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C nanoscale multilayers
SOLID STATE ELSEVIER
Solid State Ionics 101-103 (1997) 279-284
IONICS
Morphological transition in Ni/C nanoscale multilayers M. Bobeth a'*, R. Krawietz b, H. Mai c, W. Pomp&, A. Sewing a, J. T h o m a s ° "Max-Planck-Gesellschaft, Research Group on Mechanics of Heterogeneous Solids, Hallwachsstr. 3. D-01069 Dresden, Germany "Institute for Crystallography and Solid State Physics, Technical University Dresden, Mommsenstr. 13, D-01069 Dresden, Germany ~Fraunhofer-lnstitute for Materials Physics and Thin Film Technology. Helmholtzstr. 20. D-01069 Dresden, Germany dlnstitute,for Solid State and Materials Research Dresden. PF 270016. D-OIl71 Dresden, Germany
Abstract Annealing of nanoscale N i / C multilayers and C / N i / C trilayers at temperatures above about 670 K leads to the formation of separated Ni particles embedded in carbon. The agglomeration of the Ni layers starts by the formation of holes in the Ni layer and proceeds by hole growth and hole coalescence. Different thermodynamic driving forces and kinetic processes which can affect the observed agglomeration are discussed. The formation of holes by thermal grooving of Ni grain boundaries is analyzed quantitatively. Keywords: Morphological transition; Agglomeration; Multilayers; Interface diffusion Materials: Ni/C
1. Introduction Metal/carbon multilayers (e.g. metal =Ni, Co, Ru, W) with typical thicknesses of single layers of a few nanometers have been investigated extensively. This concerns, in particular, their application as dispersing elements for X-rays also at elevated temperatures up to about 700 K [1]. Besides this application, metal/carbon layers are suited for studying metal-carbon solid state reactions as for example the metal-catalyzed conversion of amorphous carbon (a-C) to graphite (g-C). The a-C phase usually forms during layer preparation by common deposition techniques. Metal/carbon multilayers are metastable owing to Corresponding author. Fax: +49-351 bobeth @tmfs.mpgfk.tu-dresden.de
463 1422; e-mail:
their high specific interfacial energy. The metal layers can decompose during annealing into separated particles embedded in a carbon matrix which has also been called agglomeration in the case of surface layers [2]. The stability of metal/carbon multilayers against agglomeration seems to be connected with the ability of the metal to form stable carbides. For example, W/C multilayers of 2 nm period were found to be stable during annealing at 773 K for 4 h [3] whereas Ni/C multilayers decomposed into Ni particles already during 20 min annealing at 673 K [4]. Similar agglomerations have been observed in Ru/C multilayers [3], as well as in C / C o / C [5] and C/Ni/C trilayers (Fig. 1). The agglomeration of the metal layers is probably supported by the graphitization of a-C. Electronmicroscopic studies revealed that a-C transforms to graphite already at relatively low temperatures when
0167-2738/97/$17.00 © 1997 Elsevier Science B.V. All rights reserved. PII S0167-2738(97)00193-8
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(a)
(b)
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Fig. 1. Plan-view TEM images with corresponding Ni diffraction patterns of C/Ni/C trilayers after annealing for 1 h at 773 K. The decomposition of Ni layers starts with the formation (a) and growth (b) of holes up to the development of separated Ni panicles (c). (Ni dark areas, layer thicknesses: (a) 5/10/5, (b) 10/3/10, (c) 2/1/2 nm.)
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a-C is in contact with metals (e.g. above 730 K for Ni [6] and above 773 K for Co [5]). It has been proposed that graphite is formed by the dissolution of C-atoms of the a-C phase in the metal, and subsequent precipitation of the dissolved carbon as graphite (see e.g. Ref. [6]). The proposed dissolution-diffusion-precipitation mechanism presumably plays an important role in the agglomeration of the metal layers. In the present paper, different thermodynamic driving forces and diffusion processes are analyzed which are related to the observed agglomeration of Ni layers in Ni/C multilayers and C / N i / C trilayers starting with the formation and growth of holes in the Ni layers up to the formation of separated Ni particles embedded in carbon.
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2. Experimental observations The Ni/C layer systems discussed in the following were prepared by pulsed laser deposition. Multilayers with 10 Ni/C layer pairs with thicknesses d(Ni) = 1.9 nm and d ( C ) = 3.5 nm were deposited on Si wafers [4]. For the study of the Ni-layer agglomeration by means of plan-view transmission electron microscopy (TEM), C / N i / C trilayers were deposited onto NaCI substrates with single layer thicknesses ranging from l to 10 nm. SNMS analysis of the as-deposited multilayers showed that thin Ni layers (--< 3 nm) contained also carbon (denoted as 'Ni:C layers' in the following) with a concentration of about 15 at% which greatly exceeds the equilibrium solubility. This results mainly by ballistic mixing due to the deposition of particles with mean kinetic energy of about 100 eV. In all Ni/C layer systems investigated, the C layers were amorphous. The Ni:C layers were amorphous (a-Ni) for thicknesses --<3 nm. Thicker Ni layers were polycrystalline with grain sizes of a few nm. TEM studies of multilayer cross sections revealed a flat interface between the Ni and C layers with a typical rmsroughness of 0.3 nm [4]. The Ni/C multilayers were annealed at 473, 573, 623 and 673 K for 20 min and the C / N i / C trilayers at 773 K for 1 h. In part, annealing of the trilayers was performed in the electron microscope in order to observe morphological changes in situ by plan-view TEM. X-ray and electron diffraction analysis of annealed layers revealed that at about 473 K the
i
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Fig. 2. Schematic of interface and volume diffusion processes which could be relevant during the formation of 'holes' (C channels) in Ni layers by thermal groovingof Ni grain boundaries or by C precipitation. metastable Ni3C phase forms which disappeared at temperatures above about 600 K where fcc nickel was detected. TEM micrographs of cross sections of Ni/C muitilayers suggested that carbon channels through 2 nm thick Ni layers (briefly called as 'holes' in the Ni layer, cf. Fig. 2) are formed at about 670 K after 20 rain annealing (cf. Ref. [4]). Planview TEM observations on C / N i / C trilayers showed that thicker Ni layers agglomerate, as expected, at higher temperatures and/or need longer annealing. The layer agglomeration starts also with the formation of holes in the Ni layer which grow with annealing time (Fig. l a and b). After 1 h annealing of a trilayer at 773 K, a I n m thick Ni layer is completely decomposed into separated Ni particles with diameters ranging from 3 to 50 nm (mean diameter= 8 nm, Fig. lc). The exceptionally large Ni particles are presumably formed on the surface of the upper C layer. This means that Ni diffuses through thin C layers (--< 2 nm) and forms large Ni particles due to the relatively fast surface diffusion of Ni on carbon. This suggestion was also supported by cross section TEM of Ni/C multilayers which showed that the upper Ni layer completely disappeared presumably also by the out-diffusion of Ni through the upper C layer (about 1 nm thick) and the formation of large Ni particles on the surface [4].
3. Theoretical analysis of morphological changes 3.1. T h e r m o d y n a m i c driving f o r c e s
The as-deposited multilayers considered here represent metastable systems with enhanced free energy
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where the main contributions are given by (i) the large interfacial area between Ni and C, (ii) the structural energy of a-C as well as of a-Ni in the case of very thin Ni layers, (iii) the grain-boundary energy in fine-grained Ni-layers, and (iv) the excess free energy of mixing in Ni:C layers with C concentrations exceeding the C solubility in Ni ( < 0.17 at% for T < 873 K [7]). Correspondingly, the reduction of interface area, the crystallization of a-C and a-Ni, and in the case of Ni:C layers also the Ni-C demixing, represent mechanisms for morphological changes of Ni/C multilayers during annealing. Besides these 'chemical' driving forces, the relief of mechanical stresses in the as-deposited layers, as well as stresses appearing due to diffusion processes, can influence the morphological development of muitilayers.
3.2. Kinetic processes The morphological changes observed in the Ni/C layer systems are limited by the mobilities of Ni and C. Atomic transport with a characteristic diffusion length la = X/Dt < 1 nm is practically frozen. Extrapolating available bulk diffusion data to the present annealing conditions, one finds that C and Ni self-diffusion are negligible. On the contrary, the diffusion of C in Ni is rather fast with Id > 100 nm for 20 min annealing at 673 K. Anisotropic diffusion data of Ni in graphite suggest that also Ni can migrate distances > 10 nm in the fast diffusion direction under the same annealing conditions. With respect to the present investigations at relatively low temperatures, mass transport along the Ni/C interface and along grain boundaries in the Ni layer should be of great importance. However, the authors are not aware of corresponding data. Data on Ni surface self-diffusion lead to a corresponding migration distance 1d > > 100 nm. These estimates suggest that special diffusion paths are sufficiently fast so that morphological changes can occur under the present annealing conditions. Because of the fast C-diffusion in Ni and the slow C self-diffusion, the dissolution of C in Ni could be the slowest, ratedetermining process.
3.3. Mechanisms of morphological changes The electronmicroscopic observations (Fig. 1) showed that the Ni/C layer systems are thermally
unstable and transform during annealing into separated Ni particles embedded in carbon. This transformation starts with the formation of holes in the Ni layer and proceeds by hole growth and coalescence as it is observed also for surface films [2]. Hole growth obviously occurs owing to capillary forces when the hole diameter is sufficiently large compared to the layer thickness. However, the mechanism of hole formation is not yet fully understood.
3.3. I. Hole .formation Concerning the N i - C system, two mechanisms could be relevant: (i) thermal grooving of Ni grainboundaries, and (ii) C precipitation in Ni layers, especially in grain-boundaries (see Figs. 2 and 3, cf. also the overview [2]). In the following, the thermal grooving of Ni grain boundaries in Ni/C layer systems is analyzed according to the study by Mullins [8] on grooving by surface diffusion. The Ni grain-boundaries are assumed to be perpendicular to the Ni/C interface. The equilibrium between the grain boundary tension, Ygb' and the Ni/C interface tension, Z, at the groove root leads to Ygb = 23~ sin fl (Fig. 3a). We consider here the case where the transport of Ni and C atoms occurs only along the Ni/C interface. The interfacial fluxes of Ni and C, JN~ and Jc, are coupled by the volume constraint ~ J N i + OcJc = 0 where ON~ and Oc are the atomic volumes. The interfacial fluxes of the components are proportional to the gradient of the corresponding chemical potentials, tz, along the
i`!iiiiiiiiiii!iiiiiiiiiiiiiiiiii
~(b) Fig. 3. Schematic of thermal grooving of" Ni grain boundaries (a) and of an equilibrium configuration of the Ni/C interface for small grain size (b).
M. Bobeth et al. / Solid State lonics 101-103 (1997) 279-284 interface which depend on the interface curvature and the normal stress component at the interface, e.g. JNi =nNi(6NiDi.Ni/kT)VtZNi where n is the atomic density, 6 the interface width and D~ the interfacial diffusivity. The derivation of appropriate formulae for the interfacial N i - C counterdiffusion (Fig. 2) under volume constraint is analogous to the approach in [9] and will be published elsewhere [10]. As a result, the evolution equation of the interface profile, h(z,t) (Fig. 3a), in the small slope approximation (Oh/Oz-~ h ' ( z , t ) < 1) is obtained as in [8] Oh O4h(z,t) Ot - B------S~, Oz
(1)
where B ~ ( J ' 2 6 D i ) e f f Y i / k T with ( ~ 6Oi)er f as an effective kinetic coefficient characterizing the overall interfacial mass transport. /2 denotes an effective atomic volume. The solution of Eq. ( 1) subject to the conditions h(z,O)=O, h ' ( O , t ) = m = t a n /3 and h'"(O,t) = 0 has been given in [8]. The last condition corresponds to a vanishing mass flux out of the grain boundary. The depth of the groove results as d = 0.78 m (Bt) 1/4 (Fig. 3a). When the groove roots from the two sides of a Ni grain boundary meet in the course of annealing (layer thickness H = 2d), a hole in the Ni layer is formed. Unfortunately, the interface energies y, and Ygb are only poorly known. A more accurate analysis should also take into account the crystal anisotropy of the interface energies. Using the above formulae, one finds that an effective interface diffusion coefficient D~>8X 10 -~9 m 2 s -~ (corresponding to B > I 0 37 m 4 s 1) is sufficient to form holes by thermal grooving for the following parameter assumptions: ~ ~-'Ygb--~-0.6 J m -2, I 2 = 9 . 2 X 10 -3o 3 I/3 m, 6=J2 , t = 1 2 0 0 s, T = 6 7 3 K, H = 3 nm (especially the assumption on Z is a crude approximation). The above analysis concerned a single grain boundary. When the groove width, w, becomes comparable with the lateral grain size, g, the presence of neighboring grain boundaries has to be taken into account (Fig. 3). Then, an equilibrium state without further grooving can be reached which is characterized by a uniform curvature of the Ni/C interface. By approximating the Ni/C interface of a Ni grain by a spherical cap, a critical grain size, go, can be estimated [2]. For grain sizes g > g c , the
283
groove roots of a grain boundary will meet so that hole formation by thermal grooving would be possible. For ~->'Ygb=0.6 J m -2, the critical grain size is -->7 times the Ni layer thickness which exceeds typical grain sizes observed (Fig. 1 a). Thus, this mechanism should not work, especially when 3~> Ygb- On the other hand, grain sizes show a large scatter and enlarge by grain ripening. Furthermore, thermal grooving is presumably more pronounced at the triple grain junctions so that, in summary, this hole formation mechanism cannot be excluded. Alternatively, holes can be formed by the precipitation of carbon within Ni layers (Fig. 2) also in cases when the Ni grain size is considerably smaller than the critical grain size g~. Preferred nucleation sites are Ni grain boundaries and, especially, triple grain junctions because of the reduced nucleation energy at those sites compared to homogeneous nucleation within Ni. A quantitative analysis of the nucleation problem has not been performed yet and is difficult because of the poor knowledge of the corresponding interface energies. 3.3.2. Hole growth After the formation of holes in the Ni layer, they grow due to the action of capillary forces as a consequence of the varying curvature along the Ni/C interface (Fig. 2). Holes in Ni/C multilayers can grow by volume diffusion of Ni through carbon to neighboring Ni layers exhibiting still a flat Ni/C interface [1 1]. However, because of the relatively slow Ni diffusion in carbon, this process is possibly less important. For predominant interface diffusion, hole growth is described by an equation similar to (1) without the small slope approximation (cf. [8]). The calculation of the growth velocity of large holes (hole radius>Ni layer thickness) is equivalent to the calculation of the shrinkage velocity of a two-dimensional Ni particle embedded in carbon (cf. Fig. 2). For an ellipsoidal shape of the Ni particle, the shrinkage velocity at the tip of the Ni particle in the elongated direction is obtained as v = 3Bc(a 2 - C2)/a 6 where a and c are the half-axes of the ellipsoid (c>a). Choosing, for example, a = 1.5 nm, c = 3 nm and the above estimate B ~ 1 0 -37 m4/s, one obtains v = - 0 . 5 nm/s. This value exceeds the hole growth velocity which is necessary to explain the observations. Also, a numerical analysis of the evolution of
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the Ni/C interface subsequent to hole formation supports that the spheroidization of fiat Ni islands occurs sufficiently fast [10]. A second driving force for hole growth acts when graphite precipitates are formed in the Ni layer (Fig. 2). According to the proposed 'dissolution-diffusion-precipitation' mechanism [6], a-C dissolves in Ni and diffuses by volume or interface diffusion to the g-C precipitates which will grow due to their lower free energy. (In the present experiments, it was however not possible to prove the graphitic structure of the tiny C precipitates below 800 K.) In the case of predominant volume diffusion of carbon through Ni as rate-determining process (assuming Ni interface transport to be relatively fast), the growth velocity of g-C precipitates can be estimated from the C flux j = nc(D c/kT)Vlzc. The chemical potential gradient is approximated in a crude manner by (pca-c-lzg-c)](H]2). Estimating /.za-C--~/.Zg-C by the heat of crystallization of a-C of 19 kJ/mol [5] and choosing a growth velocity of C precipitates, v = /2cj~0.01 nm/s, the C diffusion coefficient is obtained as D c = 2 X l 0 -17 m2/s (nc=2.6Xl025 m -3 , T=673 K, ~ c = 8 . 9 X 1 0 -3° m 3, H = 3 rim). This estimate of D c is compatible with literature data [12]. Thus, hole growth can also be explained by C volume diffusion in Ni, driven by the difference in the free energies of a-C and graphite. The final stage of separated Ni particles results obviously by the coalescence of holes and the subsequent spheroidization of flat Ni islands (Fig. lc). Further investigations are necessary in order to find out the predominant diffusion paths of C and Ni, and to elucidate whether the morphological transition is possibly reaction-controlled by the dissolution of a-C in Ni as the rate-determining process.
4. Conclusions
Ni layers in nanoscale Ni/C layer systems decompose during annealing above about 670 K. The
agglomeration starts by the formation of C channels through the Ni layer (called 'holes') which are presumably formed by C precipitation in Ni grainboundaries. Hole formation by thermal grooving of Ni grain-boundaries seems to be somewhat less likely. Because of the relatively slow Ni self-diffusion, the Ni transport occurs probably by interface diffusion or, eventually, by Ni diffusion in carbon. Concerning the C transport, besides interface diffusion, C diffusion within Ni is also sufficiently fast to explain the morphological changes. The growth of holes in the Ni layer is driven by capillary forces and leads to the formation of separated Ni particles by hole coalescence. When the holes in the Ni layers consist of graphite, the free energy difference between a-C and graphite acts as an additional driving force for hole growth.
References [1] V. Dupuis, M.F. Ravet, C. Tete, M. Piecuch, Y. Lepetre, R. Rivoira, E. Ziegler, J. Appl. Phys. 68 (1990) 5146. [2] D.J. Srolovitz, M.G. Goldiner, J. Met. 3 (1995) 31. [3] T.D. Nguyen, R. Gronsky, J.B. Kortright, Mater. Res. Soc. Symp. Proc. 230 (1992) 109. [4] R. Krawietz, B. Wehner, D. Meyer, K. Richter, H. Mai, R. Dietsch, S. Hopfe, R. Scholz, W. Pompe, Fresenius J. Anal. Chem. 353 (1995) 246. [5] T.J. Konno, R. Sinclair, Acta Metall. Mater. 43 (1995) 471. [6] R. Lamber, N.I. Jaeger, Surf. Sci. 197 (1988) 402. [7] K. Natesan, T.F. Kassner, Metall. Trans. 4 (1973) 2557. [8] W.W. Mullins, J. Appl. Phys. 28 (1957) 333. [9] C.Y. Li, J.M. Blakely, A.H. Feingold, Acta Metall. 14 (1965) 1397. [10] M. Bobeth et al., to be published. [11] F. Spaepen, in: P. Dhez and C. Weisbuch (Eds.), Physics, Fabrication, and Applications of Multilayered Structures, Plenum, New York, 1988, p. 199. [12] B.S. Berry, J. Appl. Phys. 44 (1973) 3792.
Solid State Ionics 101-103 (1997) 279-284
IONICS
Morphological transition in Ni/C nanoscale multilayers M. Bobeth a'*, R. Krawietz b, H. Mai c, W. Pomp&, A. Sewing a, J. T h o m a s ° "Max-Planck-Gesellschaft, Research Group on Mechanics of Heterogeneous Solids, Hallwachsstr. 3. D-01069 Dresden, Germany "Institute for Crystallography and Solid State Physics, Technical University Dresden, Mommsenstr. 13, D-01069 Dresden, Germany ~Fraunhofer-lnstitute for Materials Physics and Thin Film Technology. Helmholtzstr. 20. D-01069 Dresden, Germany dlnstitute,for Solid State and Materials Research Dresden. PF 270016. D-OIl71 Dresden, Germany
Abstract Annealing of nanoscale N i / C multilayers and C / N i / C trilayers at temperatures above about 670 K leads to the formation of separated Ni particles embedded in carbon. The agglomeration of the Ni layers starts by the formation of holes in the Ni layer and proceeds by hole growth and hole coalescence. Different thermodynamic driving forces and kinetic processes which can affect the observed agglomeration are discussed. The formation of holes by thermal grooving of Ni grain boundaries is analyzed quantitatively. Keywords: Morphological transition; Agglomeration; Multilayers; Interface diffusion Materials: Ni/C
1. Introduction Metal/carbon multilayers (e.g. metal =Ni, Co, Ru, W) with typical thicknesses of single layers of a few nanometers have been investigated extensively. This concerns, in particular, their application as dispersing elements for X-rays also at elevated temperatures up to about 700 K [1]. Besides this application, metal/carbon layers are suited for studying metal-carbon solid state reactions as for example the metal-catalyzed conversion of amorphous carbon (a-C) to graphite (g-C). The a-C phase usually forms during layer preparation by common deposition techniques. Metal/carbon multilayers are metastable owing to Corresponding author. Fax: +49-351 bobeth @tmfs.mpgfk.tu-dresden.de
463 1422; e-mail:
their high specific interfacial energy. The metal layers can decompose during annealing into separated particles embedded in a carbon matrix which has also been called agglomeration in the case of surface layers [2]. The stability of metal/carbon multilayers against agglomeration seems to be connected with the ability of the metal to form stable carbides. For example, W/C multilayers of 2 nm period were found to be stable during annealing at 773 K for 4 h [3] whereas Ni/C multilayers decomposed into Ni particles already during 20 min annealing at 673 K [4]. Similar agglomerations have been observed in Ru/C multilayers [3], as well as in C / C o / C [5] and C/Ni/C trilayers (Fig. 1). The agglomeration of the metal layers is probably supported by the graphitization of a-C. Electronmicroscopic studies revealed that a-C transforms to graphite already at relatively low temperatures when
0167-2738/97/$17.00 © 1997 Elsevier Science B.V. All rights reserved. PII S0167-2738(97)00193-8
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(a)
(b)
(c)
Fig. 1. Plan-view TEM images with corresponding Ni diffraction patterns of C/Ni/C trilayers after annealing for 1 h at 773 K. The decomposition of Ni layers starts with the formation (a) and growth (b) of holes up to the development of separated Ni panicles (c). (Ni dark areas, layer thicknesses: (a) 5/10/5, (b) 10/3/10, (c) 2/1/2 nm.)
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a-C is in contact with metals (e.g. above 730 K for Ni [6] and above 773 K for Co [5]). It has been proposed that graphite is formed by the dissolution of C-atoms of the a-C phase in the metal, and subsequent precipitation of the dissolved carbon as graphite (see e.g. Ref. [6]). The proposed dissolution-diffusion-precipitation mechanism presumably plays an important role in the agglomeration of the metal layers. In the present paper, different thermodynamic driving forces and diffusion processes are analyzed which are related to the observed agglomeration of Ni layers in Ni/C multilayers and C / N i / C trilayers starting with the formation and growth of holes in the Ni layers up to the formation of separated Ni particles embedded in carbon.
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2. Experimental observations The Ni/C layer systems discussed in the following were prepared by pulsed laser deposition. Multilayers with 10 Ni/C layer pairs with thicknesses d(Ni) = 1.9 nm and d ( C ) = 3.5 nm were deposited on Si wafers [4]. For the study of the Ni-layer agglomeration by means of plan-view transmission electron microscopy (TEM), C / N i / C trilayers were deposited onto NaCI substrates with single layer thicknesses ranging from l to 10 nm. SNMS analysis of the as-deposited multilayers showed that thin Ni layers (--< 3 nm) contained also carbon (denoted as 'Ni:C layers' in the following) with a concentration of about 15 at% which greatly exceeds the equilibrium solubility. This results mainly by ballistic mixing due to the deposition of particles with mean kinetic energy of about 100 eV. In all Ni/C layer systems investigated, the C layers were amorphous. The Ni:C layers were amorphous (a-Ni) for thicknesses --<3 nm. Thicker Ni layers were polycrystalline with grain sizes of a few nm. TEM studies of multilayer cross sections revealed a flat interface between the Ni and C layers with a typical rmsroughness of 0.3 nm [4]. The Ni/C multilayers were annealed at 473, 573, 623 and 673 K for 20 min and the C / N i / C trilayers at 773 K for 1 h. In part, annealing of the trilayers was performed in the electron microscope in order to observe morphological changes in situ by plan-view TEM. X-ray and electron diffraction analysis of annealed layers revealed that at about 473 K the
i
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Fig. 2. Schematic of interface and volume diffusion processes which could be relevant during the formation of 'holes' (C channels) in Ni layers by thermal groovingof Ni grain boundaries or by C precipitation. metastable Ni3C phase forms which disappeared at temperatures above about 600 K where fcc nickel was detected. TEM micrographs of cross sections of Ni/C muitilayers suggested that carbon channels through 2 nm thick Ni layers (briefly called as 'holes' in the Ni layer, cf. Fig. 2) are formed at about 670 K after 20 rain annealing (cf. Ref. [4]). Planview TEM observations on C / N i / C trilayers showed that thicker Ni layers agglomerate, as expected, at higher temperatures and/or need longer annealing. The layer agglomeration starts also with the formation of holes in the Ni layer which grow with annealing time (Fig. l a and b). After 1 h annealing of a trilayer at 773 K, a I n m thick Ni layer is completely decomposed into separated Ni particles with diameters ranging from 3 to 50 nm (mean diameter= 8 nm, Fig. lc). The exceptionally large Ni particles are presumably formed on the surface of the upper C layer. This means that Ni diffuses through thin C layers (--< 2 nm) and forms large Ni particles due to the relatively fast surface diffusion of Ni on carbon. This suggestion was also supported by cross section TEM of Ni/C multilayers which showed that the upper Ni layer completely disappeared presumably also by the out-diffusion of Ni through the upper C layer (about 1 nm thick) and the formation of large Ni particles on the surface [4].
3. Theoretical analysis of morphological changes 3.1. T h e r m o d y n a m i c driving f o r c e s
The as-deposited multilayers considered here represent metastable systems with enhanced free energy
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where the main contributions are given by (i) the large interfacial area between Ni and C, (ii) the structural energy of a-C as well as of a-Ni in the case of very thin Ni layers, (iii) the grain-boundary energy in fine-grained Ni-layers, and (iv) the excess free energy of mixing in Ni:C layers with C concentrations exceeding the C solubility in Ni ( < 0.17 at% for T < 873 K [7]). Correspondingly, the reduction of interface area, the crystallization of a-C and a-Ni, and in the case of Ni:C layers also the Ni-C demixing, represent mechanisms for morphological changes of Ni/C multilayers during annealing. Besides these 'chemical' driving forces, the relief of mechanical stresses in the as-deposited layers, as well as stresses appearing due to diffusion processes, can influence the morphological development of muitilayers.
3.2. Kinetic processes The morphological changes observed in the Ni/C layer systems are limited by the mobilities of Ni and C. Atomic transport with a characteristic diffusion length la = X/Dt < 1 nm is practically frozen. Extrapolating available bulk diffusion data to the present annealing conditions, one finds that C and Ni self-diffusion are negligible. On the contrary, the diffusion of C in Ni is rather fast with Id > 100 nm for 20 min annealing at 673 K. Anisotropic diffusion data of Ni in graphite suggest that also Ni can migrate distances > 10 nm in the fast diffusion direction under the same annealing conditions. With respect to the present investigations at relatively low temperatures, mass transport along the Ni/C interface and along grain boundaries in the Ni layer should be of great importance. However, the authors are not aware of corresponding data. Data on Ni surface self-diffusion lead to a corresponding migration distance 1d > > 100 nm. These estimates suggest that special diffusion paths are sufficiently fast so that morphological changes can occur under the present annealing conditions. Because of the fast C-diffusion in Ni and the slow C self-diffusion, the dissolution of C in Ni could be the slowest, ratedetermining process.
3.3. Mechanisms of morphological changes The electronmicroscopic observations (Fig. 1) showed that the Ni/C layer systems are thermally
unstable and transform during annealing into separated Ni particles embedded in carbon. This transformation starts with the formation of holes in the Ni layer and proceeds by hole growth and coalescence as it is observed also for surface films [2]. Hole growth obviously occurs owing to capillary forces when the hole diameter is sufficiently large compared to the layer thickness. However, the mechanism of hole formation is not yet fully understood.
3.3. I. Hole .formation Concerning the N i - C system, two mechanisms could be relevant: (i) thermal grooving of Ni grainboundaries, and (ii) C precipitation in Ni layers, especially in grain-boundaries (see Figs. 2 and 3, cf. also the overview [2]). In the following, the thermal grooving of Ni grain boundaries in Ni/C layer systems is analyzed according to the study by Mullins [8] on grooving by surface diffusion. The Ni grain-boundaries are assumed to be perpendicular to the Ni/C interface. The equilibrium between the grain boundary tension, Ygb' and the Ni/C interface tension, Z, at the groove root leads to Ygb = 23~ sin fl (Fig. 3a). We consider here the case where the transport of Ni and C atoms occurs only along the Ni/C interface. The interfacial fluxes of Ni and C, JN~ and Jc, are coupled by the volume constraint ~ J N i + OcJc = 0 where ON~ and Oc are the atomic volumes. The interfacial fluxes of the components are proportional to the gradient of the corresponding chemical potentials, tz, along the
i`!iiiiiiiiiii!iiiiiiiiiiiiiiiiii
~(b) Fig. 3. Schematic of thermal grooving of" Ni grain boundaries (a) and of an equilibrium configuration of the Ni/C interface for small grain size (b).
M. Bobeth et al. / Solid State lonics 101-103 (1997) 279-284 interface which depend on the interface curvature and the normal stress component at the interface, e.g. JNi =nNi(6NiDi.Ni/kT)VtZNi where n is the atomic density, 6 the interface width and D~ the interfacial diffusivity. The derivation of appropriate formulae for the interfacial N i - C counterdiffusion (Fig. 2) under volume constraint is analogous to the approach in [9] and will be published elsewhere [10]. As a result, the evolution equation of the interface profile, h(z,t) (Fig. 3a), in the small slope approximation (Oh/Oz-~ h ' ( z , t ) < 1) is obtained as in [8] Oh O4h(z,t) Ot - B------S~, Oz
(1)
where B ~ ( J ' 2 6 D i ) e f f Y i / k T with ( ~ 6Oi)er f as an effective kinetic coefficient characterizing the overall interfacial mass transport. /2 denotes an effective atomic volume. The solution of Eq. ( 1) subject to the conditions h(z,O)=O, h ' ( O , t ) = m = t a n /3 and h'"(O,t) = 0 has been given in [8]. The last condition corresponds to a vanishing mass flux out of the grain boundary. The depth of the groove results as d = 0.78 m (Bt) 1/4 (Fig. 3a). When the groove roots from the two sides of a Ni grain boundary meet in the course of annealing (layer thickness H = 2d), a hole in the Ni layer is formed. Unfortunately, the interface energies y, and Ygb are only poorly known. A more accurate analysis should also take into account the crystal anisotropy of the interface energies. Using the above formulae, one finds that an effective interface diffusion coefficient D~>8X 10 -~9 m 2 s -~ (corresponding to B > I 0 37 m 4 s 1) is sufficient to form holes by thermal grooving for the following parameter assumptions: ~ ~-'Ygb--~-0.6 J m -2, I 2 = 9 . 2 X 10 -3o 3 I/3 m, 6=J2 , t = 1 2 0 0 s, T = 6 7 3 K, H = 3 nm (especially the assumption on Z is a crude approximation). The above analysis concerned a single grain boundary. When the groove width, w, becomes comparable with the lateral grain size, g, the presence of neighboring grain boundaries has to be taken into account (Fig. 3). Then, an equilibrium state without further grooving can be reached which is characterized by a uniform curvature of the Ni/C interface. By approximating the Ni/C interface of a Ni grain by a spherical cap, a critical grain size, go, can be estimated [2]. For grain sizes g > g c , the
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groove roots of a grain boundary will meet so that hole formation by thermal grooving would be possible. For ~->'Ygb=0.6 J m -2, the critical grain size is -->7 times the Ni layer thickness which exceeds typical grain sizes observed (Fig. 1 a). Thus, this mechanism should not work, especially when 3~> Ygb- On the other hand, grain sizes show a large scatter and enlarge by grain ripening. Furthermore, thermal grooving is presumably more pronounced at the triple grain junctions so that, in summary, this hole formation mechanism cannot be excluded. Alternatively, holes can be formed by the precipitation of carbon within Ni layers (Fig. 2) also in cases when the Ni grain size is considerably smaller than the critical grain size g~. Preferred nucleation sites are Ni grain boundaries and, especially, triple grain junctions because of the reduced nucleation energy at those sites compared to homogeneous nucleation within Ni. A quantitative analysis of the nucleation problem has not been performed yet and is difficult because of the poor knowledge of the corresponding interface energies. 3.3.2. Hole growth After the formation of holes in the Ni layer, they grow due to the action of capillary forces as a consequence of the varying curvature along the Ni/C interface (Fig. 2). Holes in Ni/C multilayers can grow by volume diffusion of Ni through carbon to neighboring Ni layers exhibiting still a flat Ni/C interface [1 1]. However, because of the relatively slow Ni diffusion in carbon, this process is possibly less important. For predominant interface diffusion, hole growth is described by an equation similar to (1) without the small slope approximation (cf. [8]). The calculation of the growth velocity of large holes (hole radius>Ni layer thickness) is equivalent to the calculation of the shrinkage velocity of a two-dimensional Ni particle embedded in carbon (cf. Fig. 2). For an ellipsoidal shape of the Ni particle, the shrinkage velocity at the tip of the Ni particle in the elongated direction is obtained as v = 3Bc(a 2 - C2)/a 6 where a and c are the half-axes of the ellipsoid (c>a). Choosing, for example, a = 1.5 nm, c = 3 nm and the above estimate B ~ 1 0 -37 m4/s, one obtains v = - 0 . 5 nm/s. This value exceeds the hole growth velocity which is necessary to explain the observations. Also, a numerical analysis of the evolution of
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the Ni/C interface subsequent to hole formation supports that the spheroidization of fiat Ni islands occurs sufficiently fast [10]. A second driving force for hole growth acts when graphite precipitates are formed in the Ni layer (Fig. 2). According to the proposed 'dissolution-diffusion-precipitation' mechanism [6], a-C dissolves in Ni and diffuses by volume or interface diffusion to the g-C precipitates which will grow due to their lower free energy. (In the present experiments, it was however not possible to prove the graphitic structure of the tiny C precipitates below 800 K.) In the case of predominant volume diffusion of carbon through Ni as rate-determining process (assuming Ni interface transport to be relatively fast), the growth velocity of g-C precipitates can be estimated from the C flux j = nc(D c/kT)Vlzc. The chemical potential gradient is approximated in a crude manner by (pca-c-lzg-c)](H]2). Estimating /.za-C--~/.Zg-C by the heat of crystallization of a-C of 19 kJ/mol [5] and choosing a growth velocity of C precipitates, v = /2cj~0.01 nm/s, the C diffusion coefficient is obtained as D c = 2 X l 0 -17 m2/s (nc=2.6Xl025 m -3 , T=673 K, ~ c = 8 . 9 X 1 0 -3° m 3, H = 3 rim). This estimate of D c is compatible with literature data [12]. Thus, hole growth can also be explained by C volume diffusion in Ni, driven by the difference in the free energies of a-C and graphite. The final stage of separated Ni particles results obviously by the coalescence of holes and the subsequent spheroidization of flat Ni islands (Fig. lc). Further investigations are necessary in order to find out the predominant diffusion paths of C and Ni, and to elucidate whether the morphological transition is possibly reaction-controlled by the dissolution of a-C in Ni as the rate-determining process.
4. Conclusions
Ni layers in nanoscale Ni/C layer systems decompose during annealing above about 670 K. The
agglomeration starts by the formation of C channels through the Ni layer (called 'holes') which are presumably formed by C precipitation in Ni grainboundaries. Hole formation by thermal grooving of Ni grain-boundaries seems to be somewhat less likely. Because of the relatively slow Ni self-diffusion, the Ni transport occurs probably by interface diffusion or, eventually, by Ni diffusion in carbon. Concerning the C transport, besides interface diffusion, C diffusion within Ni is also sufficiently fast to explain the morphological changes. The growth of holes in the Ni layer is driven by capillary forces and leads to the formation of separated Ni particles by hole coalescence. When the holes in the Ni layers consist of graphite, the free energy difference between a-C and graphite acts as an additional driving force for hole growth.
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