Local slope evolution during thermal annealing of polycrystalline Au films

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IOP PUBLISHING JOURNAL OF PHYSICS D: APPLIED PHYSICS

J. Phys. D: Appl. Phys. 45 (2012) 435301 (9pp) doi:10.1088/0022-3727/45/43/435301

Local slope evolution during thermalannealing of polycrystalline Au filmsG M Alonzo-Medina1, A Gonzalez-Gonzalez2, J L Sacedon2, A I Oliva1

and E Vasco2

1 Centro de Investigacion y de Estudios Avanzados del IPN Unidad Merida, Depto de Fısica Aplicada,AP-73-Cordemex, 97310 Merida, Yucatan, Mexico2 Instituto de Ciencia de Materiales de Madrid, Consejo Superior de Investigaciones Cientıficas, 28049Madrid, Spain3 Dept. Fısica de la Materia Condensada, Universidad Autonoma de Madrid, 28049 Madrid, Spain

E-mail: galonzo@mda.cinvestav.mx

Received 4 May 2012, in final form 2 September 2012Published 6 October 2012Online at stacks.iop.org/JPhysD/45/435301

AbstractThe morphological evolution of thermally annealed polycrystalline gold films was studied interms of several statistical parameters of the growing surface, determined by x-ray diffractionand scanning probe microscopy, including roughness, in-plane and out-of-plane grain size andlocal slope distributions. The morphology transformations occur as a result of the balance ofattractive and repulsive interactions between surface structures emerging at different lengthscales, which comprise a competition between stress relaxation via surface currents and straingeneration. This balance is responsible for the formation of large multigrain structures via thebundling with in-plane reorientation of neighbouring grains, related to attractive interaction onthe short length scale, and the generation of grooves and surface discontinuities betweenstructures repelling each other, on longer length scales. These results shed light on the surfacephenomena occurring during post-growth annealing of T-zone structured, polycrystalline goldfilms.

(Some figures may appear in colour only in the online journal)

1. Introduction

The growth front developed during deposition of polycrys-talline films at low temperatures (T ≈ 0.2–0.4Tmelting) andmoderate film thicknesses (d ≈ 15–300 nm) is usually char-acterized by round-like shaped surface protrusions (SPs) lim-ited by local boundaries forming a surface boundary network[1–8]. Although the SP-shaped morphology is typically ob-served at the coalescence stages of growth [2], the moderaterange of thicknesses at which SPs are observed is far fromthe film post-closure stage. In that case, SPs are identifiedas the surface components of a deep competitive-type colum-nar film structure [8–10]. During thermal annealing of thinfilms, several effects produce the partial rupture of the surfaceboundary network. As an example, the annealing of Au filmsof d > 30 nm under high mobility conditions promotes the re-laxation of the SP shapes by surface diffusion, coalescence bygrain boundary migration/reorientation and bulk defect diffu-

sion, with the consequent transformation of the surface bound-ary network [3, 5]. As expected [11–13], this transformationreduces the SP boundary energy. Recently, a continuum modelof stress evolution was proposed that explains the SP shapeevolution on the basis of the balance between attractive andrepulsive forces arising between neighbouring grains in closecontact [13]. This balance depends on the local surface chemi-cal potential, the normal stress distribution, and consequently,on the local slope values m at the bottom zones of the SPflanks [13–15], and acts mainly on the surface boundary net-work. However, although the analysis of the m values duringthermal annealing of thin films can provide valuable informa-tion about the origin of the morphology evolution, this methodhas not yet been developed in detail.

In this work, experimental results concerning the trans-formation of the surface boundary network of polycrystallineAu films during annealing are discussed. The evolution ofsurface statistical parameters such as roughness, lateral size

0022-3727/12/435301+09$33.00 1 © 2012 IOP Publishing Ltd Printed in the UK & the USA

J. Phys. D: Appl. Phys. 45 (2012) 435301 G M Alonzo-Medina et al

of surface structures and distributions of local slopes obtainedat the bottom of their flanks, allow us to provide an explana-tion of the phenomena involved in the morphology transfor-mations. Here, SPs refer to the small single-crystalline struc-tures usually observed by scanning probe/electron microscopymethods at thin film surfaces, which are frequently referredto as grains [3, 7, 8]. The term grain has also been assignedelsewhere to near-single crystal components observed in poly-crystalline films by transmission electron microscopy [16–18],which have a broad range of sizes, in some cases includingmany SPs in low-misorientated configurations [18]. To avoidambiguities in the interpretation of the term SP and taking intoaccount the typical deep competitive-type columnar structureof polycrystalline films [9, 19], we will employ the term sur-face grain (SG) previously used by Porath et al [5]. Thus, aSG is defined as an SP with an out-of-plane extension in whichthe crystal coherence is maintained [20].

2. Experimental

2.1. Thin film preparation

Au films were grown by thermal evaporation on SiOx /Si(1 1 1)substrates at a temperature of Tgrowth = 85 ◦C ∼= 0.27Tmelting.The pressure of the growth chamber was maintained at10−7 mbar. The film thickness was d = 200 nm for allthe samples, and the deposition rate is 1 nm s−1. Afterdeposition, these films were immediately transferred to a pre-heated oven with Ar gas flux at PAr = 1 atm, and annealedat Ta = 100 ◦C ∼= 0.28Tmelting, for different annealing timesranged from ta = 0 s, hereafter referred to as ‘as-grown’, tota = 1.2 × 105 s.

2.2. Structural and morphological characterization

The films were cooled down to room temperature and thencharacterized by both amplitude-modulation atomic forcemicroscopy (AFM) and x-ray diffraction (XRD) techniques.For AFM measurements, high aspect ratio tips of nominalradius∼= 2 nm were used. For XRD analyses an X’Pert four-circle diffractometer with CuKα radiation was employed. Allsamples exhibited a preferred Au(1 1 1) out-of-plane texture,which is usually found in Au films grown on amorphoussubstrates [18]. The crystalline coherence length, a measureof the out-of-plane SG size λ⊥ [20], was determined by XRDθ/2θ scans through the analysis of the preferred orientationpeak (Au(1 1 1)) with Scherrer’s formalism [21].

2.3. Local in-plane texture and slope distributions

The local in-plane texture of the films was determined usingthe azimuthal φ-dependence of the local slope distributionsN(m) [22]. To obtain the N(m) distributions, a Lagrangepolynomial interpolation procedure was employed. Thisprocedure transforms the digital AFM matrix of local heightsh�r into a continuous function, being �r the position vector.The image function results in a mathematical ‘mould’ of thereal film surface. The local slope values can be calculatedas m(�r) = Tan(Cos−1(�n · k/||�n||)), with �n ≡ (nx, ny, nz) ∝

[−∇xh(�r), −∇yh(�r), 1] as the normal vector to the surfaceand k = 0, 0, 1 as the upward unit vector [23]. In order toobtain the azimuthal φ-dependence of the N(m) distributions,the generated m(�r) values are represented in polar plots.Here, φ = Tan(nx/ny) the angular coordinate, defined as theazimuth angle of the normal unit vector at each �r and m(�r) asthe radial part. The azimuthal φ-distributions N(m, φ) allowus quantifying the in-plane orientation of the m(�r) values ofcomplete images as N(m) = ∑2π

φ=0 N(m, φ).To address the atomistic scenario we will use the terrace

width values nr = (2√

2/3m) − 1/2 to represent the numberof rows of the fcc adsorption sites on a (1 1 1)-terrace, insteadof the m(�r) values [8]. The nr values in the stereographicN(nr, φ) distributions are classified into categories of widthnr ± 1/8 and φ intervals of ±5◦ [23]. As the surface boundarynetwork overcomes major morphological transformationsduring annealing, only nr values obtained at this zone willbe used for the computation of N(nr) and N(nr, φ). For thispurpose an AFM image tessellation procedure based on theidentification of local minimum and saddle points of h(�r) [23]was employed, each one defined as a SP contour point. Inorder to obtain a representative sampling of the nr values, anextended zone of four pixels at each side of a contour wasconsidered.

2.4. Lateral size of SGs and roughness

Using the contour map of each SP, the distances betweenthe intersections of the contour in the x, y directions weredetermined, assigning the maximum distance to the SPlateral size. The surface roughness w, defined as the rootmean square fluctuation of the local heights h(�r) as w =√

(1/N)∑

N (h(�r) − 〈h(�r)〉)2 where N is the number ofpixels, was determined over the complete area of the AFMimages.

3. Experimental results

3.1. Evolution of surface morphology and statisticalparameters

Figures 1(a)–(d) display the morphological evolution of thesurface of polycrystalline Au films at the annealing timeta whereas figures 1(e)–(h) show the behaviour of severalstatistical parameters, such as the surface roughness w, theaverage lateral SG size λ, the average local slope valuesobtained at SP borders m, and the crystalline coherence lengthλ⊥. For the representation of the as-grown sample, ta = 0 s,we use an arbitrary, sufficiently small, annealing time ofta = 1 × 10−3 s. Therefore, the statistical values obtained forthe as-grown sample were not considered in the computationof the scaling exponents.

The insets in figures 1(a)–(d) show contour maps; inwhich black and white regions correspond to surface boundarynetwork points and inner domains of SGs, respectively. Asobserved in figure 1(a), the surface of the as-grown sample iscovered by SGs, which are significantly smaller than the largenear-single crystalline plates often observed by transmission

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J. Phys. D: Appl. Phys. 45 (2012) 435301 G M Alonzo-Medina et al

Figure 1. (a)–(d) AFM images (1 × 1 µm2) of 200 nm thickpolycrystalline Au(1 1 1) films, annealed for different times: (a)ta = 1 × 10−3 s (as-grown), (b) 3×102 s, (c) 3×103 s and (d)1.2 × 104 s. The insets in (a)–(d) display contour maps of eachsurface. Also the x scale and a horizontal colour bar (z scale) areincluded for each sample. (e)–(h) Evolution of statisticalparameters: (e) film surface roughness, (f ) lateral SG size, (g) slopeat the SG border and (h) out-of-the-plane (in-depth) SG size.Intermediate annealing times fall inside the time interval highlightedby the vertical shaded band.

electron microscopy for Au/SiOx films [17, 18]. In thiswork, we focus only on the morphological transformationsof the SGs.

Therefore, topics inherent to plausible formation oflarge-sized single crystals as well as details of their filmmicrostructure are out of the scope of this paper. From figure 1,three morphological stages of the annealing process can beclearly differentiated:

(i) Early stage, ta � 3 × 102 s: the surface morphology isgoverned by small and round-like SGs. As ta increases,SGs expand laterally whereas both w and m suffer a slightdecrease.

(ii) Intermediate stage, 3 × 102 s < ta < 1.2 × 104s: thefilm morphology changes, as revealed by the incrementin the w, λ and m values. The value for λ is roughly

doubled, which leads to the development of larger surfacefeatures, as indicated by the contour maps in figure 1(c).These surface features will be called hereafter bundles ormultigrain structures.

(iii) Late stage, ta > 1.2 × 104 s: the film morphology iscomposed by irregular and large multigrain structuresseparated by deeper intersections. These structuresremain roughly constant their corresponding average sizeλ, whereas both w and m values increase. The average out-of-plane SG size, λ⊥ < d, shows a continuous increase forta > 3×102 s, in agreement with an in-depth growth of theSGs as annealing proceeds. Similar behaviours have beenobserved during the annealing of thinner polycrystallineAu films [3, 20].

3.2. Deep competitive-type columnar film structure

Figure 2 shows a SEM cross-section of a film after samplescratching where a columnar structure is observed as well as thedifractograms of the main (1 1 1) orientation for the as-grownand annealed gold films. The columnar structure shown infigure 2(a) is similar to the structure usually observed for hardermetal thin films [9], such as Cr, Cu and Al, for which tensile-fracture procedures are appropriate to observe the columnarstructure.

However, the high ductility of Au promotes filmstretching/thinning during tensile-fracture processes. Thecross-sections developed by this method often produceelongated-like shapes that hide the real film structure. Toovercome this disadvantage, we intentionally scratched thefilm surface with an STM tip and observed the cross-section profile on the crest of folds that developed where thefilm occasionally breaks without apparent deformation. Assuggested by both ∂tλ⊥ > 0 and λ⊥/d < 1 conditions,columns correspond in fact to a vertical stacking of grains. Thisis in qualitative agreement with the theoretical models based onfull diffusion kinetic Monte Carlo simulations for multigraincolumnar structures [19]. The corresponding normalizeddiffractograms for the as-grown and annealed films are shownin figure 2(b). A minor full-width at half-maximum is observedafter the annealing process β2, as compared with the as-grownfilm β1.

3.3. Evolution of the local in-plane texture and slopedistributions

The local in-plane texture of the films at the surface boundarynetwork zone was investigated by means of the azimuthalN(nr, φ) distributions, displayed in figure 3. The ring-like shape of the N(nr, φ) distribution of the as-grown filmobserved in figure 3(a) shows that after Au deposition onthe SiOx amorphous substrate, a random distribution of thein-plane orientations of SGs is generated. This randomdistribution is maintained during the early stage of annealing,as shown in figure 3(b). For the intermediate stage a reductionin the azimuthal dispersion at several points of the N(nr, φ)

diagram, spaced 〈�φ〉 ≈ 2π/3, is highlighted in figure 3(c).

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J. Phys. D: Appl. Phys. 45 (2012) 435301 G M Alonzo-Medina et al

Figure 2. (a) SEM cross-section of the as-grown film (Au/SiOx /Si(1 1 1), d = 200 nm). The film was intentionally scratched in order toreveal the film microstructure. Two zones are observed in SEM focus: (A) the polycrystalline film surface showing the surface protrusionsand (B) the film columnar structure, included between dashed lines. The parameters used for SEM imaging were: magnification= 2.4 × 105×, source–sample distance = 6.3 mm, accelerating voltage = 5 kV, spot size = 3 mm, landing energy = 2.5 keV and a lowvoltage high contrast detector. (b) Normalized x-ray θ/2θ -scans of the as-grown (black) and ta =1.2×104 s (red) films around the Au(1 1 1)peak.β1 and β2 are the full-width at half-maximum of each peak.

The azimuthal dispersion tends to diminish for the latestage of annealing, and even the threefold symmetry appearsto worsen, as is shown in figure 3(d). Figures 3(e)–(h) displaythe evolution of the N(nr) distribution with the annealing time,showing right-skewed curves [8, 23]. They qualitatively followa similar ta-evolution to the N(nr, φ) distributions. Eachnr < 9 fcc site rows width of the N(nr) distributions has beendeconvoluted into their normal components of the same width.Each component has an average value nr0 ranging betweennr0 ≈ 1 and seven fcc site rows. For the as-grown sample, themain normal component was observed at nr0 = 5 fcc site rows.At the early stage of annealing, a smoothening of the N(nr)distribution is evidenced as an increment in the population ofthe nr0 = 6 fcc site rows.

During the intermediate stage of the SG coalescence,understood here as a type of grain welding in a continuousfilm, a significant shift of the N(nr) distributions towardslow nr values is produced. Thus, as the populations of thecomponents of the N(nr) distributions with average valuesnr0 = 2–3 fcc site rows increase, a reduction in the N(nr)dispersion occurs. For ta � 1.2 × 104 s, the nr values decreaseslightly, as displayed by the dominant nr0 = 2 component. Therupture of the surface boundary network as a consequence ofthe SG coalescence can be caused by several phenomena, suchas surface diffusion and/or grain zipping [15, 24]. During grainzipping, grain boundaries are shifted up, with a surface energybalance γ = 2γs sin[arctan(m)] − γgb > 0, with γ , γs and γgb

the resulting, surface and grain boundary energies, respectively[13–15]. In a perfect grain zipping process, grain boundariesdisappear without the prevalence of any surface neighbourpartner, forming large single-crystal grains. Alternatively,low-misoriented grains forming shallow boundaries at theinner structure of the multigrain features are developed duringpartial grain zipping.

In contrast, the continuous increment of both the surfaceroughness and the average slope values points to an unzippingphenomenon in the remaining surface boundary network,composed by the outer perimeters of the multigrain features.During unzipping, SG boundaries are shifted down, withγ < 0and grain flanks separated from each other, through a processof surface boundary network aperture.

4. Model of interaction forces between SGs

4.1. System modelling

Adapting the model of Tello et al [13], we proposea phenomenological interpretation for the morphologicaltransformations and the consequent evolution of the statisticalparameters. For this purpose the surface morphology of theas-grown film is modelled as an array of elastic SGs attachedto the remaining film substrate characterized by a competitivecolumnar structure, as sketched in figure 4(a). Small SGs areidentified as round-like features on the film surface z = S(�r)with an average height h ∝ w, which are separated laterallyfrom each other by λ. They have a deep cylindrical-likestructure with an out-of-plane length λ⊥. The distance betweenopposing points along S(�r) located inside valleys separatingSGs is represented by δ(z). A similar initial configuration of agrowing grain ensemble and its consequent state of stress hasbeen employed to describe a kinetic continuum model of stressevolution during grain growth [13].

4.2. Short-range interactions between small SGs

The interaction forces arising between grains in close contactcan be approximated through a cohesive zone law [25], which

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J. Phys. D: Appl. Phys. 45 (2012) 435301 G M Alonzo-Medina et al

Figure 3. Azimuthal φ-dependence of the local nr distribution,N(nr, φ), for 200 nm thick polycrystalline Au films at annealingtimes of: (a) t = 10−3 s (as-grown), (b) 3 × 102 s, (c) 3 × 103 s and(d) 1.2 × 104 s. (e)–(h) display the area-normalized N(nr)distributions for the respective ta(a)–(d). Arrows indicate the meanvalues nr0 of the normal components (Gaussian-like fittings) thatintegrate the N(nr) distributions. Blue/red–shaded areas representthe �′N(nr, ta) = N(nr, 3 × 102) − N(nr, 6 × 10−3) differences((f )–(e)) and �′′N(nr, ta) = N(nr, 1.2 × 104) − N(nr, 3 × 102)((h)–(f )) indicating a gain (loss) in the population of flanks withnr0-type terraces. Dashed circles in N(nr, φ) correspond to nr0

values. Inset: fcc-(1 1 1) ball model of a SP flank zone with terracesof width nr =2 (lower) and nr =3 (upper).

in a simplified form is given by [13]

σN = t (δ)�ny · �n t (δ) = σmδ

�Exp

[1 − δ

�

], (1)

where σN the stress normal to the film surface, t (δ) is themodule of the z-dependent traction force between opposingpoints of neighbouring grains that share a local valley, �n is thesurface normal vector and �ny its y-component, δ is the widthof the local valley at a height z and σm is the maximum value oft (δ) reached at δ = �, typically in the range ≈0.1–1.0 nm [13].Figure 4(a) shows the physical meaning of the mentionedparameters. Equation (1) describes the length dependence of

the interactions between SGs: for δ < 0 SGs overlap and repeleach other, turning σN compressive, t (δ) < 0 ⇒ σN < 0,whereas for δ > 0 SGs feel attraction between them, turningσN tensile, t (δ) > 0 ⇒ σN > 0, reaching a minimum valuewhen δ → ∞. When δ = 0, the equilibrium distance betweentwo grains that meet at the grain junction is reached.

During annealing, thermally activated mass transportoccurs, driven by a curvature-dependent chemical potential,µ, of the form [13]

µ = −�(γsκ + σN), (2)

where � is the atomic volume, γs is the isotropic surfaceenergy and κ is the surface curvature. This chemical potentialleads to time-dependent downhill surface currents of adatomsJ�

s (t) that fills local valleys between grains, which can beestimated by

J�s (t) ∝ −Ds �∇sµ(t) ∝ Ds�(γs �∇sκ(t) + �∇sσN) (3)

with Ds the surface diffusion coefficient, and �∇s the surfacegradient vector. Near the surface boundary network, tractionforces �Ty = t (δ)ny [13], with ny = �ny/||�ny ||, generate anormal stress that can promote the valley closure, as sketchedin figure 4(b). The presence of surface currents J�

s (t) duringthe valley closure enables the relaxation of the applied stressσN in two complementary manners:

σN ≈ M

λ

[�

2− �

�

h

∫ ta

0J�

s (t)∂t

]≈ M

λ

(�r

2

). (4)

(i) By grain zipping, represented by the first term in the squarebrackets in equation (4), which induces a longitudinal strain��/2λ per SG within the grain bulk, with � ≈ 2�, at anenergy cost γ �

e ∝ M�2/λ, where γ �e is the interfacial elastic

energy, M = E/(1 − v) is the biaxial elastic modulus forisotropic media, E is Young’s modulus and ν is Poisson’scoefficient.(ii) By valley filling up to a height h through J�

s (t) currents,as described by the second term in the square brackets ofequation (4), as is shown in figures 4(b) and (d). The simplifiedexpression at the right-hand side of equation (4) indicates thatthe elastic contribution accommodates only a residual partM�r/2λ of the maximum stress that is not relaxed by the J�

s (t)

currents. This accommodation results in a lower elastic energycost γ �

e ∝ M�2r /λ, with �r the reduced valley width after

zipping. Thus, SGs under traction forces react by expandinglaterally and minimizing the energy cost for zipping via valleyfilling. The mechanism of valley closure comprising grainzipping and valley filling explains the formation of multigrainstructures. In addition, the mechanism explains the increasein the lateral size of the SGs, the drop in the surface roughnessand the increase in the populations of terraces of width nr0 = 6fcc site rows in the N(nr) distributions, as can be observed infigures 1(e)–(g) and 3(e)–(g).

4.3. Grain reorientation in short-range interactions

For the intermediate stage, a reduction in the azimuthaldispersion in the N(nr, φ) distributions is observed in

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J. Phys. D: Appl. Phys. 45 (2012) 435301 G M Alonzo-Medina et al

Figure 4. (a)–(c) Cross-section profiles after: (a) film growth, showing a compact array of SGs on a remaining elastic film structure. (b) SGzipping assisted by valley filling through downhill surface currents (red arrows), forming multigrain structures of two small grains. (c)Grooving of the valley between multigrain structures. Dashed profiles in (a)–(c) represent the film morphology at a previous ta-instant.(d)–(f ) Top view representations of height-constant planes of two grains that interact during closure, showing: (d) zipping by traction(straight blue lines) and valley filling (straight red lines) at a height in which SGs are in no contact. (e) SG reorientation involving graintorsion by force moments (curved blue lines) and rotation by peripheral surface currents (curved red arrows). (f ) Grooving of the outermultigrain structure boundaries supporting same-signed (clockwise) torsion. Dotted contours in figures (d)–(f ) represent SG perimeters at ata-instant after elastic grain interactions.

figure 3(c), suggesting that a grain reorientation phenomenonoccurs during the valley closure. In such a process, it canbe expected that a torque τ arises near the SG junctions bythe reduction in the angular misfit between crystal planes ofthe grains involved in the closure. Simultaneously, a decreasein the energy ascribed to the formation of incoherent grainboundaries, γgb(θ), is produced, with θ the misorientationangle [11, 12, 26]. The torque required for grain reorientationgenerates a shear stress σs(θ) on the planes contained in theemerging parts of each grain implicated in the valley closure,as illustrated in the sketch in figure 4(e). Such a stress can berelaxed during reorientation in two complementary manners,according to the equation:

σs ≈ G

[θ

2− �

4�

α2λ2

∫ ta

0J θ

s (t)∂t

]≈ G

(θr

2

). (5)

(i) By inducing a torsion strain θ/2 ∝ σs/G per grain,as represented by the first term in the square- brackets inequation (5), with an elastic energy cost γ θ

e ∝ Gh(λθ/2)2,with G the shear modulus and h ∝ w the average height. Theparameter α and θr in equation (5) represents the contact angle

between SGs, and the reduced angle of grain reorientation,respectively, and,(ii) By grain rotation through peripheral surface currentsJ θ

s (t) tangent to S(�r), as described by the second term inthe square brackets in equation (5), resulting in a minimalenergy configuration of the accommodated material as the SGboundary reorientation process progresses. The simplifiedexpression on the right-hand side in equation (5) indicatesthat the elastic contribution accommodates only a residual partG(θr/2) of the maximum stress that is not relaxed by the J�

s (t)

currents, resulting in a lower elastic energy cost of γ θe ∝ Gλθ2

r .The J�

s (t) currents involved in grain rotation can be drivenalong SG perimeters by a modified chemical potential, µ, ofthe form

µ = −�[γsκ + σN] + (αλh/2)γgb(θ). (6)

This modified chemical potential gives rise to the developmentof the time-dependent J�

s (t) currents needed to reduce themisorientation by mass transport, which can be expressed as

J θs (t) ∝ −Ds �∇sµ(t) ∝ −(αhDs) �∇θγgb(θ, t). (7)

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J. Phys. D: Appl. Phys. 45 (2012) 435301 G M Alonzo-Medina et al

Equation (7) establishes that once the reorientation processproceeds, peripheral surface currents J θ

s (t) are producedfrom highly misorientated grain boundary zones nearby thoseones with low misorientation, which have a lower energyγgb(θ). Consequently, J θ

s (t) currents produced by ∇θµ(t) aremaintained up to σs(θ) = 0 in the case of a perfect crystalalignment or toward coincidence-site-lattice configurationswith minor average misorientation angles θ → θCSL.

5. Discussion

5.1. Interplay between short and long-range interactions

The process of valley closure comprising the correlatedphenomena of valley closure and reorientation propagatesacross the film surface. Once multigrain structures formedby two small grains are developed, they can interact withneighbouring small grains or with other small multigrainstructures to produce larger multigrain structures. In any case,the surface features must be reoriented during coalescence inorder to minimize the grain boundary energy γgb(λ, θ), whichrequires an amount of elastic energy γ θ

e ∝ Gλθ2r . This effect

limits the average lateral size of grains, as was experimentallyobserved for larger annealing times in figure 1(f ). In separatework [27], an analysis of the energetic requirements for thezipping/reorientation processes was developed, in which it wasargued that grain reorientation can relax σs into lengths upto λ ≈ 500 nm. In this work, the highest normal and shearstress values generated during coalescence were estimatedto be ≈ 225 MPa and ≈0.8 GPa, respectively, both valueswithin the elastic limits reported for polycrystalline Au films[28, 29]. These results support our proposed assumption of theprominent role of the elastic phenomena in stress relaxationduring the annealing process.

For the unzipping phenomenon, the appearance of lownr values and high surface roughness for the late stage ofannealing suggests a repulsive interaction between multigrainstructures. As an example, a particular case of interactionbetween multigrain structures formed by two grains isillustrated in figure 4(f ). In this case, two oppositemechanisms with the same origin act at different lengthscales: at the boundary between the bundles, the small SGsof each bundle that face each other support the same-signrotation torques, which produce a shear stress excess on thisouter boundary and the consequent boundary aperture. Incontrast, the small SGs inside bundles support opposite-signrotation torques, promoting zipping with alignment insidethe multigrain structure. The range of values for the grainboundary energy of the as-grown film can be deduced usingthe force balance γ = 0 = 2γs sin(arctan[m]) − γgb, withγs = 1.54 J m−2 the Au(1 1 1) surface energy [15]. Thisenergy can be estimated in a rough approach from the N(nr)distributions of figure 3(a), resulting in 0.4–0.8 J m−2 for theearly stages of annealing, where nr0 = 3–7. The shearstress adds an elastic energy γes to the SG boundary. Toestimate the order of magnitude of γes we assume that about10 atoms, corresponding to about 2.5 nm at both sides ofthe grain boundary plane, are under shear stress. Estimation

gives a value of γes ≈ 0.35 J m−2, which is on the sameorder of magnitude as the as-grown equilibrium energy. Thepossible zipping of the grain boundary from the as-grownequilibrium situation produces an increment of the totalshear energy associated with the grain boundary development.Consequently, some equivalent work is needed to restore theequilibrium condition. However, the work available fromthe surface area reduction per unit of grain boundary lengthdecreases as the zipping advances and, consequently, thereduction in the surface energy does not compensate theincrement of shear energy produced by grain zipping in thiszone. As a result, unzipping takes place in order to reacha new equilibrium situation, promoting the propagation inthe depth of the valley, with higher slopes [30]. Conversely,the anti-parallel torque configuration favours the realignmentand zipping at the multigrain boundaries. If there is not anyconstraint, parallel or anti-parallel torques can be randomlyproduced. At the end of the annealing process, an incrementof the nr0 = 1, 2 fcc site row terrace population appearsin the N(nr) distributions, as shown in figures 3(c) and (d).These nr0 values match with (1 1 0) and (3 3 1) fcc facetswith a stable (1 1 1) nanofaceting as observed in the (0 1 1)-fcc surfaces [31, 32]. As measured [33], the nr0 = 1 cancorrespond also with the (1 1 3) facet stabilized by segregatedimpurities.

5.2. Possible role of nanofaceting in long-range interactions

The increment in the density of terraces with nr0 = 1, 2indicates that unzipping is favoured by a final configurationwith local minimal energy surfaces. The development ofsuch facets implies that uphill surface currents appear duringunzipping. These currents moving material from zones withexcess of shear stress at the grain boundary towards theborders of (1 1 0), (3 3 1), and (1 1 3) nano-faceted zonesdeveloped at the surface, in order to extend these localminimum-energy structures. Hence, unzipping can plausiblyevolve by a nanofaceting-driven phenomenon, where shearstress produces an excess of elastic energy at the multigrainboundaries with adjacent parallel torques. Also, metastablesmall surface configurations with lower nr terraces and lowersurface energies are developed, which are enlarged duringannealing.

5.3. Kinetics of recrystallization

Although the main goals of this work are the experimentalstudy and the interpretation of the interplay betweenzipping/unzipping and reorientation processes during anneal-ing, the growth kinetics deduced from figures 1(f ) and (h) areprovided. The in-plane SG growth follows the scaling relationλ ∝ tp=0.16, with p the coarsening exponent. This expres-sion can be approximated as λ ≈ c ln(t). Similar p valueshave been reported for Au [34] (p = 0.2) and Pb (p = 0.14)

films in experiments with comparable conditions [35]. Thelow coarsening values are interpreted as a consequence of areduced mobility produced by pinning [35] and related to im-purities and/or a delayed migration due to complex movementsof the grain boundaries [36].

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J. Phys. D: Appl. Phys. 45 (2012) 435301 G M Alonzo-Medina et al

The grain growth kinetics can be described in terms ofa grain boundary migration at the surface, driven by theminimization of the system energy across the film surface.This phenomenon enables a reduction in the total area ofboundaries and/or the formation of low-angle grain boundaries[11, 16, 17]. It is assumed that the grains grow at a rate of∂λ/∂t ∝ MGB�, where MGB and � denote the grain boundarymobility and the driving force for the SG boundary migration,respectively [16, 36–38]. For bulk (surface) grains, it holds that� ∝ κ (∝ κs) is proportional to the grain curvature [36, 39]κ ≈ λ−1, with κs ≈ ∇2

s λ−1 ∝ λ−3 the grain curvatureprojected on the film surface. The scaling exponent p = 0.16obtained from our experiments for the in-plane SG growthλ ∝ t0.16 is lower than the value predicted exponent for anideal grain growth scenario (p = 1/4), indicating that thegrain expansion is ruled by a modified surface potential drivingdiffusion. The surface process of zipping with reorientation notnecessarily corresponds to an uniform alignment of the crystalplanes along the film thickness. Within the present model, itis reasonable to assume that the grain growth at the surfaceis controlled by the progressive alignment between two SGsdetermined by J θ

s , which introduces a delay in the in-planesurface boundary migration ascribed to the grain expansion.Hence, the boundary migration rate is proportional to thevariation of the misorientation, as ∂θ/∂t = 4��J θ

s /α2λ2.Assuming that J θ

s decays as 1/λβ , we get ∂λ/∂t ∼ ∂θ/∂t ∝1/λ2+β , with β = 1 for lattice diffusion currents and withβ = 2 for perimeter currents along the grain boundary, aspredicted by the theoretical models [37]. Solving ∂λ/∂t ascaling law for the lateral expansion of the SGs as λ ∝ t1/(3+β)

can be obtained, resulting in λ ∝ tp=1/5 for the case of surfacediffusion along grain perimeters. This scaling law agrees withthe relationship experimentally obtained by Mancini et al [34],being slightly higher in a factor of �p ∼= 0.04 to the hereobtained, p = 1/7, and to the value reported for Pb films [35].Figure 1(h) shows that the out-of-plane grain size obtainedby the Scherrer formalism obeys a scaling law of the formλ⊥ ∝ ln(t) � tn=0.1, with a speed v⊥ ≡ ∂λ⊥/∂t ∝ tn−1

and n → 0, equivalent to v⊥ ∝ f (t)−1∂f (t)/∂t , with f (t)

a time-dependent scaling function f (t) ∝ tn and n > 0 agrowth exponent. The main interest of this scaling law isits time dependence rather than its functional dependence,which is in agreement with a recrystallization phenomenonthat initiates at the surface and propagates into the bulk.This dependence was also deduced [39], from annealingexperiments of polycrystalline Au films of d = 75 nm at Ta =423 K during 1 h [39]. In these experiments, the increment inλ⊥ during the annealing process follows a similar behaviourto that reported here, i.e. the λ⊥ ∝ ln(t) dependence. Thisdependence can be obtained from the linear-log representationof the data included in figure 5(b) of [39], in the rangeta ∼ 102–1.5×103 s. For higher annealing times, the datashow a tendency towards saturation when λ⊥ ≈ 55 nm andd = 75 nm. Our results must be distinguished from thoseobtained by theories and/or experimental interpretations basedon models in which the grain reorientation phenomenon isdeveloped simultaneously along the vertical grain boundaries[11, 38]. We propose a tentative model to interpret the out-of-plane grain growth speed v⊥ ∝ t−1. In this model we

used an in-depth rotation function R(t, z) and a rigid grainboundary parallel to the substrate, whose properties remainconstant during the migration process. This grain boundary isin equilibrium under a rotation gradient. Details of the modeland the development of the expression for the migration speedhave not been included in this work. A detailed discussion ofthe structure of this parallel boundary with a nanometric sectionappears out to reach for the available theory and experiments.

6. Conclusions

The surface morphology transformations during the annealingof polycrystalline Au films under moderate mobility conditionsare mainly triggered by SG zipping together with reorientationand unzipping. Scanning probe microscopies combinedwith image tessellation methods allows the determinationof the evolution of roughness, size, and perimeter slopesof SGs on the annealed surfaces. XRD gives the in-depthincrease in the diffraction coherent length. In the proposedinterpretation, elastic energies associated with zipping andcrystalline alignment are diminished by mass transport throughcurrents driven by surface chemical potential gradients.The unzipping phenomenon at multigrain boundaries is aconsequence of the relative orientation of torques generatedby the intra-multigrain crystalline realignment. This modeldescribes a recrystallization process controlled by surfacecrystalline reorientation. Surface migration kinetics and in-depth crystalline coherence propagation agrees with such amodel.

Acknowledgments

This work was supported by the projects F1-54173(bilateral program CSIC-Conacyt), 200960I182 (CSIC) andCCG10-UAM/MAT-5537 (DGUI-Comunidad de Madrid andUniversidad Autonoma de Madrid). A G-G acknowledges thesupport of the MICINN Spanish Ministry and CICYT underthe projects ESP2006-14282-C02-02, AYA 2009-14736-C02-01 and PEI201160E053. Authors thank Dr J A Aznarez foruseful discussions, Dr Gerko Oskam for his comments aboutthe manuscript, and M E Emilio Corona Hernandez for histechnical assistance.

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