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Solid–solid interface reconstruction at equilibrated Ni–Al2O3 interfaces
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Acta Materialia 60 (2012) 4359–4369 www.elsevier.com/locate/actamat
Solid–solid interface reconstruction at equilibrated Ni–Al2O3 interfaces Hila Meltzman a, Dan Mordehai b, Wayne D. Kaplan a,⇑ a b
Department of Materials Engineering, Technion-Israel Institute of Technology, Haifa 32000, Israel Department of Mechanical Engineering, Technion-Israel Institute of Technology, Haifa 32000, Israel Received 16 April 2012; received in revised form 27 April 2012; accepted 28 April 2012 Available online 31 May 2012
Abstract The atomistic structure of solid Ni–Al2O3 interfaces at equilibrium was determined by aberration-corrected transmission electron microscopy, from specimens formed during solid-state dewetting of thin Ni films on the (0 0 0 1) surface of a-Alp 2O ffiffiffi 3. It was pffiffiffi found that the interface develops a unique mechanism for misfit strain reduction, termed delocalized coherency, via a 2:5 3 2:5 3R30 reconstructed interface structure, contradicting most simulations in the literature, which assume a semi-coherent structure for this system. Based on the experimental work presented here, a structural model was also simulated, showing periodic buckling of the terminating Ni layer (i.e. interface reconstruction). The interface energy was experimentally determined from dewetted Ni particles using Winterbottom analysis, and found to be 2.16 ± 0.2 J m–2 at P(O2) = 1020 atm and T = 1623 K. Ó 2012 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. Keywords: Interface energy; Interface structure; Transmission electron microscopy; Incoherent interfaces; Reconstruction
1. Introduction The structure and properties of metal–ceramic interfaces have been the focus of many studies since the 1960s, mostly due to the potential for numerous technological applications such as metal–ceramic bonding, brazing, microelectronic packaging, thin film technology, high temperature metal-oxidation processes, photovoltaic cells, protective coatings for metals, bioactive implant coatings, and high temperature aircraft components [1–4]. The complexity of solid–solid interfaces arises not only from the difference in long-range atomistic order on both sides of the interface, but also from the wide variety of defects that can form, and the unique structural and compositional characteristics of each interface, depending on the relative orientation of the two crystals, their composition, and the temperature used for the process [5]. This degree of complexity often inhibits quantitative treatment of interfacial phenomena.
⇑ Corresponding author. Tel.: +972 4 8294591; fax: +972 4 8294580.
E-mail address: [email protected] (W.D. Kaplan).
Due to the different atomistic arrangement on both sides of the interface, misfit strain inevitably develops. If kinetically possible, the system will move to reduce this strain energy by introducing either misfit dislocations or steps (to create a semi-coherent interface), or for the more common case of a large difference in structure between the interfaceforming crystals, via the formation of what is commonly termed an incoherent interface. Due to the complexity of incoherent interfaces, most studies to date have focused on analysis of misfit dislocations and steps at semi-coherent interfaces, and only a few reports were found regarding incoherent metal–ceramic interfaces [6–8]. One of the most studied types of metal–ceramic interfaces is that between face-centered cubic (fcc) metals and a-Al2O3 (sapphire). These interfaces are usually produced either by diffusion bonding of a pre-determined orientation [4], or by thin film deposition such as molecular beam epitaxy (MBE) [9–11]. It should be noted that these preparation techniques do not necessarily produce equilibrated interfaces. The as-bonded or as-deposited interface states often reflect a metastable interface structure and orientation, rather than a thermodynamically equilibrated state. The question of whether the examined interface is equilibrated or not, and
1359-6454/$36.00 Ó 2012 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.actamat.2012.04.037
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whether it reflects the minimum energy configuration of the interface, is rarely addressed, although this is a significant driving force for microstructural evolution of interfaces. The Nb–Al2O3 interface provides a good example of the influence of interface fabrication methods on the resulting interface orientation; Mader and Ru¨hle [12] reported an orientation relationship of (1 1 0)[0 0 1]Nb||(0 0 0 1)½2 1 1 0Al2 O3 for a solid-state diffusion bonded interface, while a (1 1 1)[1 1 0]Nb|| (0 0 0 1)½2 1 1 0Al2 O3 orientation relationship was reported for the same system produced by MBE [13]. The relative thermodynamic stability of the different interfaces was not determined. One way to equilibrate metal–ceramic interfaces is to “dewet” thin metallic films on ceramic substrates, which is essentially disruption (in the liquid or solid state) of the film driven by the minimization of surface and interfacial energy [14–17]. This process of film agglomeration results in a very large number of isolated particles, which can be investigated separately to achieve very good statistics. The size of the droplets/particles (up to several micrometers in diameter) enables shorter equilibration times, and solid-state dewetting can be used in addition to liquid-state dewetting to reach equilibration for solid– solid interface studies [15]. In addition to being a good metal–Al2O3 model system, Ni–Al2O3 interfaces can be found in numerous technological applications, such as composites, microelectronic packaging, catalytic processes and applications demanding high temperature strength and oxidation resistance such as turbine blades [18–20]. In contrast to the large technological importance of this system, Ni–Al2O3 interfaces were not experimentally studied to the same extent as other metal– alumina interfaces. This is mainly because of the relative high melting temperature of the metal and/or because of the relatively large surface energy of Ni, which requires an extremely pure environment for model studies [21,22]. Due to the lack of experimental data available, numerous atomistic simulations (mostly by density functional theory (DFT)) were performed in an attempt to simulate the atomistic and electronic structures at Ni–Al2O3 interfaces, as well as other properties such as the work of separation, the work of adhesion, and interfacial energy [20,23–28]. However, most of the published simulations on this system are actually based on experimental parameters taken from other systems such as diffusion-bonded Nb–Al2O3 interfaces or Cr-doped Ni–Al2O3 interfaces [4,29]. Furthermore, due to the size-limitations of DFT, most of the simulations to date were conducted for reduced interfacial models limited to ideal interface coherency, which was not shown to exist experimentally. Long and Chen examined the structure of incoherent Ni–Al2O3 interfaces using an energy minimization technique with an effective interatomic potential that, unlike DFT, allows simulating larger systems [26]. They were able to predict the existence of a reconstructed Ni layer at the interface that absorbs the misfit strain energy. This alternative mechanism for strain energy reduction also allows the formation of shorter Ni–O bonds.
Unfortunately, the simulation was limited only to interface regions where the Ni was positioned adjacent to oxygen sites, and does not describe the entire interface structure (although this calculation method allows simulating larger systems). Nevertheless, their conclusion that interfacial reconstruction occurred is unusual, since to the best of our knowledge, there is no experimental proof of interface reconstruction at metal–ceramic interfaces. Given the importance of the Ni–Al2O3 system and the absence of experimental data, the goal of the present work is to experimentally determine the atomistic structure and interface energy of the equilibrated Ni–(0 0 0 1) sapphire interface. Samples were based on thin Ni films which were dewetted in the solid state to reach equilibrium, from which both the interface structure and energy can be determined. The results from aberration-corrected microscopy were used to construct a simplified atomistic simulation, which in turn provided details on the reconstructed and equilibrated interface. This combination of experiment and theory provided detailed information regarding the equilibrated atomistic structure of this metal–ceramic interface, showing that reconstruction is a viable process to reduce strain energy at nominally incoherent interfaces. 2. Experimental methods 2.1. Dewetting experiments (0 0 0 1) oriented sapphire (a-Al2O3) substrates of 99.99% purity were provided by Gavish Industrial Technologies & Materials (Omer, Israel). Substrates were ultrasonically cleaned in acetone and ethanol. Ni films (120–250 nm thick) were deposited on the substrates by e-beam deposition using a 99.9995% pure Ni source. The specimens were dewetted for 5 h in the solid-state at a temperature of 1623 K (0.94 Tm) to form a large number of particles with diameters ranging from 100 nm to a few micrometers (e.g. [22]), and were then cooled to room temperature at a rate of 15–20 K min–1. The process of solid-state dewetting is important to achieve pure Ni particles on sapphire, since during liquid-state dewetting Al is expected to go into solution in Ni, supplied from capillary-driven triple junction shape changes [30]. To maintain the highest local purity possible, and to control the oxygen partial pressure during equilibration, dewetting was conducted in a set of sapphire tubes that were assembled inside a conventional alumina-tube furnace. This provided an isolated inner envelope that prevents exposure of the specimen to impurities, originating from the alumina tube. A reducing atmosphere was used to fix the P(O2) at 1.6 1020 atm, which is low enough to prevent spinel formation (99.9999% Ar + 7 vol.% H2, Ptotal = 1 atm) [22,31]. The P(O2) was measured at the exit port of the furnace using a zirconia oxygen detector. The P(O2) at the hot zone of the furnace was calculated according to the temperature difference between the detector and the hot zone, and correlated to the equilibrium partial pressure of H2O (see the Appendix in Ref. [22] for details).
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2.2. Characterization Dewetting as an experimental approach also provides a very large number of equilibrated Ni–Al2O3 particles, each of which can be examined independently. This is extremely important for transmission electron microscopy (TEM) specimen preparation, which is a destructive technique and often requires several attempts. In this study, a dualbeam focused ion beam system (FIB, FEI Strata 400-S) was used to prepare TEM specimens from several complementary projections parallel to the interface plane. To improve the quality of the specimens for subsequent TEM analysis, low-voltage ion polishing (Xe+ at 0.3 eV using a IV8 Gentle Mill from Technoorg Linda, Hungary) was used for the final stage of specimen preparation. The average particle orientation with regards to the (0 0 0 1) sapphire substrates was determined by X-ray diffraction (XRD), using a conventional Bragg–Brentano X-ray diffractometer (Philips X’Pert goniometer) with Cu Ka radiation operated at 40 mA and 40 kV. TEM analysis was conducted using a monochromated and aberration-corrected TEM (FEI Titan 80–300 kV S/TEM) equipped with a post-column energy filter (Gatan Tridiem 866 ERS). This microscope has a coincident point-to-point resolution and information limit of 0.07 nm, and a focal spread of 3 nm (at 300 kV). For quantitative interpretation of the TEM data, exit-wave reconstruction was performed on a series of 11 experimental micrographs, acquired at different objective lens defocus values (using a spherical aberration coefficient Cs = 8 lm). The series was reconstructed using TrueImage software (FEI). In addition, multislice simulations (EMS) were used to simulate the contrast of single images and to determine the atomic column positions [32]. 2.3. Computational details The atomistic simulations were performed using the Large-Scale Atomic/Molecular Massively Parallel Simulator (LAMMPS) [33], based on models constructed from the TEM data. LAMMPS was used in the molecular statics mode, based on the conjugate gradient method. The interatomic interaction between the Ni atoms was described via the embedded atom method (EAM) potential developed by Mishin et al. [34]. This interatomic potential properly describes the elastic properties of Ni. The interatomic interaction between the Ni and Al2O3 atoms was described by a pair potential developed by Long and Chen [26], a potential that accurately reproduces ab initio calculated adhesive energies of the Ni–sapphire interface with an Al-terminated layer. Al–O interactions in the sapphire were omitted by freezing the Al2O3 atoms in their crystal positions. Due to the symmetries of the Ni and sapphire crystals, a parallelepiped simulation cell was used. The orientation relationship and interface plane (five macroscopic degrees of freedom) were defined from the experimental TEM investigations. Periodic boundary conditions were imposed along the Ni½1 1 0jjsapphire½1 1 0 0 and Ni½1 1 2jjsapphire½1 1 2 0
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direction. In order to avoid internal stresses due to the periodic boundary conditions, the lengths of the computational cell parallel to the interface was chosen to accommodate whole numbers of Ni and Al2O3 periodic cells, with the smallest mismatch. A hexagonal computational cell (a = 22.85 nm, corresponding to 53 and 48 repetitions of the Ni and Al2O3 unit cells, respectively) is the smallest possible cell for periodic boundary conditions, resulting in a negligible tensile strain of 0.007%. 3. Results 3.1. Interface energy The preferred orientation of the Ni was analyzed by XRD both before and after dewetting. The as-deposited film exhibited a very pronounced {1 1 1} texture. However, for solid-state dewetted Ni particles, this preferred orientation was the only one observed by XRD (unlike for liquid-state dewetting that can result in multiple particle orientations [35]). This result was further supported by particle shape analysis using scanning electron microscopy (SEM, using the dual-beam FIB), which showed that all of the particles have a (1 1 1) plane parallel to the (0 0 0 1) sapphire substrate, similar to the particle seen in Fig. 1 (for details, see Ref. [2]). Previous knowledge regarding the substrate orientation combined with Ni surface facet identification allowed for determination of the full orientation relationship of the particles from SEM micrographs. Among several hundred particles that were examined, only two orientation relationships (OR) were observed: 80% had the Nið1 1 1Þ½1 1 0jjsapphire ð0 0 0 1Þ½1 0 1 0 OR (which is referred to as OR1), and the remaining 20% had the Nið1 1 1Þ½1 1 0jjsapphireð0 0 0 1Þ ½2 1 1 0 OR (which is referred to as OR2). This was confirmed using electron diffraction in TEM. Note that the two ORs have the same interface plane but have a ±30° relative rotation of the particle about the interface normal. In this study, only particles with OR1 were examined.
Fig. 1. Secondary electron SEM micrograph of an equilibrated Ni particle, prepared by solid-state dewetting.
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In order to quantify the thermodynamic stability of the interface, the solid–solid interface energy was determined from cross-section TEM micrographs of the particles, using the Winterbottom equation [36,14]: R1 csp csv ¼ ð1Þ R2 cpv where R1 is the distance from the center of the (complete) Wulff shape (i.e. the Wulff point) to the interface with the substrate, and R2 is the distance from the Wulff point to the surface with an energy of cpv. csv and csp are the surface energy of the substrate and the interfacial energy, respectively. An example of one cross-section including the relevant parameters for Winterbottom analysis is presented in Fig. 2. For analysis, the surface energies taken from the literature were: cNi(111) = 2.07 J m–2 [21] and csapphire(0001) = 1.07 J m–2 [37]. The R1/R2 ratio measured for six particles was 0.49 ± 0.01. It was found that the Ni(1 1 1)–Al2O3 (0 0 0 1) interface energy equilibrated at P(O2) = 1020 atm and 1623 K is 2.16 ± 0.2 J m–2.
contrast was observed while examining the same interface from projection B. In order to enhance this periodicity for viewing purposes, Fig. 4b is stretched normal to the interface. The measured wavelength of the periodic con˚. trast running parallel to the interface is 23.6 A The TEM specimens were also used to search for interface dislocations using several diffraction-contrast-based TEM techniques (two-beam conditions and weak beam) employed on specimens from both projections A and B. In addition to the cross-section specimens, a significant effort was invested in preparing plan-view specimens parallel to the interface, such that the viewing direction is normal to the interface and the thin area of the specimen consists of both the Ni and the sapphire. No dislocations were found in any of the examined specimens, and thus it
3.2. TEM analysis In order to characterize the interface structure, crosssection TEM samples were extracted from several particles, identical in shape and orientation relationship (all having OR1). For full analysis the interface was characterized in two different low-index zone axes: projection A which is Ni½1 1 0jjsapphire½1 0 1 0 and projection B which is Ni½1 2 1jjsapphire½1 1 2 0. It should be emphasized that these are two complementary viewing directions for the same interface having OR1. From each projection, a defocus series was acquired and the electron exit wave was reconstructed. The amplitudes of the reconstructed waves of both projections are presented in Fig. 3. Although no reaction layer was found at the interface, a unique periodic contrast was observed in all of the specimens viewed along projection A (see Figs. 3 and 4). Surprisingly, no evidence for the existence of this periodic
Fig. 2. High angle annular dark field STEM micrograph of an equilibrated Ni particle, viewed in the [1 1 0] zone axis (a cross-section of the particle seen in Fig. 1). The parameters R1 and R2 used for the energy measurement are indicated (see Eq. (1)).
Fig. 3. Reconstructed exit wave amplitudes of (a) projection A and (b) projection B. The schematic drawing represents the sectioning directions of the Ni particle and the sapphire substrate (dashed lines), and the corresponding viewing directions for projections A and B. The Ni particle is identical in shape and orientation to the one in Fig. 1.
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Fig. 4. (a) The exit wave amplitude of the interface viewed along projection A. (b) A magnified image of the interface region showing the reconstructed Ni layer. In order to enhance the periodic contrast for viewing purposes, the image was stretched perpendicular to the interface.
was initially concluded that the interface is incoherent at equilibrium. An explanation for the presence of the periodic contrast at the interface is the existence of a characteristic atomic arrangement at the interface different from that of the ideal interface, i.e. interface reconstruction, and this is evaluated below. Since some of the HRTEM analysis was conducted on single micrographs (in addition to exit wave reconstruction), it was important to determine the atomic column positions in both the Ni and the sapphire. Fully computerized iterative digital image matching was used to compare simulated images, calculated using EMS, with experimental micrographs of the interface. This was done to determine the specific values of objective lens defocus and relative sample thickness, in order to determine the atom column positions in the bulk sapphire and Ni (away from the interface). These data were then used to determine the relative translation of the two interface forming crystals. An example of such a result is presented in Fig. 5. The imaging conditions used are: objective lens defocus (determined from iterative image matching) Df = 8 nm, Cs = 0.024 mm, A2 (two-fold astigmatism) = 44.4 nm. The best fit was found for a thickness of 42 nm. Electron energy loss spectroscopy (EELS) thickness maps were also used to experimentally determine the thickness of the examined areas of the specimens using a fully calibrated plasmon mean-free-path [38]. The thickness determined from EELS matched within 10% of the thickness determined from image simulations. 3.3. Interface structure model On the basis of the macroscopic parameters of the interface (i.e. orientation relationship and interface plane) a geometrical model of the non-relaxed interface was constructed (lattice parameters taken from room temperature ˚ , a,c(Al2O3) = 4.76 A ˚ and JCPDS data: a(Ni) = 3.52 A ˚ 12.997 A, respectively). All three surface terminations of the sapphire (0 0 0 1) were examined; oxygen-terminated, single Al1-terminated, and two Al2-terminated. The single Al1-terminated surface gave the best fit in terms of interface spacing after relaxation and interface periodicity. The hexagonal coincidence symmetry of the two lattices is immediately seen from a top view of the initial model (Fig. 6). This ˚ and is schehexagonal symmetry has a unit cell of 27.48 A matically represented on the unreconstructed interface (i.e.
before energy minimization) in Fig. 6. The projected cell parameter of this coincidence symmetry along projection ˚ , which is very similar to the measured interface A is 23.8 A ˚ . From this result it was deduced that periodicity of 23.6 A the atomistic structure that developed at the interface has a two-dimensional unit cell similar to the coincidence lattice of the terminating layers. This interfacial unit cell can be described in terms of surface (with reference pffiffiffi reconstruction pffiffiffi to the sapphire) as a 2:5 3 2:5 3R30 interface symmetry, which is similar to the surface reconstruction of (0 0 0 1) sapphire under the same experimental conditions [39]. However, it should be emphasized that in this experiment, the Ni was deposited on untreated sapphire, i.e. an initially oxygen-terminated un-reconstructed surface, and so the possibility that the surface of the substrate acted as a template for any subsequent interface reconstruction was ruled out. The periodic interface structure had to have developed during the equilibration process. There are four possible sites of the Ni atoms relative to the sapphire: 1. “on-top” sites (marked by white circles in Fig. 6) in which the Ni atoms are positioned directly above the oxygen atoms; 2. “bridge” sites (marked by a hexagonal grid) which connects the on-top sites; 3. “hollow” sites of type I (marked as “H1” only in one hexagonal cell in Fig. 6) in which the Ni is positioned above a vacancy in the oxygen sub-lattice but is actually on-top an Al site; these sites are also arrowed in Fig. 8; 4. “hollow” sites of type II (marked as “H2” only in one hexagonal cell in Fig. 6) in which the Ni is positioned above a vacancy in the oxygen sub-lattice with no Al atom below. Once the interfacial unit cell was determined from the experiments, atomistic simulations were performed in order to obtain a more complete understanding of the reconstructed Ni layer. The Ni, Al and O atoms were initially placed in their ideal crystal positions, whereas the sapphire crystal is Al1-terminated (terminated with a single layer of Al atoms). Since no misfit dislocations were identified by TEM, the Ni crystal in the simulation is defect free, i.e., the number of Ni atoms in each (1 1 1) layer parallel to the interface is the same as in the bulk. The system
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Fig. 5. Phase contrast micrograph of the Ni–sapphire interface (in projection A). The simulated Ni and sapphire images are inset and enlarged (green and red frame, respectively) and the resulting atomic positions are overlaid for both bulk phases (Ni: green, Al: blue, O: red). For imaging conditions see text.
was allowed to relax from several initial configurations, all having the same orientation relationship (OR1) but different initial distances between the Ni and sapphire lattices (different interface spacing, normal to the interface). The energy of the system was minimized via the conjugate gradient and the relaxed atomistic configuration with the lowest energy after minimization was chosen as the system that represents the equilibrated interface. Allowing the system to evolve via molecular dynamics time steps results in a similar structure to the one presented in this work. In Fig. 7 the atomistic configuration of the terminating interface layers is plotted in plan view (normal to the interface) after energy minimization. The position of the hexagonal grid is the same as in Fig. 6. In addition to a small lateral shift of the Ni atoms around their original position, some Ni atoms have shifted normal to the interface, increasing the interface spacing locally. These Ni atoms are all located in the H1 sites and are marked in yellow in Fig. 7. This means that the terminating Ni layer buckled in a periodic manner, to form delocalized interface coherency (or interface reconstruction). The reconstruction process does not change the size and position of the interfacial unit cell that existed before equilibration. This means that the simulation successfully reflects the periodicity of the interface structure as viewed from projection A. In addition to the unit cell, it also provides better understanding of the local atomic shifts lateral and normal to the interface plane. A three-dimensional schematic drawing of the reconstructed interface is presented in Fig. 8. The arrows mark the center of the H1 sites and are completely aligned with the buckling regions seen from projection A. This reconstructed structure is slightly extended to the second Ni layer from the interface.
Fig. 9 presents an experimental TEM micrograph of the interface with and without the simulated model shown in Figs. 7 and 8, superimposed on the experimental micrograph. A very good agreement exists between the dark contrast in the micrograph and the atomic positions, as expected from the image simulations (Fig. 5). Note that in the experimental micrograph, a stacking fault was identified between the second and third Ni layers from the interface. However, this stacking fault is probably not an intrinsic part of the equilibrated interface since it was not observed in other TEM specimens prepared along projection A, and so was not considered in the model. This results in a shift in the contrast relative to the model for the Ni layers farther away from the interface. In order to understand the periodicity at the interface, the strain along the first (1 1 1) Ni layer was calculated,
Fig. 6. Plan view of the terminating interface layers (Ni: green, Al: blue, O: red). (a) Before equilibration, where the coincidence symmetry is represented by the on-top sites (circles), the bridge sites (white hexagonal network) and the hollow sites of type I and II (marked as H1 and H2).
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reduction in relative surface energy of the Ni due to the presence of impurities (the Ni purity was not reported). The contribution of low surface energies and high particle aspect ratio are opposite in Eq. (1). As a result, the interface energy measured by Pilliar and Nutting (2.14 J m–2) is, by chance, very similar to the one measured in this work. 4.2. Interface structure
Fig. 7. Plan view of the terminating interface layers after equilibration and interface reconstruction, where the Ni atoms exhibit a slight lateral change of position and in addition, the Ni atoms located in the H1 sites are also shifted normal to the interface away from the sapphire and into the bulk Ni (marked in yellow).
which was found to undergo the largest deformation (the reader is referred to Appendix A for details on how the strain was calculated). Fig. 10 presents a map of the total lateral strain at the terminating Ni layer prior to reconstruction. One can see that the strain corresponds to the hexagonal interface network, with a region of maximum strain at the H1 sites. This strain is the driving force for the periodic buckling of the layer that results in a hexagonally reconstructed Ni layer. 4. Discussion 4.1. Interface energy To the best of our knowledge, only a single attempt was made in the past to experimentally measure the equilibrated Ni–Al2O3 interface energy. Pilliar and Nutting [40] also used the Winterbottom relation for the same interface at a temperature of 1273 K. However, instead of TEM they used tilted projections in SEM to determine the particle’s shape. In their work, they used lower surface energy values compared to those used here, although at a lower temperature the surface energy is known to increase. In addition, the ratio R1/R2 that was measured was higher than the one measured in this work, which is a good indication of a
The energy of a Ni–Al2O3 interface as a function of termination was calculated by Zhang et al. for a wide range of P(O2) at T = 1300 K and T = 1750 K [20]. According to the calculation, at both temperatures the most stable interface at low P(O2) is Al2-terminated. However, only the O-terminated interfaces had a change in symmetry of the terminating Ni layer (described as “buckling” of the ˚ normal to the interface). The fact that Ni layer by 0.55 A the structure of the Al1-terminated interface is different than the one determined in this work suggests that the DFT calculation may not reflect the equilibrated structure of the interface, but rather represents a metastable interface state. This is not surprising given the limited DFT cell used and given that any cell which is smaller than the cell size taken in this work will require large strains of the Ni/sapphire films, to accommodate the periodicity of both lattices in the same cell. A similar trend was reported by Wang et al. [23], who found from DFT simulations of a Ni– Al2O3 interface (that was also stretched to coherency) that the Ni terminating layer differs in structure from that of bulk Ni layers and contains vacancies and corrugations. Shi et al. [24] performed first principles calculations to demonstrate the effect of the different translations of the Ni with respect to the sapphire for the Al1 termination. Although their work was limited in size and does not reflect the complete interface reconstruction, they showed that the spacing of the layers parallel to the interface is the smallest when the Ni is positioned on top of an oxygen atom. This result is similar to the buckling effect seen in this study, which results in an increase in interface spacing at the hollow sites. The only simulation in the literature that did not strain the Ni–Al2O3 interface to coherency was done by Long and Chen [26]. In their simulation they successfully predicted reconstruction of the interface, similar to the
Fig. 8. A three-dimensional schematic representation of the reconstructed interface. The arrows mark the center of the H1 sites. The periodic buckling of the terminating Ni layer is seen at the side faces of the schematic and is perfectly aligned with the location of the H1.
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Fig. 9. (a) Final atomistic configuration from the model superimposed on the experimental micrograph of the interface along projection A. The Ni an Al atomic columns coincide very well with the dark contrast visible in micrograph (Ni: green, Al: blue, O: red). (b) The same experimental micrograph as in (a) but without the superimposed model.
experimental result we present here. Unfortunately, the unit cell used by Long and Chen was too small to reflect the complete reconstruction symmetry. In addition, they employed a set of parameters with the Rahman– Stillinger–Lemberg pairwise potential to describe bulk Ni, whereas there are more reliable interatomic potentials that capture the anisotropic elastic bulk properties of Ni (as the one employed here). The interatomic potential for the Ni–sapphire interface suggested by Long and Chen [26] (also employed in this work) indeed lacks the electronic details and preciseness of DFT calculations. Despite the simplified interatomic potential, the reconstruction of the Ni found in our simulations is in very good agreement with the structure observed experimentally and the interfacial unit cell correlates perfectly to the coincidence symmetry. We believe that this similarity implies a very small reconstruction of the sapphire crystal at the interface and that the metal elastic properties play the major role in the interface structure. These conditions may not always be fulfilled for other metal–ceramic interfaces, or even for the same system with an O-terminated interface, and such a simple approach should be used with caution for other systems. This exemplifies the need for larger DFT simulations, which include the influence of temperature.
optimize nearest neighbor bonding. The coherent regions are separated by arrays of misfit dislocations (or steps) that reduce the total elastic strain energy [41]. Incoherent interfaces by definition have no atomic matching across the interface, and the atoms at the interface are expected to essentially retain their bulk positions up to the interface layer. No misfit strain exists, since the atoms are not constrained by the other phase at the interface [41]. Unlike the atomistic structure of coherent and semi-coherent interfaces, the structure of incoherent interfaces is not fully understood. Although it is clear that no dislocations are present at incoherent interfaces, the postulation that the atoms retain their bulk positions up to the interface layer seems over-simplistic and probably does not represent equilibrated systems. Local re-arrangement of the atoms at the interface may result in a periodically reconstructed interface (similar to surface reconstruction), local relaxation of atoms to form preferred interfacial bonding, or even the formation of a diffuse interface [42]. These local changes in interface structure are all types of interface
4.3. Interface coherency The structure of interfaces is normally categorized into three types: coherent, semi-coherent, or incoherent. A coherent interface is defined as a perfect atom-by-atom match that is continuous across the interface, with no defects such as misfit dislocations, and intrinsic mismatch strain remains as an elastic strain component [41]. Semicoherent interfaces have large areas of coherency resulting from tension or compression of the interfacial layers to
Fig. 10. The strain field distributions at the first Ni atomic layer. The atoms are shaded according to the value of total strain at their position (larger compressive strains are shaded with darker atoms). Maximum compressive strain is located in the H1 sites.
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complexions [43–46], that will develop during equilibration in order to reduce interfacial energy. With the lack of a suitable definition, these interfaces are also usually considered to be incoherent. A more correct definition in this case is delocalized interface coherency, which reflects local shift of atoms at the interface that are caused by the net stresses from both sides of the interface. This shift can be periodic and result in a reconstructed interfacial layer, or non-periodic, which makes structure analysis more difficult [42,47]. Nevertheless, information on specific complexions (i.e. interface structure and chemistry at equilibrium) which are neither coherent nor semi-coherent is important, but so far comprehensive modeling and thorough experimental studies are lacking. Instead, most studies of metal–ceramic interfaces focus on the as-prepared state or other meta-stable states that may be easier to experimentally characterize, but are less fundamentally important as the relaxed and equilibrated interface for the correlation of interface thermodynamics to interface structure and chemistry [48]. The stress relief mechanism that is adopted in equilibrated Ni–Al2O3 interfaces is discussed below. 4.4. The role of reconstruction (delocalized coherency) The experimental results of this work clearly show that the Ni interfacial layer is constrained by bonding with the Ni bulk on one side and the Al and O layers on the other side. Overall, the layer conserves the fcc–(1 1 1) structure, but local shifts around the original atom positions allow for better bonding and reduced interfacial energy. The relatively large lattice misfit between Ni and sapphire is expected to be concentrated in regions of weaker atomic interactions, i.e. the H1 sites, where the Ni is positioned directly above an Al atom with no oxygen as a nearest neighbor. As a result, the Ni layers buckle in these regions and the terminating layer is relaxed locally into the Ni bulk. This unique strain (and energy) reduction mechanism was not reported for other interfaces, although it is possible that it was experimentally observed for the Cu–Al2O3 system and was misinterpreted as an incoherent interface, due to the relatively large misfit between Cu and Al2O3 and the limited resolution of the microscope used at that time [6,7]. Smaller misfit can be accommodated by misfit dislocations and/or steps, which will increase in density as the misfit increases. For semi-coherent interfaces, with increasing lattice misfit between the two interface-forming crystals, the dislocation/step density reaches a maximum and cannot absorb the strain energy. For misfit strain higher than this value, an alternative mechanism is required for strain reduction, for example the formation of delocalized interface coherency (i.e. interface reconstruction). This is analogous to the structure of grain boundaries, where for low-angle grain boundaries the misfit is accommodated by a set of lattice dislocations that increase in density as the misorientation angle increases. At a critical value of misorientation between the grains, the dislocation density reaches a maximum value, and the high-angle grain boundary develops a different
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atomistic arrangement (see Fig. 11). This arrangement is characteristic for each specific boundary, chemical environment, temperature, and crystal structure, but it is obvious that some degree of relaxation and even reconstruction occurs, and that the local atomistic structure at the boundary is not identical to that of the bulk. As such, reconstruction at nominally incoherent interfaces may occur, as was experimentally demonstrated in this work for the Ni–Al2O3 interface at equilibrium. A systematic experimental study of the influence of mismatch on interface reconstruction is a challenging task. In the interim, atomistic simulations of the influence of mismatch on interfacial reconstruction as a function of temperature and equilibrium segregation may provide unique insight into possible reconstructed interface structures, and their influence on adhesion and electronic properties at interfaces. 5. Summary and conclusions Quantitative TEM was used to determine the atomistic structure of the Nið1 1 1Þ½1 1 0jjsapphireð0 0 0 1Þ½1 0 1 0 interface, from samples equilibrated via solid-state dewetting. The experimental evidence shows that this interface develops a unique equilibrium structure in which the Ni layer adjacent to the interface buckles in a periodic manner that reflects the coincidence symmetry of the two lattices. The interface was shown to reconstruct into a pffiffiffi pffiffiexperimentally ffi 2:5 3 2:5 3R30 two dimensional interface unit cell, based on exit wave reconstruction supported by atomistic simulations. This solid–solid interface reconstruction (or delocalized coherency) is a mechanism to absorb misfit strain energy and reduce the interface energy, and may occur at other interfaces normally considered to be incoherent. The measured interface energy (using Winterbottom analysis) is 2.16 ± 0.2 J m–2 at P(O2) = 1020 atm and T = 1623 K. Acknowledgements The authors gratefully acknowledge P. Wynblatt and T. Besmann for enlightening discussions. A. Heuer is acknowledged for critical comments and for suggesting the term “delocalized interface coherency”. The authors thank D. Chatain for in-depth discussions and critical comments. The United States–Israel Binational Science Foundation (BSF Grant 2004068) and the Russell Berrie Nanotechnology Institute at the Technion are acknowledged for partial support of this study. H.M. acknowledges support from the Women in Science program of the Israel Ministry of Science and an Ilan Ramon Scholarship. Appendix A. Strain calculation at the interface In order to analyze the deformation of the (1 1 1) atomic layer of Ni at the interface, the atomic-level strain along the layer was calculated within the framework of the atomistic simulations. Since only the in-plane strain is of interest, the
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Fig. 11. Schematic drawing demonstrating the analogy between grain boundaries and interface structures with regards to coherency. Top: At grain boundaries increasing the misorientation angle results in an increase in dislocation density. For large misorientations the boundary assumes a characteristic structure, i.e. reconstruction. Bottom: The increase in lattice mismatch at interfaces will increase the misfit dislocation density. For large misfit values, the interface may adopt a characteristic (reconstructed) structure. For reconstructed grain boundaries and interfaces, the characteristic structure is not limited to one lattice plane.
Fig. 12. The strain calculation. The (a) deformed and (b) non-deformed {1 1 1} layer of Ni atoms. The triangle for which the strain is calculated and its corresponding virtual membrane are plotted. The atomic-level strain is calculated at the point marked in (a) in the middle of the triangle.
where h0 is a 2 2 matrix composed of the elements h0ij . As opposed to the classic definition of reduced coordination, the relative coordinates are not limited to values between 0 and 1. After relaxation the atoms acquire new positions ~ r0 , which correspond to a local strain field and a translation of the center of each three atoms. Accordingly, the virtual membrane is deformed by the same strain and translated by the same distance (Fig. 12b). If the translation component ~ D is deduced from the atomic positions of the three atoms, the strain contribution, i.e. the relative position of each atom on the deformed virtual membrane, is the same as the non-deformed relative positions: s ¼ ðr0 DÞh1
analysis is focused on the part of the strain tensor in the x–y plane of the interface. The method presented here yields similar results to other techniques to calculate the whole strain tensor in the bulk (e.g. [49]), in which all nearest neighbors are in fcc positions. First the strain field at the center point of each three neighboring atoms that forms a triangle is calculated (for instance, the strain at the point within the triangle plotted in Fig. 12a). Each three atoms are assumed to be “glued” to a rectangular virtual membrane, defined by two independent two-dimensional vectors h0ij , where i is the vector number and j is the indices of the ith vector. The position of each atom on the membrane ~ r can be written in relative coordinates: s¼
rh1 0
ð2Þ
ð3Þ
where h is a 2 2 matrix composed of the vectors defining the deformed virtual membrane. The solution of Eqs. (2) and (3) simultaneously for the three atoms yields the vectors defining the deformed virtual membrane (~ r0 is also considered as an unknown parameter). Since the initial virtual membrane can be chosen arbitrarily, two perpendicular unit vectors are selected, i.e. h0 is a unity matrix 1. The two-dimensional strain tensor at the center of the triangle of each three neighboring atoms is then 1 e ¼ ðhT h 1Þ 2
ð4Þ
Finally, the local strain at each atomic position is the average strain of all the six triangles surrounding it.
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Ru¨hle M. J Eur Ceram Soc 1996;16:353. Ru¨hle M, Evans AG. Mater Sci Eng, A 1989;107:187. Finnis MW. J Phys: Condens Matter 1996;8:5811. Gutekunst G, Mayer J, Ru¨hle M. Scr Metall Mater 1994;31:1097. Ernst F. Mater Sci Eng 1995;14:97. Dehm G, Scheu C, Ru¨hle M, Raj R. Acta Mater 1998;46:759. Sasaki T, Matsunaga K, Ikuhara Y, Ohta H, Hosono H, Yamamoto T. Sci Technol Adv Mater 2003;4:575. Sasaki T, Matsunaga K, Ohta H, Hosono H, Yamamoto T, Ikuhara Y. Mater Trans 2004;45:2137. Oh SH, Scheu C, Wagner T, Tchernychova E, Ru¨hle M. Acta Mater 2006;54:2685. Scheu C, Klein S, Tomsia AP, Ru¨hle M. J Microsc – Oxford 2002;208:11. Scheu C. Interf Sci 2004;12:127. Mader W, Ru¨hle M. Acta Metall 1989;37(3):853–66. Mayer J, Flynn CP, Ru¨hle M. Ultramicroscopy 1990;33(1): 51–61. Shaffir E, Riess I, Kaplan WD. Acta Mater 2009;57:248. Sadan H, Kaplan WD. J Mater Sci 2006;41:5099. Sadan H, Kaplan WD. J Mater Sci 2006;41:5371. Chatain D, Ghetta V, Wynblatt P. Interf Sci 2004;12:7. Mercer C, Faulhaber S, Yao N, McIlwrath K, Fabrichnaya O. Surf Coat Technol 2006;201:1495. Murut AE, White CL. Acta Mater 1997;45:2547. Zhang W, Smith JR, Evans AG. Acta Mater 2002;50:3803. Brillo J, Egry I. J Mater Sci 2005;40:2213. Meltzman H, Chatain D, Avizemer D, Besmann TM, Kaplan WD. Acta Mater 2011;59:3473. Wang X-G, Smith JR, Evans AG. Phys Rev B 2006;74:081403. Shi S, Tanaka S, Kohyama M. J Am Ceram Soc 2007;90:2429.
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Acta Materialia 60 (2012) 4359–4369 www.elsevier.com/locate/actamat
Solid–solid interface reconstruction at equilibrated Ni–Al2O3 interfaces Hila Meltzman a, Dan Mordehai b, Wayne D. Kaplan a,⇑ a b
Department of Materials Engineering, Technion-Israel Institute of Technology, Haifa 32000, Israel Department of Mechanical Engineering, Technion-Israel Institute of Technology, Haifa 32000, Israel Received 16 April 2012; received in revised form 27 April 2012; accepted 28 April 2012 Available online 31 May 2012
Abstract The atomistic structure of solid Ni–Al2O3 interfaces at equilibrium was determined by aberration-corrected transmission electron microscopy, from specimens formed during solid-state dewetting of thin Ni films on the (0 0 0 1) surface of a-Alp 2O ffiffiffi 3. It was pffiffiffi found that the interface develops a unique mechanism for misfit strain reduction, termed delocalized coherency, via a 2:5 3 2:5 3R30 reconstructed interface structure, contradicting most simulations in the literature, which assume a semi-coherent structure for this system. Based on the experimental work presented here, a structural model was also simulated, showing periodic buckling of the terminating Ni layer (i.e. interface reconstruction). The interface energy was experimentally determined from dewetted Ni particles using Winterbottom analysis, and found to be 2.16 ± 0.2 J m–2 at P(O2) = 1020 atm and T = 1623 K. Ó 2012 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. Keywords: Interface energy; Interface structure; Transmission electron microscopy; Incoherent interfaces; Reconstruction
1. Introduction The structure and properties of metal–ceramic interfaces have been the focus of many studies since the 1960s, mostly due to the potential for numerous technological applications such as metal–ceramic bonding, brazing, microelectronic packaging, thin film technology, high temperature metal-oxidation processes, photovoltaic cells, protective coatings for metals, bioactive implant coatings, and high temperature aircraft components [1–4]. The complexity of solid–solid interfaces arises not only from the difference in long-range atomistic order on both sides of the interface, but also from the wide variety of defects that can form, and the unique structural and compositional characteristics of each interface, depending on the relative orientation of the two crystals, their composition, and the temperature used for the process [5]. This degree of complexity often inhibits quantitative treatment of interfacial phenomena.
⇑ Corresponding author. Tel.: +972 4 8294591; fax: +972 4 8294580.
E-mail address: [email protected] (W.D. Kaplan).
Due to the different atomistic arrangement on both sides of the interface, misfit strain inevitably develops. If kinetically possible, the system will move to reduce this strain energy by introducing either misfit dislocations or steps (to create a semi-coherent interface), or for the more common case of a large difference in structure between the interfaceforming crystals, via the formation of what is commonly termed an incoherent interface. Due to the complexity of incoherent interfaces, most studies to date have focused on analysis of misfit dislocations and steps at semi-coherent interfaces, and only a few reports were found regarding incoherent metal–ceramic interfaces [6–8]. One of the most studied types of metal–ceramic interfaces is that between face-centered cubic (fcc) metals and a-Al2O3 (sapphire). These interfaces are usually produced either by diffusion bonding of a pre-determined orientation [4], or by thin film deposition such as molecular beam epitaxy (MBE) [9–11]. It should be noted that these preparation techniques do not necessarily produce equilibrated interfaces. The as-bonded or as-deposited interface states often reflect a metastable interface structure and orientation, rather than a thermodynamically equilibrated state. The question of whether the examined interface is equilibrated or not, and
1359-6454/$36.00 Ó 2012 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.actamat.2012.04.037
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whether it reflects the minimum energy configuration of the interface, is rarely addressed, although this is a significant driving force for microstructural evolution of interfaces. The Nb–Al2O3 interface provides a good example of the influence of interface fabrication methods on the resulting interface orientation; Mader and Ru¨hle [12] reported an orientation relationship of (1 1 0)[0 0 1]Nb||(0 0 0 1)½2 1 1 0Al2 O3 for a solid-state diffusion bonded interface, while a (1 1 1)[1 1 0]Nb|| (0 0 0 1)½2 1 1 0Al2 O3 orientation relationship was reported for the same system produced by MBE [13]. The relative thermodynamic stability of the different interfaces was not determined. One way to equilibrate metal–ceramic interfaces is to “dewet” thin metallic films on ceramic substrates, which is essentially disruption (in the liquid or solid state) of the film driven by the minimization of surface and interfacial energy [14–17]. This process of film agglomeration results in a very large number of isolated particles, which can be investigated separately to achieve very good statistics. The size of the droplets/particles (up to several micrometers in diameter) enables shorter equilibration times, and solid-state dewetting can be used in addition to liquid-state dewetting to reach equilibration for solid– solid interface studies [15]. In addition to being a good metal–Al2O3 model system, Ni–Al2O3 interfaces can be found in numerous technological applications, such as composites, microelectronic packaging, catalytic processes and applications demanding high temperature strength and oxidation resistance such as turbine blades [18–20]. In contrast to the large technological importance of this system, Ni–Al2O3 interfaces were not experimentally studied to the same extent as other metal– alumina interfaces. This is mainly because of the relative high melting temperature of the metal and/or because of the relatively large surface energy of Ni, which requires an extremely pure environment for model studies [21,22]. Due to the lack of experimental data available, numerous atomistic simulations (mostly by density functional theory (DFT)) were performed in an attempt to simulate the atomistic and electronic structures at Ni–Al2O3 interfaces, as well as other properties such as the work of separation, the work of adhesion, and interfacial energy [20,23–28]. However, most of the published simulations on this system are actually based on experimental parameters taken from other systems such as diffusion-bonded Nb–Al2O3 interfaces or Cr-doped Ni–Al2O3 interfaces [4,29]. Furthermore, due to the size-limitations of DFT, most of the simulations to date were conducted for reduced interfacial models limited to ideal interface coherency, which was not shown to exist experimentally. Long and Chen examined the structure of incoherent Ni–Al2O3 interfaces using an energy minimization technique with an effective interatomic potential that, unlike DFT, allows simulating larger systems [26]. They were able to predict the existence of a reconstructed Ni layer at the interface that absorbs the misfit strain energy. This alternative mechanism for strain energy reduction also allows the formation of shorter Ni–O bonds.
Unfortunately, the simulation was limited only to interface regions where the Ni was positioned adjacent to oxygen sites, and does not describe the entire interface structure (although this calculation method allows simulating larger systems). Nevertheless, their conclusion that interfacial reconstruction occurred is unusual, since to the best of our knowledge, there is no experimental proof of interface reconstruction at metal–ceramic interfaces. Given the importance of the Ni–Al2O3 system and the absence of experimental data, the goal of the present work is to experimentally determine the atomistic structure and interface energy of the equilibrated Ni–(0 0 0 1) sapphire interface. Samples were based on thin Ni films which were dewetted in the solid state to reach equilibrium, from which both the interface structure and energy can be determined. The results from aberration-corrected microscopy were used to construct a simplified atomistic simulation, which in turn provided details on the reconstructed and equilibrated interface. This combination of experiment and theory provided detailed information regarding the equilibrated atomistic structure of this metal–ceramic interface, showing that reconstruction is a viable process to reduce strain energy at nominally incoherent interfaces. 2. Experimental methods 2.1. Dewetting experiments (0 0 0 1) oriented sapphire (a-Al2O3) substrates of 99.99% purity were provided by Gavish Industrial Technologies & Materials (Omer, Israel). Substrates were ultrasonically cleaned in acetone and ethanol. Ni films (120–250 nm thick) were deposited on the substrates by e-beam deposition using a 99.9995% pure Ni source. The specimens were dewetted for 5 h in the solid-state at a temperature of 1623 K (0.94 Tm) to form a large number of particles with diameters ranging from 100 nm to a few micrometers (e.g. [22]), and were then cooled to room temperature at a rate of 15–20 K min–1. The process of solid-state dewetting is important to achieve pure Ni particles on sapphire, since during liquid-state dewetting Al is expected to go into solution in Ni, supplied from capillary-driven triple junction shape changes [30]. To maintain the highest local purity possible, and to control the oxygen partial pressure during equilibration, dewetting was conducted in a set of sapphire tubes that were assembled inside a conventional alumina-tube furnace. This provided an isolated inner envelope that prevents exposure of the specimen to impurities, originating from the alumina tube. A reducing atmosphere was used to fix the P(O2) at 1.6 1020 atm, which is low enough to prevent spinel formation (99.9999% Ar + 7 vol.% H2, Ptotal = 1 atm) [22,31]. The P(O2) was measured at the exit port of the furnace using a zirconia oxygen detector. The P(O2) at the hot zone of the furnace was calculated according to the temperature difference between the detector and the hot zone, and correlated to the equilibrium partial pressure of H2O (see the Appendix in Ref. [22] for details).
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2.2. Characterization Dewetting as an experimental approach also provides a very large number of equilibrated Ni–Al2O3 particles, each of which can be examined independently. This is extremely important for transmission electron microscopy (TEM) specimen preparation, which is a destructive technique and often requires several attempts. In this study, a dualbeam focused ion beam system (FIB, FEI Strata 400-S) was used to prepare TEM specimens from several complementary projections parallel to the interface plane. To improve the quality of the specimens for subsequent TEM analysis, low-voltage ion polishing (Xe+ at 0.3 eV using a IV8 Gentle Mill from Technoorg Linda, Hungary) was used for the final stage of specimen preparation. The average particle orientation with regards to the (0 0 0 1) sapphire substrates was determined by X-ray diffraction (XRD), using a conventional Bragg–Brentano X-ray diffractometer (Philips X’Pert goniometer) with Cu Ka radiation operated at 40 mA and 40 kV. TEM analysis was conducted using a monochromated and aberration-corrected TEM (FEI Titan 80–300 kV S/TEM) equipped with a post-column energy filter (Gatan Tridiem 866 ERS). This microscope has a coincident point-to-point resolution and information limit of 0.07 nm, and a focal spread of 3 nm (at 300 kV). For quantitative interpretation of the TEM data, exit-wave reconstruction was performed on a series of 11 experimental micrographs, acquired at different objective lens defocus values (using a spherical aberration coefficient Cs = 8 lm). The series was reconstructed using TrueImage software (FEI). In addition, multislice simulations (EMS) were used to simulate the contrast of single images and to determine the atomic column positions [32]. 2.3. Computational details The atomistic simulations were performed using the Large-Scale Atomic/Molecular Massively Parallel Simulator (LAMMPS) [33], based on models constructed from the TEM data. LAMMPS was used in the molecular statics mode, based on the conjugate gradient method. The interatomic interaction between the Ni atoms was described via the embedded atom method (EAM) potential developed by Mishin et al. [34]. This interatomic potential properly describes the elastic properties of Ni. The interatomic interaction between the Ni and Al2O3 atoms was described by a pair potential developed by Long and Chen [26], a potential that accurately reproduces ab initio calculated adhesive energies of the Ni–sapphire interface with an Al-terminated layer. Al–O interactions in the sapphire were omitted by freezing the Al2O3 atoms in their crystal positions. Due to the symmetries of the Ni and sapphire crystals, a parallelepiped simulation cell was used. The orientation relationship and interface plane (five macroscopic degrees of freedom) were defined from the experimental TEM investigations. Periodic boundary conditions were imposed along the Ni½1 1 0jjsapphire½1 1 0 0 and Ni½1 1 2jjsapphire½1 1 2 0
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direction. In order to avoid internal stresses due to the periodic boundary conditions, the lengths of the computational cell parallel to the interface was chosen to accommodate whole numbers of Ni and Al2O3 periodic cells, with the smallest mismatch. A hexagonal computational cell (a = 22.85 nm, corresponding to 53 and 48 repetitions of the Ni and Al2O3 unit cells, respectively) is the smallest possible cell for periodic boundary conditions, resulting in a negligible tensile strain of 0.007%. 3. Results 3.1. Interface energy The preferred orientation of the Ni was analyzed by XRD both before and after dewetting. The as-deposited film exhibited a very pronounced {1 1 1} texture. However, for solid-state dewetted Ni particles, this preferred orientation was the only one observed by XRD (unlike for liquid-state dewetting that can result in multiple particle orientations [35]). This result was further supported by particle shape analysis using scanning electron microscopy (SEM, using the dual-beam FIB), which showed that all of the particles have a (1 1 1) plane parallel to the (0 0 0 1) sapphire substrate, similar to the particle seen in Fig. 1 (for details, see Ref. [2]). Previous knowledge regarding the substrate orientation combined with Ni surface facet identification allowed for determination of the full orientation relationship of the particles from SEM micrographs. Among several hundred particles that were examined, only two orientation relationships (OR) were observed: 80% had the Nið1 1 1Þ½1 1 0jjsapphire ð0 0 0 1Þ½1 0 1 0 OR (which is referred to as OR1), and the remaining 20% had the Nið1 1 1Þ½1 1 0jjsapphireð0 0 0 1Þ ½2 1 1 0 OR (which is referred to as OR2). This was confirmed using electron diffraction in TEM. Note that the two ORs have the same interface plane but have a ±30° relative rotation of the particle about the interface normal. In this study, only particles with OR1 were examined.
Fig. 1. Secondary electron SEM micrograph of an equilibrated Ni particle, prepared by solid-state dewetting.
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In order to quantify the thermodynamic stability of the interface, the solid–solid interface energy was determined from cross-section TEM micrographs of the particles, using the Winterbottom equation [36,14]: R1 csp csv ¼ ð1Þ R2 cpv where R1 is the distance from the center of the (complete) Wulff shape (i.e. the Wulff point) to the interface with the substrate, and R2 is the distance from the Wulff point to the surface with an energy of cpv. csv and csp are the surface energy of the substrate and the interfacial energy, respectively. An example of one cross-section including the relevant parameters for Winterbottom analysis is presented in Fig. 2. For analysis, the surface energies taken from the literature were: cNi(111) = 2.07 J m–2 [21] and csapphire(0001) = 1.07 J m–2 [37]. The R1/R2 ratio measured for six particles was 0.49 ± 0.01. It was found that the Ni(1 1 1)–Al2O3 (0 0 0 1) interface energy equilibrated at P(O2) = 1020 atm and 1623 K is 2.16 ± 0.2 J m–2.
contrast was observed while examining the same interface from projection B. In order to enhance this periodicity for viewing purposes, Fig. 4b is stretched normal to the interface. The measured wavelength of the periodic con˚. trast running parallel to the interface is 23.6 A The TEM specimens were also used to search for interface dislocations using several diffraction-contrast-based TEM techniques (two-beam conditions and weak beam) employed on specimens from both projections A and B. In addition to the cross-section specimens, a significant effort was invested in preparing plan-view specimens parallel to the interface, such that the viewing direction is normal to the interface and the thin area of the specimen consists of both the Ni and the sapphire. No dislocations were found in any of the examined specimens, and thus it
3.2. TEM analysis In order to characterize the interface structure, crosssection TEM samples were extracted from several particles, identical in shape and orientation relationship (all having OR1). For full analysis the interface was characterized in two different low-index zone axes: projection A which is Ni½1 1 0jjsapphire½1 0 1 0 and projection B which is Ni½1 2 1jjsapphire½1 1 2 0. It should be emphasized that these are two complementary viewing directions for the same interface having OR1. From each projection, a defocus series was acquired and the electron exit wave was reconstructed. The amplitudes of the reconstructed waves of both projections are presented in Fig. 3. Although no reaction layer was found at the interface, a unique periodic contrast was observed in all of the specimens viewed along projection A (see Figs. 3 and 4). Surprisingly, no evidence for the existence of this periodic
Fig. 2. High angle annular dark field STEM micrograph of an equilibrated Ni particle, viewed in the [1 1 0] zone axis (a cross-section of the particle seen in Fig. 1). The parameters R1 and R2 used for the energy measurement are indicated (see Eq. (1)).
Fig. 3. Reconstructed exit wave amplitudes of (a) projection A and (b) projection B. The schematic drawing represents the sectioning directions of the Ni particle and the sapphire substrate (dashed lines), and the corresponding viewing directions for projections A and B. The Ni particle is identical in shape and orientation to the one in Fig. 1.
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Fig. 4. (a) The exit wave amplitude of the interface viewed along projection A. (b) A magnified image of the interface region showing the reconstructed Ni layer. In order to enhance the periodic contrast for viewing purposes, the image was stretched perpendicular to the interface.
was initially concluded that the interface is incoherent at equilibrium. An explanation for the presence of the periodic contrast at the interface is the existence of a characteristic atomic arrangement at the interface different from that of the ideal interface, i.e. interface reconstruction, and this is evaluated below. Since some of the HRTEM analysis was conducted on single micrographs (in addition to exit wave reconstruction), it was important to determine the atomic column positions in both the Ni and the sapphire. Fully computerized iterative digital image matching was used to compare simulated images, calculated using EMS, with experimental micrographs of the interface. This was done to determine the specific values of objective lens defocus and relative sample thickness, in order to determine the atom column positions in the bulk sapphire and Ni (away from the interface). These data were then used to determine the relative translation of the two interface forming crystals. An example of such a result is presented in Fig. 5. The imaging conditions used are: objective lens defocus (determined from iterative image matching) Df = 8 nm, Cs = 0.024 mm, A2 (two-fold astigmatism) = 44.4 nm. The best fit was found for a thickness of 42 nm. Electron energy loss spectroscopy (EELS) thickness maps were also used to experimentally determine the thickness of the examined areas of the specimens using a fully calibrated plasmon mean-free-path [38]. The thickness determined from EELS matched within 10% of the thickness determined from image simulations. 3.3. Interface structure model On the basis of the macroscopic parameters of the interface (i.e. orientation relationship and interface plane) a geometrical model of the non-relaxed interface was constructed (lattice parameters taken from room temperature ˚ , a,c(Al2O3) = 4.76 A ˚ and JCPDS data: a(Ni) = 3.52 A ˚ 12.997 A, respectively). All three surface terminations of the sapphire (0 0 0 1) were examined; oxygen-terminated, single Al1-terminated, and two Al2-terminated. The single Al1-terminated surface gave the best fit in terms of interface spacing after relaxation and interface periodicity. The hexagonal coincidence symmetry of the two lattices is immediately seen from a top view of the initial model (Fig. 6). This ˚ and is schehexagonal symmetry has a unit cell of 27.48 A matically represented on the unreconstructed interface (i.e.
before energy minimization) in Fig. 6. The projected cell parameter of this coincidence symmetry along projection ˚ , which is very similar to the measured interface A is 23.8 A ˚ . From this result it was deduced that periodicity of 23.6 A the atomistic structure that developed at the interface has a two-dimensional unit cell similar to the coincidence lattice of the terminating layers. This interfacial unit cell can be described in terms of surface (with reference pffiffiffi reconstruction pffiffiffi to the sapphire) as a 2:5 3 2:5 3R30 interface symmetry, which is similar to the surface reconstruction of (0 0 0 1) sapphire under the same experimental conditions [39]. However, it should be emphasized that in this experiment, the Ni was deposited on untreated sapphire, i.e. an initially oxygen-terminated un-reconstructed surface, and so the possibility that the surface of the substrate acted as a template for any subsequent interface reconstruction was ruled out. The periodic interface structure had to have developed during the equilibration process. There are four possible sites of the Ni atoms relative to the sapphire: 1. “on-top” sites (marked by white circles in Fig. 6) in which the Ni atoms are positioned directly above the oxygen atoms; 2. “bridge” sites (marked by a hexagonal grid) which connects the on-top sites; 3. “hollow” sites of type I (marked as “H1” only in one hexagonal cell in Fig. 6) in which the Ni is positioned above a vacancy in the oxygen sub-lattice but is actually on-top an Al site; these sites are also arrowed in Fig. 8; 4. “hollow” sites of type II (marked as “H2” only in one hexagonal cell in Fig. 6) in which the Ni is positioned above a vacancy in the oxygen sub-lattice with no Al atom below. Once the interfacial unit cell was determined from the experiments, atomistic simulations were performed in order to obtain a more complete understanding of the reconstructed Ni layer. The Ni, Al and O atoms were initially placed in their ideal crystal positions, whereas the sapphire crystal is Al1-terminated (terminated with a single layer of Al atoms). Since no misfit dislocations were identified by TEM, the Ni crystal in the simulation is defect free, i.e., the number of Ni atoms in each (1 1 1) layer parallel to the interface is the same as in the bulk. The system
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Fig. 5. Phase contrast micrograph of the Ni–sapphire interface (in projection A). The simulated Ni and sapphire images are inset and enlarged (green and red frame, respectively) and the resulting atomic positions are overlaid for both bulk phases (Ni: green, Al: blue, O: red). For imaging conditions see text.
was allowed to relax from several initial configurations, all having the same orientation relationship (OR1) but different initial distances between the Ni and sapphire lattices (different interface spacing, normal to the interface). The energy of the system was minimized via the conjugate gradient and the relaxed atomistic configuration with the lowest energy after minimization was chosen as the system that represents the equilibrated interface. Allowing the system to evolve via molecular dynamics time steps results in a similar structure to the one presented in this work. In Fig. 7 the atomistic configuration of the terminating interface layers is plotted in plan view (normal to the interface) after energy minimization. The position of the hexagonal grid is the same as in Fig. 6. In addition to a small lateral shift of the Ni atoms around their original position, some Ni atoms have shifted normal to the interface, increasing the interface spacing locally. These Ni atoms are all located in the H1 sites and are marked in yellow in Fig. 7. This means that the terminating Ni layer buckled in a periodic manner, to form delocalized interface coherency (or interface reconstruction). The reconstruction process does not change the size and position of the interfacial unit cell that existed before equilibration. This means that the simulation successfully reflects the periodicity of the interface structure as viewed from projection A. In addition to the unit cell, it also provides better understanding of the local atomic shifts lateral and normal to the interface plane. A three-dimensional schematic drawing of the reconstructed interface is presented in Fig. 8. The arrows mark the center of the H1 sites and are completely aligned with the buckling regions seen from projection A. This reconstructed structure is slightly extended to the second Ni layer from the interface.
Fig. 9 presents an experimental TEM micrograph of the interface with and without the simulated model shown in Figs. 7 and 8, superimposed on the experimental micrograph. A very good agreement exists between the dark contrast in the micrograph and the atomic positions, as expected from the image simulations (Fig. 5). Note that in the experimental micrograph, a stacking fault was identified between the second and third Ni layers from the interface. However, this stacking fault is probably not an intrinsic part of the equilibrated interface since it was not observed in other TEM specimens prepared along projection A, and so was not considered in the model. This results in a shift in the contrast relative to the model for the Ni layers farther away from the interface. In order to understand the periodicity at the interface, the strain along the first (1 1 1) Ni layer was calculated,
Fig. 6. Plan view of the terminating interface layers (Ni: green, Al: blue, O: red). (a) Before equilibration, where the coincidence symmetry is represented by the on-top sites (circles), the bridge sites (white hexagonal network) and the hollow sites of type I and II (marked as H1 and H2).
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reduction in relative surface energy of the Ni due to the presence of impurities (the Ni purity was not reported). The contribution of low surface energies and high particle aspect ratio are opposite in Eq. (1). As a result, the interface energy measured by Pilliar and Nutting (2.14 J m–2) is, by chance, very similar to the one measured in this work. 4.2. Interface structure
Fig. 7. Plan view of the terminating interface layers after equilibration and interface reconstruction, where the Ni atoms exhibit a slight lateral change of position and in addition, the Ni atoms located in the H1 sites are also shifted normal to the interface away from the sapphire and into the bulk Ni (marked in yellow).
which was found to undergo the largest deformation (the reader is referred to Appendix A for details on how the strain was calculated). Fig. 10 presents a map of the total lateral strain at the terminating Ni layer prior to reconstruction. One can see that the strain corresponds to the hexagonal interface network, with a region of maximum strain at the H1 sites. This strain is the driving force for the periodic buckling of the layer that results in a hexagonally reconstructed Ni layer. 4. Discussion 4.1. Interface energy To the best of our knowledge, only a single attempt was made in the past to experimentally measure the equilibrated Ni–Al2O3 interface energy. Pilliar and Nutting [40] also used the Winterbottom relation for the same interface at a temperature of 1273 K. However, instead of TEM they used tilted projections in SEM to determine the particle’s shape. In their work, they used lower surface energy values compared to those used here, although at a lower temperature the surface energy is known to increase. In addition, the ratio R1/R2 that was measured was higher than the one measured in this work, which is a good indication of a
The energy of a Ni–Al2O3 interface as a function of termination was calculated by Zhang et al. for a wide range of P(O2) at T = 1300 K and T = 1750 K [20]. According to the calculation, at both temperatures the most stable interface at low P(O2) is Al2-terminated. However, only the O-terminated interfaces had a change in symmetry of the terminating Ni layer (described as “buckling” of the ˚ normal to the interface). The fact that Ni layer by 0.55 A the structure of the Al1-terminated interface is different than the one determined in this work suggests that the DFT calculation may not reflect the equilibrated structure of the interface, but rather represents a metastable interface state. This is not surprising given the limited DFT cell used and given that any cell which is smaller than the cell size taken in this work will require large strains of the Ni/sapphire films, to accommodate the periodicity of both lattices in the same cell. A similar trend was reported by Wang et al. [23], who found from DFT simulations of a Ni– Al2O3 interface (that was also stretched to coherency) that the Ni terminating layer differs in structure from that of bulk Ni layers and contains vacancies and corrugations. Shi et al. [24] performed first principles calculations to demonstrate the effect of the different translations of the Ni with respect to the sapphire for the Al1 termination. Although their work was limited in size and does not reflect the complete interface reconstruction, they showed that the spacing of the layers parallel to the interface is the smallest when the Ni is positioned on top of an oxygen atom. This result is similar to the buckling effect seen in this study, which results in an increase in interface spacing at the hollow sites. The only simulation in the literature that did not strain the Ni–Al2O3 interface to coherency was done by Long and Chen [26]. In their simulation they successfully predicted reconstruction of the interface, similar to the
Fig. 8. A three-dimensional schematic representation of the reconstructed interface. The arrows mark the center of the H1 sites. The periodic buckling of the terminating Ni layer is seen at the side faces of the schematic and is perfectly aligned with the location of the H1.
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Fig. 9. (a) Final atomistic configuration from the model superimposed on the experimental micrograph of the interface along projection A. The Ni an Al atomic columns coincide very well with the dark contrast visible in micrograph (Ni: green, Al: blue, O: red). (b) The same experimental micrograph as in (a) but without the superimposed model.
experimental result we present here. Unfortunately, the unit cell used by Long and Chen was too small to reflect the complete reconstruction symmetry. In addition, they employed a set of parameters with the Rahman– Stillinger–Lemberg pairwise potential to describe bulk Ni, whereas there are more reliable interatomic potentials that capture the anisotropic elastic bulk properties of Ni (as the one employed here). The interatomic potential for the Ni–sapphire interface suggested by Long and Chen [26] (also employed in this work) indeed lacks the electronic details and preciseness of DFT calculations. Despite the simplified interatomic potential, the reconstruction of the Ni found in our simulations is in very good agreement with the structure observed experimentally and the interfacial unit cell correlates perfectly to the coincidence symmetry. We believe that this similarity implies a very small reconstruction of the sapphire crystal at the interface and that the metal elastic properties play the major role in the interface structure. These conditions may not always be fulfilled for other metal–ceramic interfaces, or even for the same system with an O-terminated interface, and such a simple approach should be used with caution for other systems. This exemplifies the need for larger DFT simulations, which include the influence of temperature.
optimize nearest neighbor bonding. The coherent regions are separated by arrays of misfit dislocations (or steps) that reduce the total elastic strain energy [41]. Incoherent interfaces by definition have no atomic matching across the interface, and the atoms at the interface are expected to essentially retain their bulk positions up to the interface layer. No misfit strain exists, since the atoms are not constrained by the other phase at the interface [41]. Unlike the atomistic structure of coherent and semi-coherent interfaces, the structure of incoherent interfaces is not fully understood. Although it is clear that no dislocations are present at incoherent interfaces, the postulation that the atoms retain their bulk positions up to the interface layer seems over-simplistic and probably does not represent equilibrated systems. Local re-arrangement of the atoms at the interface may result in a periodically reconstructed interface (similar to surface reconstruction), local relaxation of atoms to form preferred interfacial bonding, or even the formation of a diffuse interface [42]. These local changes in interface structure are all types of interface
4.3. Interface coherency The structure of interfaces is normally categorized into three types: coherent, semi-coherent, or incoherent. A coherent interface is defined as a perfect atom-by-atom match that is continuous across the interface, with no defects such as misfit dislocations, and intrinsic mismatch strain remains as an elastic strain component [41]. Semicoherent interfaces have large areas of coherency resulting from tension or compression of the interfacial layers to
Fig. 10. The strain field distributions at the first Ni atomic layer. The atoms are shaded according to the value of total strain at their position (larger compressive strains are shaded with darker atoms). Maximum compressive strain is located in the H1 sites.
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complexions [43–46], that will develop during equilibration in order to reduce interfacial energy. With the lack of a suitable definition, these interfaces are also usually considered to be incoherent. A more correct definition in this case is delocalized interface coherency, which reflects local shift of atoms at the interface that are caused by the net stresses from both sides of the interface. This shift can be periodic and result in a reconstructed interfacial layer, or non-periodic, which makes structure analysis more difficult [42,47]. Nevertheless, information on specific complexions (i.e. interface structure and chemistry at equilibrium) which are neither coherent nor semi-coherent is important, but so far comprehensive modeling and thorough experimental studies are lacking. Instead, most studies of metal–ceramic interfaces focus on the as-prepared state or other meta-stable states that may be easier to experimentally characterize, but are less fundamentally important as the relaxed and equilibrated interface for the correlation of interface thermodynamics to interface structure and chemistry [48]. The stress relief mechanism that is adopted in equilibrated Ni–Al2O3 interfaces is discussed below. 4.4. The role of reconstruction (delocalized coherency) The experimental results of this work clearly show that the Ni interfacial layer is constrained by bonding with the Ni bulk on one side and the Al and O layers on the other side. Overall, the layer conserves the fcc–(1 1 1) structure, but local shifts around the original atom positions allow for better bonding and reduced interfacial energy. The relatively large lattice misfit between Ni and sapphire is expected to be concentrated in regions of weaker atomic interactions, i.e. the H1 sites, where the Ni is positioned directly above an Al atom with no oxygen as a nearest neighbor. As a result, the Ni layers buckle in these regions and the terminating layer is relaxed locally into the Ni bulk. This unique strain (and energy) reduction mechanism was not reported for other interfaces, although it is possible that it was experimentally observed for the Cu–Al2O3 system and was misinterpreted as an incoherent interface, due to the relatively large misfit between Cu and Al2O3 and the limited resolution of the microscope used at that time [6,7]. Smaller misfit can be accommodated by misfit dislocations and/or steps, which will increase in density as the misfit increases. For semi-coherent interfaces, with increasing lattice misfit between the two interface-forming crystals, the dislocation/step density reaches a maximum and cannot absorb the strain energy. For misfit strain higher than this value, an alternative mechanism is required for strain reduction, for example the formation of delocalized interface coherency (i.e. interface reconstruction). This is analogous to the structure of grain boundaries, where for low-angle grain boundaries the misfit is accommodated by a set of lattice dislocations that increase in density as the misorientation angle increases. At a critical value of misorientation between the grains, the dislocation density reaches a maximum value, and the high-angle grain boundary develops a different
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atomistic arrangement (see Fig. 11). This arrangement is characteristic for each specific boundary, chemical environment, temperature, and crystal structure, but it is obvious that some degree of relaxation and even reconstruction occurs, and that the local atomistic structure at the boundary is not identical to that of the bulk. As such, reconstruction at nominally incoherent interfaces may occur, as was experimentally demonstrated in this work for the Ni–Al2O3 interface at equilibrium. A systematic experimental study of the influence of mismatch on interface reconstruction is a challenging task. In the interim, atomistic simulations of the influence of mismatch on interfacial reconstruction as a function of temperature and equilibrium segregation may provide unique insight into possible reconstructed interface structures, and their influence on adhesion and electronic properties at interfaces. 5. Summary and conclusions Quantitative TEM was used to determine the atomistic structure of the Nið1 1 1Þ½1 1 0jjsapphireð0 0 0 1Þ½1 0 1 0 interface, from samples equilibrated via solid-state dewetting. The experimental evidence shows that this interface develops a unique equilibrium structure in which the Ni layer adjacent to the interface buckles in a periodic manner that reflects the coincidence symmetry of the two lattices. The interface was shown to reconstruct into a pffiffiffi pffiffiexperimentally ffi 2:5 3 2:5 3R30 two dimensional interface unit cell, based on exit wave reconstruction supported by atomistic simulations. This solid–solid interface reconstruction (or delocalized coherency) is a mechanism to absorb misfit strain energy and reduce the interface energy, and may occur at other interfaces normally considered to be incoherent. The measured interface energy (using Winterbottom analysis) is 2.16 ± 0.2 J m–2 at P(O2) = 1020 atm and T = 1623 K. Acknowledgements The authors gratefully acknowledge P. Wynblatt and T. Besmann for enlightening discussions. A. Heuer is acknowledged for critical comments and for suggesting the term “delocalized interface coherency”. The authors thank D. Chatain for in-depth discussions and critical comments. The United States–Israel Binational Science Foundation (BSF Grant 2004068) and the Russell Berrie Nanotechnology Institute at the Technion are acknowledged for partial support of this study. H.M. acknowledges support from the Women in Science program of the Israel Ministry of Science and an Ilan Ramon Scholarship. Appendix A. Strain calculation at the interface In order to analyze the deformation of the (1 1 1) atomic layer of Ni at the interface, the atomic-level strain along the layer was calculated within the framework of the atomistic simulations. Since only the in-plane strain is of interest, the
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Fig. 11. Schematic drawing demonstrating the analogy between grain boundaries and interface structures with regards to coherency. Top: At grain boundaries increasing the misorientation angle results in an increase in dislocation density. For large misorientations the boundary assumes a characteristic structure, i.e. reconstruction. Bottom: The increase in lattice mismatch at interfaces will increase the misfit dislocation density. For large misfit values, the interface may adopt a characteristic (reconstructed) structure. For reconstructed grain boundaries and interfaces, the characteristic structure is not limited to one lattice plane.
Fig. 12. The strain calculation. The (a) deformed and (b) non-deformed {1 1 1} layer of Ni atoms. The triangle for which the strain is calculated and its corresponding virtual membrane are plotted. The atomic-level strain is calculated at the point marked in (a) in the middle of the triangle.
where h0 is a 2 2 matrix composed of the elements h0ij . As opposed to the classic definition of reduced coordination, the relative coordinates are not limited to values between 0 and 1. After relaxation the atoms acquire new positions ~ r0 , which correspond to a local strain field and a translation of the center of each three atoms. Accordingly, the virtual membrane is deformed by the same strain and translated by the same distance (Fig. 12b). If the translation component ~ D is deduced from the atomic positions of the three atoms, the strain contribution, i.e. the relative position of each atom on the deformed virtual membrane, is the same as the non-deformed relative positions: s ¼ ðr0 DÞh1
analysis is focused on the part of the strain tensor in the x–y plane of the interface. The method presented here yields similar results to other techniques to calculate the whole strain tensor in the bulk (e.g. [49]), in which all nearest neighbors are in fcc positions. First the strain field at the center point of each three neighboring atoms that forms a triangle is calculated (for instance, the strain at the point within the triangle plotted in Fig. 12a). Each three atoms are assumed to be “glued” to a rectangular virtual membrane, defined by two independent two-dimensional vectors h0ij , where i is the vector number and j is the indices of the ith vector. The position of each atom on the membrane ~ r can be written in relative coordinates: s¼
rh1 0
ð2Þ
ð3Þ
where h is a 2 2 matrix composed of the vectors defining the deformed virtual membrane. The solution of Eqs. (2) and (3) simultaneously for the three atoms yields the vectors defining the deformed virtual membrane (~ r0 is also considered as an unknown parameter). Since the initial virtual membrane can be chosen arbitrarily, two perpendicular unit vectors are selected, i.e. h0 is a unity matrix 1. The two-dimensional strain tensor at the center of the triangle of each three neighboring atoms is then 1 e ¼ ðhT h 1Þ 2
ð4Þ
Finally, the local strain at each atomic position is the average strain of all the six triangles surrounding it.
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