effects of substrate shape, curvature and roughness on thin heteroepitaxial films of pt on au(111)

Surface Science 602 (2008) 2863–2875

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Surface Science

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Effects of substrate shape, curvature and roughness on thinheteroepitaxial films of Pt on Au(111)

Anant Mathur, Jonah Erlebacher *

Department of Materials Science and Engineering, Johns Hopkins University, 102 Maryland Hall, 3400 North Charles Street, Baltimore, MD 2118, United States

a r t i c l e i n f o a b s t r a c t

Article history:Received 17 March 2008Accepted for publication 14 July 2008Available online 3 August 2008

Keywords:Thin film growthPlatinumGoldHeteroepitaxy

0039-6028/$ - see front matter � 2008 Elsevier B.V. Adoi:10.1016/j.susc.2008.07.024

* Corresponding author. Tel.: +1 410 5166077.E-mail address: [email protected] (J. Erleb

The growth of thin (1–10 nm) films of Pt on Au(111) was studied in order to understand and clarify dif-ferences in growth mode observed in ultra-high vacuum (UHV) studies and in electrochemical depositionstudies. It was found that on flat Au(111), Pt grows in a layer-by-layer growth mode, but if the gold sub-strate is exposed to an acidic environment prior to Pt deposition, then the substrate becomes nanoscop-ically rough (islanded) and Pt growth follows a pseudo-Stranski–Krastanov (SK) growth mode in whichan initially thin wetting layer becomes rougher with increasing film thickness. An analysis of curvatureeffects on epitaxial growth mode shows that thermodynamic curvature effects involving surface stressare negligible for the Pt/Au(111) system. Rather, the apparent SK growth is linked to kinetic effects asso-ciated with inhomogeneous in-plane elastic relaxation of Pt films on rough surfaces that drive Pt atomsfrom pits to the tops of islands in the early stages of growth. Implications for the control of epitaxial filmroughness are discussed.

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1. Introduction

1.1. Motivation for studying platinum heteroepitaxy on gold

Within the heterogeneous catalysis community, significantattention is being paid to core-shell nanostructured materials, inwhich the properties of a thin (1–5 nm) shell are modified by elec-tronic or mechanical interaction with a thicker core [1–3]. Much ofthis attention is being driven by theoretical predictions. For exam-ple, in the development of carbon oxygen reduction catalysts forfuel cells, theoretical predictions and experiments have suggestedthat a monolayer of Pt on Pd(111) is superior as a catalyst for thischemical reaction than Pt alone [4], the current material of choice.As a result of these and similar studies, a large effort is being madeto fabricate both core-shell nanoparticles possessing industriallyuseful high surface areas, as well as model ultra-thin epitaxial filmson substrates [5]. Comparisons between the nanoparticulate mate-rials, thin films, and theoretical models are quite challenging. Theyare complicated by the polyfaceted nature of nanoparticles, ther-modynamic curvature effects, and also by differences betweenthe (electro)chemical environments and/or the ultra-high vacuum(UHV) environments that are chosen for their manufacture. In thispaper, we describe a study of 1–10 nm thick epitaxial films of Pt onsmooth and chemically roughened planar Au(111).

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acher).

The motivation for this work comes from an observation madewhile making Pt-plated nanoporous gold (Pt-NPG) [6,7]. NPG is abicontinuous mesoporous metal containing open porosity fabri-cated by chemical dealloying of silver from silver/gold alloys, typ-ically in nitric acid [8]. The structure that remains after dealloyingAu35Ag65, for example, is an open porous metal, nearly pure gold incomposition, possessing high surface area (�5 m2/g) with poresand ligaments each approximately 15 nm in diameter. As grainssizes in the alloy are typically >10 microns wide, the pore size ofNPG is significantly smaller than the grain size, and thus when con-sidering its microscopic properties and morphology, NPG can beconsidered an extended porous single crystal. For electrochemicalapplications, NPG is useful because it contains the high surfacearea of nanoparticles, but in an interconnected, mechanically ro-bust network in which every point on the surface is electricallyaccessible. NPG can be made into a core-shell catalytic materialby electrolessly reducing Pt onto its surface, creating thin films�1–5 nm in thickness, using methods described in Refs. [6,7].Micrographs of the morphology of Pt-NPG are also found in thesereferences. Briefly summarizing those results, we found that1 nm (average)-thick films appeared conformal and smooth, butthe 5 nm (average)-thick films appeared ‘‘bumpy”, consisting of is-lands 3–4 nm in diameter and few nanometers high. In both cases,however, the films were epitaxial, as indicated by the obvious lat-tice fringes in high resolution transmission electron microscope(TEM) micrographs, as well as sharp electron diffraction patterns.

In Ref. [7], it was hypothesized that the observed growth of Pton NPG was a manifestation of Stranski–Krastanov (SK) growth

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2864 A. Mathur, J. Erlebacher / Surface Science 602 (2008) 2863–2875

of Pt on highly curved substrates. However, there is no report of SKgrowth in the literature on Pt/Au for films greater than one mono-layer (ML) thick (a literature review is provided in Section 1.2). Thebase-level gross morphological difference between NPG and planargold is the curvature of the substrate, of order 0.1 nm�1 in NPG andzero for planar substrates, and it was thought that the high curva-ture of the ligaments of NPG changed the surface stress and the lat-tice mismatch conditions, leading to SK growth. Surface stresseffects on NPG are of current interest, especially as they influenceactuation in electrochemical environments and the mechanicalproperties of NPG [9]. However, changes in substrate latticeparameter with curvature due to surface stress turn out not to belarge enough to affect Pt growth on NPG, as we show in Appendix1.

A second hypothesis regarding the growth of Pt on NPG wasthat approximating nanoporous gold as a smoothly curved surfacewas not a good approximation to the real morphology, and that abetter structural model (suggested by simulation [10] but not ob-served, as yet, by any microscopy) was that the surface of NPG iscovered with small facets just a few nanometers wide. Becausegrowth in this picture is not occurring on infinitely wide substratesas it would for planar deposition, perhaps the elastic boundaryconditions for injections of misfit dislocations and pseudomorphicgrowth are different from the assumption in the usual Matthews–Blakeslee model [11]. Growth on small facets might allow in-planerelaxation, perhaps preferentially driving SK growth over pseudo-morphic growth. To test this hypothesis, we developed highly ori-ented (111) single crystals of NPG that exhibited ‘‘mesa-like”porosity, i.e., a flat topped surface with a variety of terrace widths.If the growth mode were affected by finite terrace widths, then thefilm morphology on the narrower terraces should be bumpy,exhibiting SK growth, while growth of the wider terraces shouldbe layer-by-layer and remain planar. The sample preparation andmicroscopy of this study are interesting and will be reported else-where, but the results of this study were negative – planar growthwas seen over all accessible terrace widths from �50 nm to�500 nm.

These observations combine to suggest that there was some-thing special about surface of gold in the chemical environmentin which we made Pt-NPG, and the bulk of this paper concerns thisissue. Ultimately, our conclusions are that chemically induced sur-face roughness on Au forms during dealloying in acid and does notanneal away quickly upon taking the sample out of the electro-chemical environment; rather, surfaces of gold exposed to acidare rough, specifically consisting of nanoscopic islands that tem-plate the apparent SK growth of Pt on Au. We find that this effectcan be mimicked in the UHV environment, and may be useful tomake films with controlled roughness on the nanometer length-scale. The primary purpose of this paper is to describe this thin filmwork, but our results are also applicable to growth on porous met-als. In hindsight, perhaps these observations are not surprising –the observation of surface roughness induced on gold due to adsor-bates (especially water) has been seen before [12]. Our contribu-tion is to extend an examination of this effect to subsequent thinfilm growth.

1.2. Pt/Au growth: literature review

The earliest report on Pt/Au growth we have found is byMatthews and Jesser from 1967 [13]. They grew Pt thin films onAu surfaces of various low-index orientations in UHV including(001) and (111) using molecular beam epitaxy (MBE) and usedex-situ TEM to study the films after dissolving away soluble sub-strates on which gold substrate films were initially grown. Growthat 40 �C and 300 �C yielded identical results for all orientations – no3D nuclei were formed at any stage of growth, the Pt films remained

planar with misfit dislocations appearing at a critical thickness of�1 nm, and the spacing between misfit dislocations was �25 nm.They concluded, therefore, that Pt/Au growth occurred in theFrank-van-der–Merwe (FM) mode. In contrast, Sachtler et al. in1981 [14] examined MBE growth of Pt on Au(100) using in-situLEED and AES. Noting that the Auger signal during growth variesuniquely for each growth mode, they concluded that their resultsfor the Auger signal intensity change during Pt/Au(100) growthat temperatures between 150 �C and 200 �C were best fitted by aVolmer–Weber growth mode. Sugawara and Nittono in 1991 [15]found, in agreement with Matthews and Jesser and at variance withSachtler et al., that growth occurred in the FM mode. Their experi-mental regime was similar to Ref. [13] except that Pt deposition wasconducted by ion-beam sputtering rather than evaporation. Theyimaged Pt/Au bilayers in TEM and found misfit dislocations whenthe Pt film thickness exceeded 1 nm.

In 1997, Uosaki and coworkers [16] electrochemically depositedPt thin films on Au(111) surfaces at room temperature using elec-trolytes containing the [PtCl6]2� complex. They characterized mor-phological evolution using in-situ STM and presented clearevidence of layer-by-layer FM growth albeit with some increasein surface roughness. On the other hand, Waibel et al. [17] per-formed a very similar study in 2002 using both [PtCl4]2� and[PtCl6]2� containing electrolytes, and instead found evidence forVW growth. Support for VW growth of Pt/Au was also presentedby Chang and Carter [18] using simulation methods. However, an-other simulation study of this system by Haftel et al. [19] predicteda fairly complex evolution of the Pt film which may be best de-scribed as roughened FM growth.

Finally, in another experimental work in 1999 Pedersen andcoworkers [20] studied, using in-situ STM, the very initial growth(up to 2.5 ML) of Pt grown with MBE in UHV on Au(111) at100 �C. They observed marginal surface alloying that was limitedto the first monolayer; subsequent monolayers grew as pure Ptwith a flat morphology conformal to the substrate, without nucle-ation of islands. It can be concluded from their work that growthoccurred in the FM mode.

This literature review is summarized in Table 1.

2. Experimental

Flat Au(111) surfaces were prepared by growing thick goldfilms (200–400 nm) on mica, using standard procedures [21–23]. Muscovite mica discs (Muscovite Mica Grade V-1, SPI Sup-plies, Westchester, PA), freshly cleaved in air, were placed in aUHV chamber with base pressure 1 � 10�10 Torr. After a fewhours of pre-annealing the mica at 500 �C, Au (99.99%, Sigma-Al-drich) was deposited at �2 nm/min from an effusion cell at1600 �C. Post-deposition, the Au/mica discs were annealed (inUHV) at 300 �C for 4 h. The flat Au surfaces obtained were rough-ened by exposure to concentrated nitric acid by removing the an-nealed Au/mica samples and immersing them in concentratednitric acid (70%, Sigma-Aldrich) for 1 h. The samples were thenrinsed thoroughly in ultraclean running water (Millipore Milli-Q,18.2 MX, 2 ppb total organic content (TOC)) multiple times, alter-nately in long periods of immersion and under a direct waterstream. Care was taken that the acid did not de-wet the Au sur-face during transfer from nitric acid to water, eliminating the pos-sibility of surface residue left behind by a receding contact line.After rinsing, the samples were covered and dried in air. In an ef-fort to fully characterize the resulting surface, the nitric-exposedAu samples were imaged in an atomic force microscope (AFM),and examined in X-ray photoelectron spectroscopy (XPS) to studysurface composition with particular interest in detecting any sur-face impurities and adsorbates.

Table 1A summary of experimental studies on Pt/Au heteroepitaxy available in the literature. The growth modes are indicated in the rightmost column

Author (year) Growth method Substrate temperature(�C)

Growth rate (Å/min)

Au orientation Analytical tool Growth mode

1 Matthews and Jesser(1967)

MBE/UHV 40, 300 2 (111) andother

TEM (ex-situ) FM

2 Sachtler et al. (1981) MBE/UHV 150–200 0.05 (100) LEED, AES (in-situ)

VW

3 Sugawara and Nittono(1991)

Ion-beam sputtering/UHV

190 2.3 (100) TEM (ex-situ) FM

4 Uosaki et al. (1997) Electro-deposition 25 0.2 (111) STM (in-situ) FM (rough)5 Pedersen et al. (1999) MBE/UHV 100 (?) 0.14 (111) STM (in-situ) FM (with monolayer

alloying)6 Waibel et al. (2002) Electro-deposition 25 ? (111) STM (in-situ) VW

FM, Frank–van der–Merwe (layer-by-layer); VW, Volmer–Weber (islanding).

A. Mathur, J. Erlebacher / Surface Science 602 (2008) 2863–2875 2865

The roughened Au/mica samples were placed in UHV where Ptwas deposited from a ‘mini e-beam’ source (eVap 3000, MDC Vac-uum) charged with pure Pt (99.999%, Alfa Aesar). The vacuumchamber pressure was allowed to recover to base level (usuallyovernight) before Pt deposition was initiated. Prior to each deposi-

Fig. 1. AFM micrographs and typical corresponding linescans of Au/mica films: (a) as-gronitric acid for 1 h, (c) Au/mica exposed to nitric acid for 1 h followed by UHV for 24 h, (d) Aat 300 �C for 3 h. The linescan z-scale (6 nm) and x-dimension (1 lm) is identical for all

tion run the e-beam source was switched on and allowed to outgasfor several hours at a low beam current. All Pt growth was per-formed at 2 Å/min measured using a calibrated Quartz CrystalMicrobalance (QCM; XTM/2, Inficon). A residual gas analyzer(RGA; Transpector 2, Inficon) was used to monitor the partial pres-

wn Au film deposited on cleaved mica, (b) Au/mica surface exposed to concentratedu/mica exposed to nitric for 1 h followed by UHV for 24 h and then annealed in UHVscans.


sures of the residual atmospheric components in the UHV chamberduring growth.

Fig. 2. AFM micrographs of Pt films on as-grown Au film on mica: (a) 1 nm, (b)2 nm, (c) 4 nm Pt films. The morphology is essentially identical in each case andindistinguishable from the Au substrate in Fig. 1a – terraces are flat, step edges areclearly resolved, but no Pt island formation is observed, suggesting Pt grows in alayer-by-layer or planar mode. The z-scale (10 nm) and x-dimension (4 lm) areidentical for all scans.

3. Results

3.1. Characterization of UHV grown Au films: as-grown and roughenedin nitric acid

The thick Au films (>200 nm) grown on mica possessed large(111)-oriented grains exhibiting atomically flat terraces greaterthan 100 nm wide, with RMS roughness <0.2 nm on individual ter-races. A typical AFM scan of the as-grown Au surface is shown inFig. 1a where flat terraces several hundred nanometers wide arebounded by mono- and bi-layer high steps. In comparison, the im-age in Fig. 1b of a nitric-acid exposed surface shows a markedlyrougher structure. A large density of island-type features are visi-ble on the surface of each terrace. Typical linescans of the AFMmicrographs (also shown in Fig. 1) show features that are typically<1 nm in height and about 10–20 nm in lateral width. On nitric-acid roughened surfaces, step edges can still be identified, althoughthey are much less distinct and appear to be significantly rougherthan on the as-grown surface. Fig. 1c presents a scan of an Au sur-face exposed to nitric acid for 1hr and then subjected to UHV expo-sure for 24 h. The morphology appears identical to Fig. 1b and theroughness characteristics remain unchanged. A marked change inthe surface is observed in Fig. 1d, which shows a scan of a nitric-ex-posed Au surface annealed in UHV at 300 �C for 3 h. A two-dimen-sional pattern appears on this surface; instead of the random anduniform roughness of Fig. 1b and c, ‘mesas’ 20–50 nm in widthare observed, separated by ‘pits’ of similar size. The mesas form acontinuous 2D interconnected structure on the surface, but theirvertical height is nearly unchanged from those of the islands inscan Fig. 1c.

Analysis of images in Fig. 1 reveals that the Au(111) surfaceundergoes a significant roughness transition as a result of nitricacid exposure. Islands of 1–2 monolayers (MLs) height and 10–20 nm width appear uniformly over the surface. Further corrobora-tion of this assessment is provided in image Fig. 1d where thechange in roughness characteristics is consistent with in-planecoarsening of mesas and pits – at the moderate annealing temper-ature (i.e., 300 �C), strong step-edge diffusion barriers in Au [24,25]play an impeding role in classical surface smoothening, and as a re-sult a coarsened rather than smoothened surface is formed. Largerislands grow at the expense of smaller islands and correspond-ingly, larger pits (i.e., vacancy islands) grow at the expense of smal-ler ones. This process explains the morphology transition from Fig.1c and d, and also provides additional evidence that the roughnessmanifested in Fig. 1b and c indeed consists of Au islanding, and isnot merely an artifact of experiment or imaging. Importantly, italso indicates the absence of adsorbates that would have otherwisestymied Au adatom mobility.

Surface impurities can affect epilayer growth significantly;hence their possible presence on the nitric-roughened Au surfacewas examined very closely. Both as-grown and nitric-roughenedAu/mica samples were examined in XPS; the scans for both (notshown) were virtually identical and displayed no evidence of ad-sorbed impurities beyond typical air-adsorbed C and O.

3.2. UHV grown Pt/Au(111)

Thin Pt films of thickness 2 nm and 4 nm were grown on top ofas-grown Au/mica surfaces (i.e., always in UHV), while films 2 nm,4 nm, 10 nm and 20 nm thick were grown on nitric-exposed Au/mica samples. Kalff et al. [26] demonstrated that adsorbate mole-cules such as CO on step edges during growth can affect film mor-

phology significantly. In light of this, we paid close attention to theamount of CO in our UHV system before, during and after Pt depo-sition. An upper bound of 5 � 10�13 Torr for the CO partial pressurewas observed with the residual gas analyzer throughout the


growth processes and no spikes or dips in CO were detected thatcould be attributed to adsorption or desorption during growth.Thus the presence of CO was not considered to be a major factorin our experiments. Besides, our growth results differ qualitativelyfrom the documented influence of CO in Pt growth.

Fig. 2 shows an AFM scan of a 4 nm Pt film grown on an as-grown flat Au/mica sample. In terms of morphological characteris-tics the scan is virtually identical to that in Fig. 1a, i.e., the Pt film isflat, featureless and conformal to the Au substrate. We note againthat the larger features in the linescans (�2 nm and above) repre-sent underlying features (e.g. steps) on the Au surface, not the film.The Pt film morphology itself is best assessed by focusing on indi-vidual terraces.

AFM scans of 2, 4, 10 and 20 nm thick Pt films on nitric acidroughened Au are shown in Fig. 3a–d. An evolution of the Pt filmwith increasing thickness is clearly observable: roughness appearsto initially increase and then decrease, whereas the average ‘island’size initially remains constant and then increases until finally atthe highest thickness (20 nm) the islands more or less disappear.A detailed examination of early film growth is presented in Fig.

Fig. 3. AFM micrographs and corresponding typical linescans of Pt films grown on Aurespectively, deposited on the representative roughened Au surface in Fig. 1b. Scan size

4a–c, where smaller area (0.5 lm � 0.5 lm) scans of a roughenedAu substrate, a 2 nm Pt film on a roughened Au film, and a 4 nmPt film on a roughened gold substrate are shown. Compared tothe roughened Au surface of Fig. 4a.1, a fine 2D ‘network-like’ Pt is-land structure is apparent in Fig. 4b.1, whereas Fig. 4c.1 shows in-plane coarsening of the islands in Fig. 4b.1. Two characteristicparameters that can be measured from the linescans are the verti-cal peak-to-valley height H of the islands, and the peak-to-peakspacing k between islands. The following transformation is ob-served in the images: H increases from <0.5 nm in Fig. 4a.2 toH � 1 nm for the 2 nm film in Fig. 4b.2 and remains unchangedon the 4 nm film in Fig. 4c.2. On the other hand, the island spacingk remains initially constant at �15 nm for the bare roughened Au,Fig. 4a.2, and the 2 nm Pt film in Fig. 4b.2 but then increases mark-edly to k = 65 nm for the 4 nm film, Fig. 4c.2 This change is ob-served from the broadening of the peaks in the linescans, and isalso reflected in the autocorrelation images, Fig. 4b.3 and Fig.4c.3 of the corresponding AFM scans. Some degree of symmetryin the surface structure is revealed by the periodicity of the dark

/mica surfaces roughened in nitric acid for 1 h: (a–d) 2, 4, 10 and 20 nm Pt films,is 1 lm; linescan z-range is 6 nm.

Fig. 4. About 0.5 lm AFM scans of (a.1) nitric-exposed Au surface, (b.1) 2 nm Pt film and (c.1) 4 nm Pt film grown on the roughened Au surface. Corresponding linescans areshown with z-range 3 nm. Images (b.3) and (c.3) are autocorrelation images of typical single terrace zones in (b.1) and (c.1). Typical island spacings k and peak-to-valleyisland heights H are identified.


and bright spots, where the increase in the lateral spacing of thesecorresponds to the increased magnitude of k.

To summarize, the following key morphological characteristicswere identified during Pt film growth on rough Au substrates asPt thickness was increased:

1. Mean island height initially increases and subsequentlydecreases in early growth (i.e., up to 4 nm film thickness).

2. Mean island spacing remains initially constant and subse-quently increases (beyond 2 nm film thickness).

4. Discussion

4.1. Platinum Growth on Au – discussion

From the absence of any island formation in Fig. 2 it is clear thatPt growth on as-grown Au/mica surface occurs with planar mor-phology, and therefore the Frank-van-der–Merwe growth modeis exhibited in this system. Both Pt and Au are fcc materials withslightly differing lattice parameters: a = 3.92 Å for Pt and 4.08 Åfor Au, so that a 4.08% lattice mismatch exists between the two.Consequently a coherent Pt film on Au accumulates strain energywhich, when critical thickness is achieved, is reduced by formationof misfit dislocations (MDs) at the interface. Using a multi-beamoptical stress sensor (MOSS) installed in our UHV chamber for in-situ film stress measurement [27,28], we measured a critical thick-ness of 11 Å (see Appendix 2). Our observations are in excellentagreement with previous studies of Pt/Au growth [13,15] that alsonoted FM growth mode and measured, by ex-situ TEM, an MD crit-ical thickness of 10 Å.

In contrast to the FM mode on flat Au, Pt growth on roughenedAu exhibits a markedly different morphology evolution. If FMgrowth occurred on rough Au, a uniformly thick Pt film conformal

to the underlying Au would be expected to grow so that mean is-land height and spacing would remain unchanged, although RMSsurface roughness could be expected to decrease slowly owing toan increase in average island size. Actual observations are in factto the contrary: island height increases significantly and so doessurface roughness. It is only in later growth that these parametersdecrease. Thus the growth mode is not merely FM type on a roughsubstrate. Nor is it Volmer–Weber – a key characteristic of VWgrowth is the ‘de-wetting’ of film from the substrate which is notobserved here. Even on Pt films of 2 nm nominal thickness (Fig.4a) the measured island height is only �1 nm, i.e., the Pt is notentirely agglomerated in the islands as would be expected inVW. Instead, the film is continuous over the entire surface andyet non-uniform in thickness.

In the Stranski–Krastanov growth mode, a thin strained epilayergrows conformally on the substrate after which a strain- and sur-face-energy mediated islanding instability results in periodic is-land growth, so that a continuous but non-uniformly thick film isformed. While Pt on roughened Au has empirically similar charac-teristics, the growth mode is not classical SK because film islandingis closely linked to pre-existing features on the rough Au substrate.That is, the islands on the rough Au substrate present a templatefor preferential growth of Pt film islands. This idea is examinedin detail in the following sections.

4.2. Growth on rough surfaces

In idealized layer-by-layer growth on flat substrates, step-edgebarriers are negligible, so that film atoms arriving on existing two-dimensional islands can diffuse onto lower terraces and conse-quently one layer is completed before a new layer begins to form.An enhanced step-edge barrier however inhibits downward diffu-sion of atoms and it becomes more likely for a stable nucleus toform on top of existing islands before the layer underneath is com-


pleted. A rough growth surface presents a template with large arealstep density, and therefore presents an opportunity for step-edgebarriers to affect growth morphology. Indeed, experimental andsimulation studies on homoepitaxial rough-substrate growth[29–31] have demonstrated that surface roughness increases dur-ing growth as a direct result of the step-edge or Ehrlich–Schwoebelbarrier.

In principle, because Pt growth on flat Au is layer-by-layer, dur-ing initial growth on rough Au a uniform Pt monolayer should formon the Au islands and pits. Subsequent Pt growth effectively occurson top of this uniform Pt monolayer and it has been shown that Pthomoepitaxial growth occurs layer-by-layer even at low tempera-tures [32]. This is attributed to the marginal ES barrier on Pt(111):EES = 0.06 eV [33,45,46] compared to a diffusion barrier ofED = 0.26 eV on that surface. Thus, in spite of a large step density

Fig. 5. An illustration of lattice-mismatched heteroepitaxial growt

on the rough Au surface, Pt should grow conformally to the Au sur-face and quickly smoothen out, resulting in a flat Pt film that car-ries no ‘memory’ of the underlying rough Au surface. This iscontrary to our observations.

A feature of Pt/Au growth that we have ignored in the aboveanalysis is the effect of mismatch strain. The Pt monolayers atopAu are stretched to match the larger lattice parameter of Au, andthe �4% strain may affect the surface diffusion and ES barriers.Unfortunately, it is very difficult to rigorously assess the impactof strain on these quantities, and reliable values are not to be foundin the literature. Even so, it is highly unlikely that strain on Pt(111)will substantially increase the difference EES � ED (i.e., the addi-tional barrier to step-descent for an adatom). It is therefore safeto assume that ES barriers on strained Pt(111) do not inhibitlayer-by-layer growth.

h on rough substrate. Please refer to the text for explanation.


Strain in the Pt over layer, however, does play a critical role infilm morphology evolution and we examine those effects in somedetail in the remainder of this paper.

4.3. A physical model for Lattice–Mismatched heteroepitaxy on roughsurfaces

The chemical potential lðxÞ on a surface in one dimension maybe written as

lðxÞ ¼ l0 þ cXjðxÞ þ ð1=2ÞSijklrijðxÞrklðxÞX ð1Þ

where l0 is the equilibrium chemical potential of a flat, unstressedsurface, c is the surface energy, X is atomic volume, j(x) is the cur-vature at point x on the surface, Sijkl is the compliance tensor and r(x)describes the stress state at x. While the second term on the rightrepresents curvature effects (Gibbs–Thomson effects), the thirdterm accounts for effects of stress on the surface chemical potential.From the Nerst–Einstein relation, we know that the flux density J ofatoms diffusing along a surface is J ¼ �ðDS=kTÞðol=oxÞ, where Ds isthe surface diffusion coefficient and T is temperature.

If a solid surface is uniformly stressed, i.e., no gradients in strainenergy exist, then ðo=oxÞðrijðxÞrklðxÞÞ ¼ 0 and the occurrence ofstress does not contribute to any diffusive flux. However, if surfacestrain energy gradients are present, they can drive a diffusion fluxfrom positions of higher strain to those of lower strain. We arguebelow that such strain energy gradients during Pt growth on roughAu lead to the morphology evolution we have observed.

The roughened Au surface is schematically illustrated in Fig. 5a,where 2 ML high islands (island height denoted by H) with separa-tion k exist on the Au surface prior to Pt deposition. We assume thevery initial layers of Pt are governed by the following behavior: onmesas, the large Schwoebel barrier on Au(111) prevents down-ward adatom flow, and in pits step flow occurs from the walls ofthe roughened Au surface. This hypothesis cannot be distinguishedfrom a preferential nucleation in pits or on mesas with subsequentsurface diffusion (an effect which may be important in strain-med-iated heteroepitaxy [34]), but regardless the end result should bethat a uniform monolayer is formed on top of the Au islands as wellas in the pits. It is important to note that this monolayer is fullycoherent to the lattice of the Au substrate and is therefore undertensile strain of �4%.

As subsequent monolayers are formed, it becomes apparent thatthe coherently strained films atop the Au islands have markedly dif-ferent boundary conditions from those in the pits. The island-film(i.e., that part of the film growing on the island on the substrate)has finite stress-free boundaries due to the free edges on the sides,which allows the film to partially relax in-plane while maintainingcoherency. Thus owing to the free boundaries, the strained film canreduce its strain energy without having to form energy-costing mis-fit dislocations. Indeed, it has been shown that growth on small is-lands or ‘mesas’ can be an effective method for reducing threadingdislocation density in lattice-mismatched systems to the extentthat on small enough mesas ‘infinitely’ thick coherent films canbe grown without misfit-dislocations [35–37]. Theoretical analysesby several authors [35,36,38–40] show that stress in an island-filmdecays exponentially in the vertical dimension as

r / Meme�pz=l; ð2Þ

where r, M, and em are the stress, film biaxial modulus and misfitstrain respectively, z is the vertical distance away from the film-sub-strate interface and l is the lateral dimension of the mesa. As a result,unlike growth on flat ‘semi-infinite’ surfaces where strain energybuilds up monotonically with thickness, strain energy in the island-film tapers off with growth. Since recovery of background strain en-ergy is the thermodynamic driving force for MD formation, when l issmall enough the film can be relaxed without being dislocated.

On the other hand, the pit-film (i.e., that part of the film growingon the surface region between islands) may be unable to relax at allbecause in order to do so it would have to either de-wet from theside-walls and thereby create new surfaces which costs surface en-ergy, or form prohibitively expensive dislocations. For instance, ifthe pits are small enough in width (smaller than the equilibriumMD spacing in planar Pt/Au) forming dislocations by injecting extrahalf-planes in the film lattice which is constrained by fixed sidewalls will simply reverse the stress (from tensile to compressive)and will therefore be energetically unfavorable [41]. It is noted thatunlike the island-film which is coherent yet relaxed (partially), thepit-film is coherent and fully strained. This difference in the film’sstrain between adjacent islands and pits yields a non-vanishingstrain energy gradient along the surface, which provides the drivingforce for atoms to migrate from positions of high energy i.e., pits, tothose of lower strain energy i.e., islands. The resulting surface diffu-sion flux causes preferential growth of Pt atop the Au islands, lead-ing to an increase in island height shown in Fig. 5f and a consequentincrease in surface roughness. Island spacing remains unchangedbecause growth occurs preferentially on top of existing islands.

As growth continues, the islands increase in height as well aswidth due to step-edge adhesion; inter-layer transport from thepits to the island surface is impeded by the increasing diffusionlength necessary owing to increased island height. While islandspacing is thus far unchanged, as the islands grow and impingeupon each other they begin to coalesce – this process results inthe increase in apparent island spacing accompanied by a decreasein surface roughness. Also, upon coalescence the film strain gradi-ents begin to disappear and it starts to smoothen. This is indeedobserved in the Pt films of 10 nm and 20 nm thickness, whereout-of-plane height variations are seen to return to the <0.5 nmrange and surface roughness decreases significantly (Fig. 3d ande). It is noted however that although the film became smootherin our experiments, it did not smoothen completely due to the lim-ited kinetics of surface diffusion at the growth temperatures.

Rough estimates of the magnitude of the energy terms involvedin this model may be obtained from some simple calculations.Assuming complete elastic relaxation of Pt ML#2 on the islandsand no relaxation in the pit, we may compute the strain energy dif-ference per atom in the island- and pit-films. Using continuum lin-ear elasticity, the strain energy per unit area of a biaxially strainedfilm is: W ¼ Mf e2

f hf , where the quantities on the right are the biaxialmodulus, strain, and the height of the film respectively. For a singlemonolayer hf = 0.28 nm (diameter of a Pt atom), while Mf = 271 GPafor Pt and mismatch strain ef � 0.04. Thus the area density of strainenergy in the Pt monolayer W = 0.121 J/m2. Assuming 6.9 � 1018

atoms/m2 in a Pt(111) plane coherently strained to the underlyingAu(111), the strain energy per Pt atom of the unrelaxed film in thepit is Wa � 0.11 eV/atom. An additional energy term for the strainedfilm arises from surface stress effects [42,43]. To elastically strain afilm by e, work must be done against the surface stress f, which inthe case of Pt(111) is compressive with a magnitude fPt � 2.5 J/m2

[44]. Assuming strain-independence of f, work done per unit areais simply Wf = fe, which comes to Wf � 0.1 J/m2 or about 0.1 eV/atom. Thus, atoms in the strained pit-film have total energy greaterthan those in the island-film by W = Wa + Wf = 0.21 eV. This amountis comparable to the Pt(111) in-plane surface diffusion barrier[45,46]. Fig. 6 schematically shows the effect of strain on the sur-face’s potential energy landscape; the adatom diffusion potentialin a pit-film is shifted up relative to the island-film by an amountW. The trapping potential at step-edges (not shown) is also dimin-ished in magnitude. Because the free energy of the pit-films is high-er than the island-films, the chemical potential gradient drives a Ptadatom flux from the pits to the islands.

In summary, strain-energy gradients precipitated by substratemorphology drive Pt atoms from the pits to the islands and as a re-

Fig. 6. Owing to the coherency strain in the pit-film, the mean potential energy is shifted up by an amount W compared to the relaxed island-film. See text for details. Forsimplicity, the potential energy function at the step edge is not shown. An adatom in the pit may lower its energy by migrating to a nearby island.

Fig. A1. A NPG ligament is approximated as a solid cylinder for purposes ofmechanics calculations. Cylindrical polar coordinates are centered on one of thefaces of the cylinder. Surface stress f (compressive for Au) acts longitudinally andtangentially as shown.


sult island height initially increases and so does surface roughness.While the film does grow in the pits, it grows preferentially on theislands. With increasing thickness, the islands grow and coarsen sothat the mean island spacing increases, and roughness begins todecrease as well. The size and spacing of the roughened gold is-lands are easily within the dimensions of NPG used to grow Pt-NPG, and thus strongly suggest that the apparent SK growth seenin Pt-NPG is due to pre-existing surface roughness left over fromthe initial acid-dealloying step.

5. Conclusions

The physical model described here explains the unique mor-phological evolution in growth of Pt thin films on Au(111) sub-strates possessing nanoscale roughness. The model, based onmismatch strain energy gradient arguments, utilizes some simpli-fying assumptions which need further examination. For instance,strain effects on surface diffusion and step-edge barriers may besignificant and may have to be taken into account. Also, for sim-plicity we have ignored the marginal mixing that occurs at thePt/Au interface [20] and assumed a sharp interface instead. Thepresence of a couple of mixed monolayers though should only re-sult in the equal out-of-plane gradation of strain in the pit- and is-land-films, leaving the in-plane strain energy gradients more orless unaffected.

While a rigorous theoretical examination using atomistic mod-eling techniques, for instance, is necessary in order to fully accom-modate the nuances of surface and growth phenomena, ourdiscussion presents a novel interrelation between substrate andfilm morphologies, where the former controls the relaxationmodes available to the growing epilayer and significantly impactsits mode of growth. Our observations and analysis clarify the spe-cifics of the Pt/Au(111) system and are used here to explain theparticular observation of apparent SK growth of Pt on nanoporousgold. However, these ideas may be readily extended to rough-sub-strate growth in other lattice-mismatched systems.

Acknowledgements

We gratefully acknowledge support for this work by the USDepartment of Energy, Office of Basic Energy Sciences under grantDE-FG02-05ER15727, and helpful discussions with R.C. Cammarata.

Appendix 1. Effects of curvature and surface stress on Pt/Auheteroepitaxy

In this section, the impact of the surface stress of Au on the lat-tice mismatch between Pt and Au ligaments in NPG is assessedusing some simple mechanics calculations. In any finite solid

bound by surfaces, surface stress imposes a strain state that typi-cally vanishes rapidly into the bulk of the structure. But whenlengthscales become small enough, a non-zero strain state may ex-ist throughout the bulk of the solid leading to abnormal mechani-cal properties and behavior. Several effects have been predictedand observed [47–51], from changes in elastic modulus of ultra-thin films to structural and phase transitions in nanostructures.In the context of epitaxy on nanoscaled NPG or of epitaxy on rough(non-flat) surfaces generally ligaments, we may hypothesize thefollowing: the bulk strain arising in the NPG ligaments as a resultof the surface stress of Au will alter the lattice mismatch betweenthe Au substrate and an epitaxial species such as Pt, compared tothe bulk (flat) surface case. In principle, the changed epilayercoherency stress may sufficiently change strain energy behaviorto affect growth mode.

An NPG ligament may be idealized as a solid cylinder of radius R(so that curvature j = 1/R) and length L. Surface stress f is assumedto be isotropic and strain-independent; we recognize a priori thatfor the Au surface f is compressive. Fig. A1 illustrates the geometryand the action of the surface stress, which seeks to shrink the sur-face by contracting the ligament in length along its axis as well asin its radial dimension. We assume that the stress distribution in-side the ligament is uniform, so that no strain gradients exist in itsbody. This approximation is necessary for simplicity; in realitystrain gradients do exist, particularly in the vicinity of the two endsof the cylinder where the curved and the flat surfaces join orthog-onally. This latter problem is somewhat alleviated by assuming L >> R. Since the two end surfaces are far from each other we ignoretheir interactions altogether; in other words, the surface stresses


acting on them are presumed to have little impact on the samplebulk towards the middle of the ligament.

We imagine that the cylinder is formed by carving out of a large,stress-free solid. The cylinder is initially stress-free, but theappearance of surfaces exerts forces on its interior. To achievemechanical equilibrium with these surface-generated forces, anon-vanishing stress state rij with concomitant strain eij appearsin the cylinder. Again, these are assumed to be uniform throughthe body of the ligament for the sake of simplicity. The energy togenerate this strain state is provided by a corresponding changein surface area DA and a balance between the strain energy andsurface energy terms yields a measure of the strains in the liga-ment. With knowledge of the strains we may assess how latticemismatch with an epilayer might be affected.

We consider an isolated system of fixed volume consisting ofthe ligament and its surroundings. The thermodynamic availabilityof the system is written as

A ¼ U � T0Sþ P0V ; ðA1:1Þ

where U is the internal energy of the system, T0 and P0 are the tem-perature and pressure of the surroundings, respectively, and S and Vare the system entropy and volume, respectively. At thermody-namic equilibrium, the system availability is minimized, so that

dA ¼ dU � T0dSþ P0dV ¼ 0: ðA1:2Þ

Assuming that the system and the surroundings are at absolutezero temperature and noting that system volume is fixed by defini-tion, the condition for thermodynamic equilibrium of this systemis

dA ¼ dU ¼ 0: ðA1:3Þ

Within the system, we may identify three ‘phases’: they cylin-der itself (we label it a), the surroundings (labeled b), and the inter-face between the two, i.e., the cylinder surface (labeled s). Withthis notation, we may rewrite Eq. (2.4) as

dU ¼ dUa þ dUb þ dUs ¼ 0: ðA1:4Þ

We can now write the variations in the internal energy of eachphase separately:

dU¼0¼ðTadSa�PadVaþlai dNa

i ÞþðTbdSb�PbdVbþlb

i dNbi Þþðf dAsÞ;ðA1:5Þ

where li and Ni are the chemical potential and number of moles ofthe ith component, and f and A are the surface stress and surfacearea, respectively. Because the system is at absolute zero (as as-sumed above) and no mass exchange occurs between the twophases, we may rewrite Eq. (A1.5) as

dU ¼ 0 ¼ ð�PadVaÞ þ ð�PbdVbÞ þ ðfdAsÞ: ðA1:6Þ

We note now that because the system volume is fixed, the vari-ations in the volume of phases a and b are equal and opposite, i.e.,dVa ¼ �dVb. Finally, we can set the pressure in the b phase as zero(i.e., vacuum) and denote Pa in the cylinder simply as P to arrive at

dU ¼ �PdV þ fdA ¼ 0: ðA1:7Þ

The P � V term in this relation represents mechanical work andcan be replaced by a more general term incorporating the strainenergy of the system to yield

dU ¼ �VZ

rijdeij þ fAdes ¼ 0; ðA1:8Þ

where V is the total volume of the ligament, A is the total surfacearea, and des is the surface area strain. We may integrate Eq.(A1.8) from the reference state (zero strains in volume and surface),and substitute for strain energy to get

DU ¼ �V12

kejjekk þ leijeij

� �þ fADes ¼ 0: ðA1:9Þ

That is,

DU ¼ �VðDUV Þ þ AðDUsÞ ¼ 0: ðA1:10Þ

The term in the brackets in Eq. (A1.9) represents the strain en-ergy density (see, for example, Ref. [52] p.106) of an isotropic linearelastic medium in terms of strain components, k and l are theLame’ constants for the material. Summation occurs over repeatedindices.

The natural coordinates for this system are the cylindrical coor-dinates ðr; h; zÞ, which are also the principal axes for which shearstresses and strains vanish. The effects of surface stress are thus re-solved into two components: the component along the circularperimeter of the cross-section generates a ‘hoop stress’ in thecross-sectional plane, whereas the effect of the longitudinal com-ponent is approximated as a compressive stress applied uniformlyacross the cross-section. We recognize the following boundaryconditions (BC’s):

1. From a simple geometric consideration it can be shown that fora uniform radial strain err, the angular strain ehh = err.

2. The longitudinal compressive stress may be written as

rzz ¼ �2pRf

pR2 ¼ �2fR; ðA1:11Þ

where the numerator represents the magnitude of the total force onthe perimeter acting normal to the cross-section, and the denomi-nator is simply the area over which it is assumed to be uniformlydistributed. The negative sign explicitly recognizes the compressivenature of the stress.

Using Hooke’s law for linear elasticity

rij ¼ kekkdij þ 2leij; ðA1:12Þ

where dij is the Kroenecker delta, we may write the following forrzz:

rzz ¼ kðerr þ ehh þ ezzÞ þ 2lezz: ðA1:13Þ

Substituting the boundary conditions, we get

2fR¼ kð2err þ ezzÞ þ 2lezz; ðA1:14Þ

so that

ezz ¼ �ðkerr þ 2f=RÞðkþ 2lÞ ðA1:15Þ

The strain tensor may therefore be written as:

eij ¼err 0 00 err 00 0 � ðkerrþ2f=RÞ

ðkþ2lÞ

0B@

1CA: ðA1:16Þ

The strain energy density for this strain state is determinedusing the relation (as previously used in Eq. (A1.9))

DUV ¼12

kejjekk þ leijeij;

and so the total strain energy in the ligament volume is found to be

VðUV Þ ¼ Vkð2Rðkþ lÞerr � f Þ

Rðkþ 2lÞ þ 4l e2rr �

f � Rkerr

Rðkþ 2lÞ

� �2 !

:

ðA1:17Þ

This expression represents the first term on the right in Eq.(A1.9). We now evaluate the second term, which involves sur-face area strain. The strained surface area may be written as


A0 ¼ Að1þ ehhÞð1þ ezzÞ, so that to first order, the strain in the sur-face area is simply Des ¼ ðehh þ ezzÞ. Substituting boundary condi-tion (1) and Eq. (A1.15), we get

Des ¼ err �ðkerr þ 2f=RÞðkþ 2lÞ

� �ðA1:18Þ

Finally, the volume V and area A per unit length of the ligamentalong z axis are V ¼ pR2 and A ¼ 2pR, respectively. Combiningthese results together in the equilibrium condition (Eq. (A1.10)),we have

0 ¼ �pR2 kð2Rðkþ lÞerr � f ÞRðkþ 2lÞ þ 4lðe2

rr �f � Rkerr

Rðkþ 2lÞ Þ2

� �

þ 2pRf err �ðkerr þ 2f=RÞðkþ 2lÞ

� �ðA1:19Þ

Although rather unwieldy, Eq. (A1.19) can, in fact, be solved ex-actly for err, but the exact solution is not presented here. Instead,we computed the radial strain err in a typical NPG ligament of10 nm diameter, where f = 1.5 J/m2 [53], k = 203.7 GPa, andl = 27.8 GPa [54] for Au. This yielded err ¼ �0:0003. Using bound-ary condition (1) and Eq. (A1.9), we conclude that for these liga-ment parameters that ehh ¼ �0:0003, and ezz ¼ �0:0018. Thesetwo quantities represent the strains in the surface lattice parame-ter of the Au ligament along the perimeter and the length respec-tively, and are found quite small. The difference in theirmagnitudes simply reflects the strain anisotropy on the surface.

A much less cumbersome assessment of lattice strain due tosurface stress may be made by considering a small sphere of radiusR instead of a cylinder. Although the sphere geometry is much lessrepresentative of an NPG ligament, our objective is merely to studymaterial response at a comparable scale, so it is a reasonable exer-cise. The sphere’s surface exerts a Laplace pressure DP ¼ 2f

R on itsinterior, and this pressure causes a volumetric strain, which, usingHooke’s law, is given by eV ¼ �DP

K , where K is the bulk modulus. Fora 10 nm diameter Au sphere, using K = 220 GPa [54], we findeV = �0.0027. This is the volume strain eV ¼ DV=V; we can easilydetermine the radial strain which is found to be err = �0.0009. Thisquantity is indeed of the same order as the strains found in the pre-ceding computation for a cylindrical ligament shape and serves asa quick validity check.

The preceding calculations are used to conclude that surfacestress causes negligibly small lattice strain in NPG with ligamentsor local surface curvatures 10 nm diameter and larger such as seenin the Pt heteroepitaxial growth experiments. It must be acknowl-edged that the actual strain state in a real sample will be morecomplex than what is calculated here using a uniform strain ap-proach. Also, the surface strain state will certainly be different fromthe bulk, unlike the assumption in the present analysis. A moresophisticated molecular dynamics model has recently been madeby Crowson et al., and yields strains the same order of magnitudeas calculated here [55].

The computation here provides a sense of the marginal magni-tude of the lattice strain due to surface stress. The consequentchanges in the mechanics of Pt growth are also expected to be min-iscule: because of the small lattice strain, the lattice mismatch be-tween Pt and Au substrate remains nearly unchanged compared tobulk, and for all purposes the mismatch strain energy is identical tothe case of planar growth.

Appendix 2. Measurement of the critical thickness for Pt filmson smooth Au(111)

The characteristic length scale of the FM mode is the criticalfilm thickness at which misfit dislocations become energeticallyfavorable and begin to accommodate the lattice mismatch strain.

In the original Matthews–Blakeslee criterion for the critical filmthickness in FM growth, the effect of surface stress was neglected,as well as the effect of anisotropic linear elasticity. These effectswere added by Sieradzki and Cammarata [42], who find the criticalthickness hcrðFMÞ is a modified Matthews–Blakeslee criterion:

b8pð1þ mfÞhcrðFMÞ

ln2hcrðFMÞ

r0� f

ð1� mf Þlfð1þ mf ÞhcrðFMÞ

� jemj: ðA2:1Þ

Using b � af=2, r0 ¼ b=2, mf ¼ 0:38, lf ¼ 60:9 GPa, f = 2.49 J/m2,and em ¼ 0:04, hcr(FM) was found to be less than 5 Å for Pt/Au. Thissuggests that misfit dislocations are thermodynamically feasibleeven after the first couple monolayers of growth. However, Mat-thews and Jesser [13] and Sugawara and Nittono [15] observedMDs directly (using TEM) only in Pt films greater than 1 nm thick.The reason for the apparent discrepancy between theory andexperiment has to do with the fact that Eq. (A2.1) is based onlyon thermodynamic considerations, and does not take into accountkinetic factors such as energy barriers to dislocation nucleation andmotion which tend to delay the onset of MDs.

In the present work, the critical thickness was measured bymonitoring film stress evolution during growth using the multi-ple-beam optical stress sensor (MOSS) technique described in Refs.[27,28]. The Stoney equation relates the measured curvature of thesubstrate to the film stress as:

jðtÞ ¼ 6½rmðtÞhfðtÞ þ DFðtÞ�Msh2

s

; ðA2:2Þ

so that if DFðtÞ is known, the product of film stress and thickness,S � rmðtÞhfðtÞ, can be easily computed from the measured curva-ture jðtÞ and known parameters Ms and hs.

The change in surface stress DFðtÞ is hard to determine in theinitial stages of growth when a complex evolution of a mixed inter-facial region appears, including changes in surface crystallography,e.g., lifting of reconstructions. In Si0.8Ge0.2 growth on Si [27], forexample, the DFðtÞ term represents a surface segregation of Geand is reflected by a non-linearity in curvature evolution until�1 nm film thickness is grown. Unlike Si/Ge, however, Pt/Au is anon-mixing system due to a large positive enthalpy of mixingdetermined by both theory and experiment [56–58]; surface alloy-ing has been shown to occur only in a single monolayer [20], withno surface segregation. Nevertheless, the behavior of DFðtÞ duringthe initial stages (<2 monolayers) of growth in this system is un-known at present. We therefore made an approximation out ofnecessity to ignore this term in our calculations altogether, a sim-plification whose impact is alleviated by the following assessment:during initial growth the Pt film is coherently strained to match theAu substrate, so that the mismatch stress in the film is rm ¼ rcoh¼Mfem ¼ ð271 GPaÞð0:04Þ ¼ 10:8 GPa (where Mf ¼ E=ð1� mÞ ¼ 271GPa is the biaxial modulus of Pt). Using surface stress values ofAu(111) and Pt(111) from Ref. [53] the value of DF is expectedto be �1 J/m2, so that a comparison between the two quantitiesin the square brackets in Eq. (A2.2) shows that even for a 5 Å thickPt/Au film, the rmhf term dominates over the DF term, and the lat-ter can be reasonably ignored. This, of course, is due to the verylarge strain in the Pt film; in other systems with smaller latticemismatch ignoring DF could lead to significant errors.

Finally, the substrates in these experiments were �400 nmthick Au films grown on �100 lm thick mica discs. Because themica was �250 times the thickness of the Au film, the mechanicalresponse of the Au/mica substrate during Pt growth was domi-nated by the mechanical properties of mica, and so the substrateproperties were approximated by those of the mica alone, i.e.,Ms � Mmica and hs � hmica. For the sake of further simplicity, isotro-pic elastic behavior was assumed for mica, with Young’s modulusE = 100 GPa and m ¼ 0:25 [59]; in reality mica is anisotropic [60].

Fig. A2.1. Plots of (a) stress-thickness S vs. time, and (b) Pt film thickness vs. time. (the time axis is identical in both plots). In Zone I, S increases linearly with increasing filmthickness; in Zone II, S deviates from linearity indicating onset of strain relaxation via misfit dislocations; in Zone III growth is ceased but strain relaxation continues.


As will be seen shortly, accounting for the exact mechanical behav-ior of the substrate is unnecessary for the present analysis and thisapproximation is rather useful.

A plot of the product of mismatch stress and thickness i.e.,S ¼ rmhf , vs. time is presented in Fig. A2.1a and b shows a plotof film thickness hf vs. time. Due to coherency during initial growththe mismatch stress was constant, i.e., rm ¼ rcoh. Therefore, if thegrowth rate was constant and hf increased linearly with time (as itdid), S would also increase linearly in magnitude. This is indeed ob-served in Zone I of the plot. The positive sign of S reflects the factthat the Pt film is in tension and tries to shrink in plane, inducinga concave curvature to the substrate.

A deviation from linearity in the S vs. time plot during constantgrowth rate indicates a change in the magnitude of rm. Just such adeviation is observed at the beginning of Zone II (linear behavior isshown by dashed line for reference) which indicates that beyondthis point rm < rcoh. Reduction in film stress during growth mayoccur as a result of island formation in SK or misfit dislocation for-mation in FM. Since no island formation was observed in AFM ofPt/Au the departure from linearity is associated with MD onset atthe Pt/Au interface. Consequently, the Pt film thickness at thebeginning of Zone II is identified as the FM critical thickness forPt/Au – it can be simply read from the plot and is found to behcrðFMÞ ¼ 1:1 nm. The derivative dS=dhf in Zone I yields the coher-ency stress and is found to be rcoh ¼ 7:8 GPa; this is smaller inmagnitude than the expected value rcoh ¼ 10:8 GPa from theory.The likely causes of the discrepancy are the various approxima-tions that were made in the analysis. However, it is important topoint out that while the magnitude and slope of S is affected bythe approximations, the point of departure from linearity is not,and so the measure of critical thickness is accurate. The measuredhcrðFMÞ ¼ 1:1 nm is in excellent agreement with the critical thick-ness of 1 nm found experimentally by Matthews and Jesser [13],as well as Sugawara and Nittono [15].

Platinum growth was ceased at the end of Zone II in Fig. A2.1aby turning off and shuttering the Pt evaporation source after a filmof thickness hf ¼ 50:4 Å had been deposited. The plot of S in ZoneIII illustrates film stress evolution at this constant film thickness.

The decreasing magnitude of film stress represents ongoing nucle-ation of new strain-relieving misfit dislocations, albeit at a decreas-ing rate. This trend was continued in data recorded for 10 h beyondwhat is shown in the plot here and presents some interesting infor-mation regarding the kinetics of dislocation motion at room tem-perature, not addressed in this paper.

The possibility that the curvature change observed by MOSScould be due to temperature rise of the Au/mica substrate duringPt vapor deposition was also examined. Substrate deformation insuch a scenario could be driven by differential thermal expansionbetween the Au film and the mica, causing a thermal mismatchstress. To verify this, a Au/mica sample mounted in UHV was sub-jected to a temperature cycle between 30 �C and 45 �C using a sam-ple heater integral to the stage (the Pt source was not operated),and the sample curvature was monitored during this excursion.The Au film was 400 nm thick and the mica disc was 72 lm thick;the Stoney equation was used (with Mmica ¼ 133 GPa, as before) toextract stress–thickness S.

In Fig. A2.2, S vs. time is co-plotted with the temperature pro-file. It is immediately apparent that S follows the temperaturequite closely, which is to be expected since film thickness is con-stant, and film stress rm ¼ MAu½ðaAu � amicaÞDT� is linear in T, whereaAu and amica are the coefficients of thermal expansion (CTEs) of Auand mica, respectively, and MAu is the biaxial modulus of Au. Not-ing that aAu = 14.2 � 10�6 K�1 [22], amica = 9 � 10�6 K�1 [30] andMAu ¼ 143 GPa, mismatch stress rm ¼ �7:3� 10�4DT GPa. ForDT ¼ 15 �C and hf = 400 nm, we find from theory that S ¼ rmhf ¼�43:8 GPa-Å, which is very close to the actual value of S duringthe dwell at 45 �C as measured from the plot.

The most important observation in this temperature excursionexperiment, however, is that the sign of the mismatch stress in thiscase is opposite to that in case of Pt deposition (Fig. A2.1a), whichmeans that whereas a coherent Pt film on Au is in tension andtherefore causes concave curvature in the substrate, a temperatureincrease alone causes convex curvature owing to the higher CTE ofAu than mica. It is therefore clear that a rise in temperature of thesubstrate during Pt growth did not contribute to the curvature ob-served by MOSS in Fig. A2.1a. In a final experiment to verify this

Fig. A2.2. Plots of temperature vs. time and stress-thickness S vs. time for a Au-mica sample cycled between 30 �C and 45 �C. S is observed to closely follow the temperatureprofile; it is noted that the sign of S is negative owing to the larger CTE of Au than mica. This eliminates the possibility that curvature detected by MOSS during Pt growth wasdue to a temperature rise of a thermally mismatched substrate.


point, the Pt source was heated to deposition temperature but withan empty crucible (i.e., no film growth occurred) in order to ob-serve curvature changes due to radiative heating from the source;none was measured over an observation period of 2 h.

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effects of substrate shape, curvature and roughness on thin heteroepitaxial films of pt on au(111)

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