM photomicrographs acquired either as digital im-ages (1024×1024 pixels) and/or by conventional pho-tographic techniques. Figure 1 is a typical secondaryelectron image of a sample which shows that crystal-lites range in radius from less than 500 nm to about5 µm.

3. Orientation Relationships of Copperon Sapphire

The crystallographic orientation of the as-depositedfilms and of the annealed samples have been deter-mined by taking pole figures (Siemens D5000 X-Raydiffractometer). Before annealing, copper films showa Cu(111) fiber texture, with an almost isotropic dis-tribution in the surface plane. After heat treatment, the

resulting copper crystals adopt a variety of orientationswith respect to the α-alumina substrate. The most fre-quently observed orientations were those where eithera {111}, {100} or a {211} Cu plane was approximatelyoriented parallel to the substrate surface, with {221}and {233} planes appearing less frequently. The major-ity of crystallites are oriented (111)Cu//(1120)α-alumina.After annealing, the crystallite distribution in the planeof the substrate surface is no longer random, but is re-lated to the sapphire orientation. However, according tothe pole figures, it appears that for the longest anneal-ing treatments, the epitaxy is less marked. This changecould result either if some copper crystallites undergoorientation changes during annealing, or if preferentialdisappearance of epitaxed crystallites takes place, lead-ing to an apparently more random distribution of theones that remain. The latter hypothesis is more likely,given that the smallest crystallites are probably betterepitaxed and that they also evaporate more quickly thanthe larger ones.

The most frequently observed orientation is(111)Cu// (1120)α-alumina with (110)Cu//(0001)α-alumina.Other orientations, which correspond to a rotation ofthe copper lattice around the 〈110〉 axis, while keep-ing the (110)Cu//(0001)α-alumina relation unchanged,are possible, with (112), (221) or (332) parallel to the{1120} sapphire plane.

Since solid copper poorly wets sapphire [18] its ECSis truncated by the substrate well below its Wulff point.Under these conditions, perimeters of the copper ECScorresponding to sections through its Wulff point maybe viewed along an axis perpendicular to the sapphireplane and along directions tilted by up to about 15◦

from this axis.In the scanning electron microscope used for this

study, samples may be rotated and tilted in one direc-tion. Thus, copper crystals oriented with {100} surfacesapproximately parallel to the substrate could be ro-tated and tilted so as to be viewed under conditionswhere the crystal perimeter traces out the 〈100〉 zone,which contains {100} and {110} facets, as shown inFig. 2(a). Crystals with {211} planes parallel to the sub-strate could be reoriented with respect to the electronbeam so as to view the crystals along a 〈110〉 direc-tion. From this direction, the perimeter of the crystaltraces out a 〈110〉 zone which contains {100}, {311},{111} and {110} facets viewed on edge, as shown inFig. 2(b). The {311} facets are so small that they arenot distinguishable on the profile but they are visibleon the crystal close to the {100} of the edge. We will


Figure 2. Copper crystals viewed along (a) a 〈100〉 direction, (b) a 〈110〉 direction, and (c) a 〈112〉 direction. The right hand side panels areschematic crystal shapes produced using Wulffman freeware [19].

refer to the orientations represented by Fig. 2(a) and(b) by the viewing directions, namely, as the 〈100〉-and 〈110〉-orientations, respectively. In Fig. 2(c), wealso display a crystal viewed along a 〈211〉 axis; in thisviewing direction the crystal perimeter contains {111},{110} and {311} facets.

4. Copper Crystal Shape and SurfaceEnergy Anisotropy

4.1. Image Analysis

Information on crystal shape was extracted from theperimeter of crystals such as those displayed in Fig. 2.

To a first approximation, the crystal shape may be de-scribed as a sphere truncated by facets of particularsurface orientations. Quantitative shape data were ac-quired by digital image analysis methods, which com-bined the use of both commercial software (Adobe Pho-toshop), and generally available freeware (NIH Image).

Some image correction was necessary prior to anal-ysis, because of an elongation of the images relatedto sample charging in the SEM, in spite of coating ofthe samples with a thin carbon film. Tests of imageelongation performed on conducting samples showednegligible distortion, but similar tests on our samplesshowed a systematic elongation of samples primarilyalong the vertical direction, the magnitude of which


amounts to a few percent. Thus, images were correctedby contracting them in the vertical direction, until theangles between {100} and {111} facets were consis-tent with crystallographic requirements. This approachyielded satisfactory results for crystals viewed alongthe 〈100〉 and 〈110〉 axes (Fig. 2(a) and (b)). For anECS profile viewed along a 〈211〉 direction (Fig. 2(c)),the small size of the {110} facets on such a perimeterprevents suitable image correction required for quan-titative analysis. However the presence of facets canbe detected by constructing a qualitative γ -plot (seebelow).

4.2. Relationship Between ECS and Anisotropyof Surface Energy

The equilibrium shape of a crystal can be calculatedfrom a knowledge of the anisotropy of surface energyby means of the Wulff construction. When all surfaceorientations are present on the ECS, it is also possible toperform the Wulff construction in reverse, and thus ex-tract the anisotropy of surface energy. The mathematicsunderlying the Wulff construction are equivalent to aLegendre transform [20].

We use the following construction to obtain theanisotropy of surface energy from the ECS (or Wulffshape). Let the 2-D shape of the crystal perimetersin photomicrographs such as Fig. 2 be described asyW (xW ), with an origin (Wulff point) defined by thecenter of gravity of the perimeter, and a facet perpen-dicular to the y-axis as shown in Fig. 3. Then, any pointon the Wulff shape (xW , yW ) represents a particular sur-

θ

xw,yw

xγ,yγ

y

x

γ/γ(0)

ECS

Figure 3. Schematic of inverse Wullf construction. One quarter ofthe ECS is shown. The point (xW , yW ) on the ECS is associated witha surface energy γ , given by the distance of the tangent at that pointfrom the center of the ECS (Wulff point).

face orientation, defined by the angle θ . At θ = 0, thedistance to the facet is yW (0). The radial line at the an-gle θ is perpendicular to the tangent to the ECS at (xW ,yW ) and crosses it at the point (xγ , yγ ). The distanceto the tangent between the Wulff point and the point(xγ , yγ ) is proportional to the surface energy at θ . Thesurface energy, γ , as a function of orientation, referedto the energy at the orientation θ = 0, may then beobtained from simple geometric considerations as:

γ (θ )/γ (0) = (xW sin θ + yW cos θ )/yW (0) (1)

The absence of sharp edges on the facets in SEM im-ages such as Fig. 2 indicates that all facets are smoothlyconnected to the curved portions of the ECS, and thatall possible orientations are present on the copper ECS.This is discussed in more detail in Section 4.3.

4.3. Equilibrium Anisotropy Alongthe 〈100〉- and 〈110〉-Zones

We report here measurements of the anisotropy of sur-face energy along the 〈100〉- and 〈110〉-zones of cop-per, i.e. along all three edges of the standard stereo-graphic triangle. Figures 4(a) and (b) show the resultsobtained along these two zones. The figures representaverages over images of 7 and 6 crystals, respectively,equilibrated for periods of between 78 and 105 hours.The estimated error in the reported anisotropy is of thesame order as the size of the symbols used in those twofigures. All of the crystals used in computing the aver-age behavior had radii in the range from 3 to 4.5 µm.The reason for selecting this size range for the deter-mination of the equilibrium anisotropy will becomeclear from the discussion of equilibration kinetics givenbelow.

Figure 4(a) shows deep cusps at the {100} orienta-tions located at 0 and 90◦, indicating the presence of{100} facets at those orientations. Also, the fact that ori-entations very close to 0 and 90◦ appear in this figureprovides evidence for the existence of all possible ori-entations vicinal to {100} on the ECS. This result is con-sistent with the absence of an edge around {100} facetson the images of crystals. Similarly, Fig. 4(b) showsa deep cusp near 55◦ where a {111} facet is expected.Energies can be computed for orientations very close to{111}, thus confirming that all orientations near {111}exist on the ECS. Again, this is consistent with the ab-sence of an edge around {111} facets in the images ofcrystals. It is interesting to compare these results with


1

1.004

1.008

1.012

1.016

0 15 30 45 60 75 90

γ /

γ (1

00)

angle (°) from (100) along <100>-zone

{110}

(a)

1

1.005

1.01

1.015

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γ /

γ (11

1)


{311}

(b){111}

1

1,005

1,01

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1,025

1,03

0 15 30 45 60 9075

γ /

γ (11

1) -

qua

litat

ive

scal

e


{311}

(c)

Figure 4. Anisotropy of surface energy of copper along (a) the〈100〉-zone (data averaged over 7 large crystals), and (b) the 〈110〉-zone (data averaged over 6 large crystals). (c) is a qualitative plot ofanisotropy along the 〈211〉-zone of one crystal, which illustrates thepresence of a cusp (or local minimum) at the {311} orientation.

those obtained previously for other fcc metals. Cer-tain details of the lead ECS have undergone extensiverecent study by scanning tunneling microscopy [21].At temperatures where shape equilibration is possible,

lead like copper shows {100} and {111} facets and ex-hibits all possible surface orientations, i.e. the vicinalsurfaces connect tangentially to the {100} and {111}facets. However, in the case of gold which also dis-plays {100} and {111} facets [4, 9], there are missingorientations in the vicinal range around those facets,with the result that the facets are separated from thesurrounding curved parts of the ECS by sharp edges.

Figure 4(a) for 〈100〉-zone shows that there is a sharpcusp at 45◦ from the two {100} facets at 0 and 90◦,which corresponds to the presence of {110} facets onthe ECS. This is the first time that {110} facets havebeen reported on the ECS of pure fcc metals, at tem-peratures where equilibration can be achieved in rea-sonable times. In lead crystals, {110} facets have beenreported on growth forms [22], but these facets disap-pear by a roughening transition below the temperaturesat which equilibration is possible. Thus, the rougheningtransition temperature for the {110} facets in copper oc-curs at a higher homologous temperature than in otherfcc metals. A slightly less pronounced cusp can alsobe seen along the 〈110〉-zone in Fig. 4(b), at 90◦ from{100}, where the {110} orientations are present.

We have also observed small {311} facets in someimages (see Fig. 2(a)). In Fig. 4(b), there is a small in-dentation in the vicinity of 25◦ from {100} where {311}facets are expected. This indentation could represent asmall cusp. Better evidence for the presence of a {311}cusp can be seen on Fig. 4(c) after treatment of theshape of a crystal viewed along the 〈211〉-zone axis; aclear minimum around 31.5◦ from {110} is displayedon a fairly flat region of this γ -plot. However, it shouldbe recalled that 〈211〉-zone images cannot be correctedfor quantitative shape analysis (see earlier discussion).The fact that {311} facets are not always clearly visiblein images may indicate that the equilibration temper-ature of 1240 K is close to the roughening transitiontemperature of {311} facets.

The maximum surface energy anisotropies of cop-per, γ max /γ {100} ∼ 1.014 along the 〈100〉-zone, andγ max /γ {111} ∼ 1.020 along the 〈110〉-zone, foundhere are similar to those found by Hondros and McLean[13, 14] for pure copper at 1200 K. The facets energiesrank in the same order: γ {111} < γ {100} < γ {110}.However, the value of the ratios γ {110}/γ {100} andγ {100}/γ {111} are different: we have 1.012 and 1.004,whereas they report 1.006 and 1.008, respectively. Itshould be noted that all of these values lie around ananisotropy of 1% or less. Such differences could per-haps result from the 40 K difference in temperature


between our respective measurements, but are moreprobably related to the different techniques used. Themethod used by Hondros and McLean generates dataover the whole stereographic triangle, and relies on ex-trapolation and interpolation to identify the maximumand minimum values of surface energy. The presentapproach does not attempt to measure the anisotropyfor all orientations, but rather focuses on the orien-tations along the edges of the stereographic triangle,which include the cusp orientations. Thus our resultsare likely to provide a better measurement of the max-imum anisotropy along the edges of the stereographictriangle.

5. Equilibration Kinetics

In order to establish the conditions under which coppercrystals can reach their equilibrium shapes, several as-pects of equilibration kinetics were examined. In a firstset of experiments aimed at investigating shape changeas a function of equilibration time, it was found that theshapes from which the copper crystals evolve towardsequilibrium is controlled by surface-active impurities.A second set of experiments, performed to assess thesize dependence of the crystal shape, provides evidencefor the mechanisms of equilibration, and for the exis-tence of energy barriers which prevent the equilibriumshape from being achieved in certain size ranges. Theseexperiments are now described in more detail.

5.1. Shape Change Controlled by the Removalof Adsorbed Impurities

At short equilibration annealing time, the ECS of cop-per is affected by the presence in the copper of residualsurface-active impurities. Evidence of surface contam-ination is provided by Fig. 5. Figure 5(a) shows a seg-ment of a crystal after annealing at the equilibrationtemperature (1240 K) for 4 hours in a He-5%H2 mix-ture. It can be seen that the large {111} facet in the figurehas a sharp edge (unlike the facets of Fig. 2) and that thesurrounding {321} facets are themselves surrounded bya series of faceted rings which are a clear sign of surfaceimpurities. Figures 5(b) and (c) show two segments ofcrystal profiles near the {111} and {100} facets, takenfrom a sample annealed for 54 hours at 1240 K in a He-5%H2 mixture. These figures illustrate that the facetedrings have almost disappeared, and that the edges of thefacets are no longer sharp. After annealing for 78 hours,

Figure 5. Evidence of surface contamination and decontaminationwith annealing time at 1240 K, in a He-5% H2 atmosphere. (a) Af-ter annealing for 4 hours, crystals show {111} and {321} facets withsharp edges surrounded by faceted rings, indicative of surface con-tamination. (b) and (c) After annealing for 54 hours, images in thevicinity of {111} and {100} facets show the absence of sharp edgesand the disappearance of faceted rings; also {321} facets have disap-peared, although some faint rings are still visible in the vicinity of{321} poles.

all evidence of faceted rings have disappeared from thecrystals, which have achieved smooth curved regionssurrounding the facets, as shown in Fig. 2. We takethis to provide adequate evidence that the surfaces of


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(b)

{111}

Figure 6. Anisotropy of surface energy of copper along (a) the〈100〉-zone for one crystal 3.05 µm in radius and (b) the 〈110〉-zonefor a crystal 2.8 µm in radius, after annealing for 4 hours at 1240 Kin a He-5% H2 atmosphere.

samples treated for 78 hours or longer are free of anysignificant surface contamination.

Figure 6(a) and (b) display the anisotropy of sur-face energy along the 〈100〉- and 〈110〉-zones of coppercrystals after annealing for 4 hours at 1240 K in a He-5%H2 mixture. When compared to the plots of Fig. 4(a)and (b), the maximum anisotropies,γ max /γ {100} andγ max /γ {111}, are found to be about twice as large(1.022 and 1.045, respectively) due to the presence ofresidual surface-active species. However, from the rel-ative energies displayed by the various facets, com-pared to those at pure copper surfaces, it seems thatthe {110} orientations are less prone to contamination.From these figures, and the crystal morphology dis-played in Fig. 5(a), it can be concluded that duringthe first few hours of equilibration, impurities adsorb

on the facets of the densest orientations which rapidlyreach a larger size than they do on pure copper crystals.In order to reach their equilibrium size during the sub-sequent annealing which cleans the copper surfaces,these facets will have to decrease in size.

It is well known from UHV studies of surface seg-regation in binary Cu-rich alloys, that even high puritycopper contains surface-active residual sulfur [23]. InUHV studies, sulfur is generally removed by sputter-annealing cycles that eventually deplete the sulfurpresent in the bulk of the sample, thereby eliminatingsegregated sulfur at the surface. In the present study,sputter cleaning of the surface is, of course, not a fea-sible option. However, the equilibration procedure ofheating copper in flowing He-H2, gradually removesthe residual sulfur into the gas phase as H2S, therebydepleting it from the bulk of the micro-crystals in muchthe same manner as might be accomplished by sputter-cleaning. This process has its own kinetics, which in-directly control the equilibration time by imposing anincubation period during which micro-crystals are pro-gressively decontaminated of the impurity.

The effects of controlled surface-active oxygen im-purities on the ECS of Cu have been documented in pre-vious studies [13, 14]. As mentioned earlier, we havealso conducted experiments to determine the effects ofoxygen partial pressure on the ECS of copper. Althoughthe details will be presented elsewhere, it is importantfor interpretation of the present results to note that theoverall surface energy anisotropy of copper does notchange at oxygen partial pressures p(O2) of 10−13 Pa orless, but is significantly increased by oxygen adsorptionat higher p(O2). In the presence of adsorbed oxygen thetangentially connected facets which are present on thepure copper ECS develop sharp edges, indicating thatnot all surface orientations are stable when Cu surfacesare covered with oxygen. Furthermore, new facets, notseen on pure copper, are stabilized by oxygen and alsodisplay sharp edges. Thus, the results reported abovefor the effects of presumed sulfur surface contamina-tion are qualitatively similar to those observed in thepresence of controlled oxygen adsorption [14].

5.2. Equilibration by Surface Diffusion, and StepNucleation Kinetics

Before presenting the results obtained on the size de-pendence of the crystal shape, it is useful to recall thetwo main factors which control the kinetics of shapeequilibration of pure crystals. For crystals with an


initially arbitrary shape (e.g. a spherical shape inheritedfrom the solidification of a spherical drop) it is possi-ble to use the formalism of Nichols and Mullins [24]to estimate the relaxation time needed to form facetsby transporting material away from the regions wherethey develop. This provides an estimate of equilibrationtime by diffusional processes. However, the existenceof a possible mechanism for kinetic inhibition of theequilibration process has recently been put forward byMullins et al. [25]. This work has shown that crystalswhich lack step propagating defects (such as disloca-tions with a screw component intercepting the facetedsurfaces) will not equilibrate in reasonable times dueto the existence of nucleation barriers which cannot beovercome by thermal activation. Both of these factorsseem to play a role in the present experiments.

We first examine the kinetics of equilibration of crys-tals as treated by Nichols and Mullins [24]. Here it isassumed that surface diffusion is the only rate limitingmechanism of shape change. The relaxation time, τ ,for evolution of the shape from a sphere to the facetedshape by surface diffusion, is proportional to particleradius, r , to the fourth power:

τ = r4

24 B(2)

where B = Dsγ ν�2/kT, Ds is the surface diffusioncoefficient, γ is the surface energy, ν is the number ofatoms per unit surface area, � is the atomic volume, kis the Boltzmann constant and T the absolute temper-ature. The position of the facet at time t , as comparedto its equilibrium position, is given by:

δ(t) = δ(0)e−t/τ (3)

These relationships indicate that the time for crys-tals to come to equilibrium from their initial shape isstrongly dependent on their size, with small crystalsequilibrating faster than large ones. We have estimatedrelaxation times from Eq. (2), using the following val-ues of the parameters: the lattice constant of copper,a = 0.3615 nm, γ = 1.52 J/m2 [26], ν = 2/a2,� = a3/4, and Ds (at 1240 K) = 6.6 × 10−9 m2/s[27]. For the crystals used in Section 4.3 for reportingthe measured anisotropy, with radii ranging from 3 to4.5 µm, we obtain estimated relaxation times from 0.75to 3.5 hours. Equation (3) implies that facets would beexpected to be within 1% of their equilibrium positionsafter periods exceeding 5 relaxation times.

Although the kinetics of contamination removal can-not readily be evaluated, due to lack of informationon the relevant reaction rates, the photomicrographs ofFig. 5 indicate that, under a gas mixture containing 5%of hydrogen, the time for impurity removal at the equi-libration temperature is of the order of 50 hours. Af-ter particle surfaces become clean, it should thereforetake less than 20 hours (based on the above estimatesof relaxation time) for the shapes of all crystals to beequilibrated.

In order to complete the kinetic analysis, we nowaddress the inhibition of facet growth or shrinkage bynucleation barriers proposed by Mullins et al. [25].That recent work has shown that a facet of diameterd f , which is free of step-generating defects, will beunable to achieve its equilibrium size if:

d f > 80kT/(aγ f π tan2 α) (4)

where γ f is the energy of the facet orientation, and α isthe angle subtended by the facet. If we take γ f ≈ γ , andα ≈ 12◦ (from our experimental results on the {100}facets) we obtain d f > 18 nm. Thus, if all of the crystalsin the samples studied here were free of dislocationswith a screw component emerging at their surfaces, wewould expect all of the crystals to have non-equilibriumshapes. However, the probability that a given crystal isfree of dislocations decreases as its size increases. Thenumber of dislocations, n, in a spherical crystal is givenapproximately by:

n ∼ 2ρr2 (5)

where ρ is the dislocation density and r the particleradius. If we take the dislocation density in a well-annealed metal crystal to be of the order of 1011 m−2, wefind that a crystal with a radius of 1 µm should contain1 dislocation on average, and that a crystal with a radiusof 3.5 µm would contain an average of 10 dislocations.On this basis, therefore, we would expect that crystalslarger than 3.5 µm can achieve equilibrium sized facetsin our experiment (subject to diffusional constraints),whereas crystals smaller than 1 µm should have non-equilibrium sized facets, and that a region of transitionwould prevail between these two size regimes.

One final aspect of nucleation inhibition of equili-bration needs to be described, before we can identifythe particle size range expected to have equilibratedshapes in our experiments. Mullins et al. [25] haveshown that facet size in perfect particles with smaller


than equilibrium-sized facets can increase up to ∼0.5to 0.75 of the equilibrium size before facet growth is ar-rested by the nucleation barrier. In contrast, for particleswith larger than equilibrium size facets, facet shrinkageis essentially entirely inhibited.

5.3. Crystal Size Dependence of Anisotropy Alongthe 〈100〉-Zone

The maximum anisotropy of surface energy(γ max /γ {100}) for 〈100〉-oriented crystals de-termined from figures such as Fig. 4 is plotted inFig. 7, as a function of crystal radius. This figurehas been constructed from the analysis of 73 crystalsranging in radius from 0.25 to 4.7 µm. These crystalswere equilibrated for different annealing times (78to 105 hrs), under different atmospheres (5, 10 or30% of hydrogen in helium). These experiments wereperformed to determine the optimal experimentalconditions required for achieving the ECS of purecopper. We will show that the highest values ofanisotropy correspond to samples that could not reachequilibrium facet sizes as a result of some of thekinetic effects described above.

Two dashed lines have been added to Fig. 7, as aguide for the following discussion. The figure shows

Figure 7. Maximum anisotropy for 〈100〉-zone crystals as a func-tion of particle radius. Filled circles and diamonds correspond to crys-tals equilibrated under an He-5% H2 mixture for 78 and 105 hours, re-spectively; open squares and circles correspond to crystals annealedunder an He-10% H2 mixture for 79 hours; and cross-in square pointscorrespond to crystals annealed under an He-30% H2 mixture for76 hours.

three regions. In region 1, where the crystals have thelargest radii, ranging from 3 to 4.7 µm, the maximumanisotropy is roughly constant around 1.014. In region2, where the crystal radii range between 2 and 3 µm,the anisotropy is scattered between ∼1.012 and 1.028,and tends to increase gradually with decreasing crystalradius for crystals annealed under 5% of hydrogen.Finally, in region 3, the smallest crystals below a size ofabout 1.5 µm have an anisotropy that scatters between1.025 and 1.035.

First, it should be emphasized that from the stand-point of surface diffusion kinetics, all the crystalswithin the size range studied here were heat treatedfor sufficiently long times to reach their equilibriumshapes. Thus, any failure to reach equilibrium mustresult from kinetic effects associated with either de-contamination or nucleation inhibition of facet growthand shrinkage.

Region 2 is one where we expect a transition fromlarge dislocated crystals to dislocation-free small crys-tals. The larger crystals in region 1 contain sufficientnumbers of dislocations to avoid the nucleation inhi-bition of facet motion, and are therefore able to reachtheir equilibrium shapes, whereas the small crystals inregion 3 contain essentially no dislocations, and can-not achieve equilibrium sized facets. We interpret thehigher anisotropy of the small crystals of region 3 asfollows. Because of their small size, these crystals canchange their shapes very rapidly by surface diffusion.During the early stages of the heat treatment the sur-faces of all particles have substantial levels of adsorbedimpurities, which (as discussed above) will tend to in-crease anisotropy. Thus, these small crystals developlarger facets, even though the facet size may be arrestedat ∼0.75 of the equilibrium facet size in the presenceof adsorption. Since nucleation inhibition of facet mo-tion is much more severe for facet shrinkage than facetgrowth, these small crystals remain trapped with largerfacets, even though removal of impurities occurs duringthe later stages of the heat treatment.

In region 2, the large scatter in maximum anisotropyprovides evidence of the competition between removalof surface impurities and shape equilibration. Considerthe behavior for example of 2 µm crystals. For caseswhere the heat treatment atmosphere contained higherhydrogen concentrations (10 and 30%) removal of ad-sorbing impurities occurs faster, and crystallites tendto have close to equilibrium shapes. These crystalscan be purified before their facets reach larger sizesdue to adsorbed impurities, whereas in 5% hydrogen


atmospheres, the crystals have higher anisotropy be-cause they form large facets by rapid surface diffusionbefore surface impurities are removed.

There is additional evidence which supports the hy-pothesis that the large anisotropy of small particles re-sults from the fact that their {100} facets are unable toshrink back from the large size acquired early in the an-nealing process, when impurity adsorption is high. Thisevidence comes from the fact that the curved surfacesof small particles in region 3 are in “local” equilibrium.Figure 8(a) shows the raw data (in pixels) of the arcsbetween {100} facets for three particles ranging from

0

50

100

150

200

250

300

350

0 50 100 150 200 250 300 350

Y (

pixe

ls)

X (pixels)

(a)

1

1.005

1.01

1.015

1.02

1.025

1.03

1.035

0 15 30 45 60 75 90

γ /

γ (10

0)


1.25 µm

2.5 µm

3.45 µm

(b)

Figure 8. (a) Shapes of the curved surfaces between two {100}facets for 3 crystals viewed along the 〈100〉-zone; the crystal radiiare (from inside to out) 0.5, 1.5 and 4.5 µm and their maximumanisotropies are 1.035, 1.026 and 1.013 respectively. (b) Anisotropyof surface energy along the 〈100〉-zone for 3 crystals ranging in radiusfrom 1.25 to 3.45 µm.

0.5 to 4.5 µm in radius (and from 1.035 to 1.013 inmaximum anisotropy). These arcs differ only in scale,i.e. they are geometrically similar. The shape similarityimplies that the small crystals in region 3 can adjust theshapes of the curved parts of their perimeters between{100} facets to the dictates of local equilibrium, eventhough their {100} facets remain larger. Such effectshave been observed in the case of Pb crystals embed-ded in Al by Gabrisch et al. [28]. Thus, the increasein anisotropy along the 〈100〉-zone shown in Fig. 8(b),with decreasing crystal size, results directly from thegreater size of the {100} facets of the smaller crystals.It is interesting to note that the nucleation inhibitionwhich prevents the motion of {100} facets does not ap-pear to inhibit the motion of the {110} facets (that arepresent at 45◦ from the axes in Fig. 8(a). This interest-ing difference between {110} and {100} facets couldbe connected either with the much more open struc-ture of {110} surfaces, or with the lower adsorption ofimpurities on these orientations.

6. Summary and Conclusions

The ECS of copper, as well as the orientation relation-ships between copper crystals and the {1120}-orientedsapphire substrate on which the crystals were grown,were investigated. The most frequently observed ori-entation relationship is (111)Cu//(1120)α-alumina with(110)Cu//(0001)α-alumina. Other orientations, whichcorrespond to a rotation of the copper lattice around the〈110〉 axis, while keeping the (110)Cu//(0001)α-alumina

relation unchanged, are possible, with (112), (221) or(332) parallel to the (1120)α-alumina plane.

In the study of the copper ECS, only crystals withradii in the range of 3 to 4.5 µm are able to achieveequilibrium shape under a gas mixture containing 5%of hydrogen. If the fraction of hydrogen is increasedto 10% or more, crystals of radius as low as 2 µm canequilibrate. Smaller crystals acquire non-equilibriumshapes due to the coupling of two kinetic processes:the rate of removal of adsorbed impurities, and the in-hibition of facet motion by step nucleation phenomena.The latter process does not affect the equilibration of3 to 4.5 µm particles because they contain sufficientlylarge numbers of dislocations to ensure a supply ofgrowth-promoting steps at the surface.

The anisotropy of surface energy compares well withthat found by Hondros and McLean [13, 14]. However,there is some discrepancy of about 0.5% in the relativeenergies of the facets. Such differences could possibly


result from the 40 K difference in equilibration tem-perature, but more probably from the techniques used.The present approach does not attempt to measure theanisotropy for all orientations, but rather focuses on theorientations along the edges of the stereographic trian-gle, which include the minimum and maximum surfaceenergy orientations along the principal crystallographiczones. Thus, our results are likely to provide a bettermeasurement of the maximum anisotropy along thosezones.

The maximum anisotropy of copper is significantlysmaller than that of the other fcc metals, lead and gold,for which reliable ECS data are available. Additionaldifferences between copper and those other metals in-clude the presence of {110} facets and the possible pres-ence of {311} facets on the ECS. Lead and gold only dis-play {111} and {100} facets at temperatures where equi-libration can be achieved in reasonable times. Thus, theroughening temperatures of {110} and {113} occur atrelatively higher homologous temperatures in copper.Finally, since all facets in copper are connected tangen-tially to the surrounding curved surfaces, its ECS dis-plays all possible surface orientations. This is similarto the behavior of lead, but different from that of gold,where some orientations vicinal to {111} and {100} areunstable.

Acknowledgments

We thank J.C. Heyraud and S. Nitsche from CRMC2 inMarseille and J. Fouletier from LEPMI in Grenoble forhelpful discussions. One of us (PW) wishes to acknowl-edge with thanks the support of his work by the NationalScience Foundation, under grant DMR9820169.

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Equilibrium Shape of Copper Crystals Grown on Sapphire