J.H. van der Merwe

Structure of epitaxial crystal interfaces

Abstract Theoretical work on the structural influences of misfit, bonding and crystal dimension in epitaxial bicrystals is reviewed. The main contents relate to the model in which the inter-action between the crystal halves is represented by a periodic force acting at the interface, the crystals are approximated by elastic continua and the structure is assumed to be governed by lowest energy principles. In this model the misfit is accommodated by sequences of misfit dislocations located at the interface and/or overall lattice strains. The degree to which one or the other mechanism dominates depends on the size and shape of the crystal in addition to the elastic and bonding properties. Realistic approximations for various special cases, e.g. monolayers, thick crystals and intermediate thicknesses, are considered. An exact solution covering the entire spectrum of interest does not exist.

Misfit accommodation by compliant substrates

Experimentally it has been found that twist bonding a thin layer of epitaxially grown material to a thick substrate of the same material has the properties of being compliant with respect to the growth of a heteroepitaxial layer onto the thin layer. The benefit of the compliant substrate is that the heteroepitaxial layer is of much higher perfection when compared to the same growth but on a bulk substrate, primarily due to a very low density of threading dislocations present in the heteroepitaxial deposit. This concept has been investigated experimentally and discussed in terms of what is the operative mechanism of the compliant substrate. However, there is still much to be explained about the mechanism by which the compliant substrate accommodates misfit between itself and the heteroepitaxial layer. How the structure of the interface between the compliant substrate and the thick substrate can be tailored to derive the maximal benefit in the epilayer is the subject of the present article. The importance o...

The role of structural ledges at phase boundaries—I. Interfaces with rectangular atomic nets

Abstract Misfitting crystal interfaces are examined as to the mechanisms of minimizing interfacial energy. A comparison between the energetics of planar and stepped interphase boundaries is made. Structural ledge interfaces with terrace patches of (dimensions lx × ly) are examined by employing, with increased sophistication, the rigidlike or purely geometrical energetics, followed by the relaxed energetic properties. For the stepped interface the mismatch which builds up along each terrace pair patch is compensated for by a relative displacement of the atomic patterns of the two terraces (pattern advance) at the next step. In this manner the average misfit parallel to the interface is minimized and the necessity for the introduction of misfit dislocations to accommodate this misfit eliminated. Since the atomic plane spacings parallel to the terraces are different on the two sides of the interface, misfit also exists in a direction normal to the terraces. This misfit is accommodated by a tilt type misfit dislocation in every superperiod. The present paper proposes models for calculating the energies associated with (i) the interaction between opposing terrace patches (including elastic relaxation), (ii) the line energy of misfitting opposing risers at the steps and (iii) the energy of the tilt misfit dislocation in a superperiod. This energy is compared with the energy of a planar interface containing conventional misfit dislocations. The calculations predict the regimes of crystal parameters for which misfit accommodation by either planar or stepped interfaces are energetically favored. In general it is shown that steps become more favorable as the misfit decreases. The calculated resolved and normal stresses associated with steps are a significant fraction of the shear modulus and may contribute to the plasticity of metals.

The role of structural ledges at phase boundaries—III. F.C.C.-B.C.C. interfaces in Kurdjumov-Sachs orientation

Abstract The ledge mode of misfit accommodation is extended to {111} f.c.c.∥{110} b.c.c. interfaces with the Kurdjumov-Sachs (KS) orientation relationship. As with the Nishiyama-Wassermann (NW) relationship the geometric quantities are uniquely related by the misfit ratio r of atomic diameters. A rigid lattice analysis shows that the driving forces for a KS orientation relationship are significantly larger than those for the NW-x and even more so than those for the NW-y relationships. The rigid analysis also predicts that the terrace width which satisfies the periodicity conditions for a stepped interface are well within the terrace widths which allow significant energy gains. The models developed for the NW analysis are modified for KS configurations and employed to sum energies for the terrace patches, tilt misfit dislocations and the riser interface energy. Comparing this energy with that of a planar interphase boundary comprised of only misfit compensating dislocations shows that a stepped interphase boundary is energetically favored at r values near and greater than rKS but less than rNW-y.

Misfit accomodation in epitaxial monolayers on (111) F.C.C. and (110) B.C.C. substrates I: Analytical approach

Abstract The model introduced previously for describing epitaxial systems with rectangular interfacial atomic meshes was modified to suit the skew atomic meshes at interfaces between f.c.c. and b.c.c. overlayers on (111) f.c.c. and (110) b.c.c. substrates and was employed in an analytical approach to determine the distribution of the misfit between dislocations and misfit strain in the case of monolayers. Three main dislocation-type solutions, framed in different coordinate systems and subjected to restrictions of varying stringencies on Poissons ratio and the angle between unit cell axes, were considered for the governing equations. The stability of the corresponding dislocation configurations was deduced. On using the critical misfit at which a coherent monolayer becomes unstable as a basis of comparison it is shown that the coherent monolayer is equally stable against a transition via edge-type or mixed-type dislocations and tolerate a misfit which is 2 1 2 times larger than the critical misfit tolerated by systems with rectangular interfacial atomic meshes. The effect of angular misfit between the unit cell axes of the overlayer and substrate is demonstrated. In order for the model to be analytically tractable it has certain shortcomings. An improved model will be used in Part II in a computational onslaught on the problem.

The role of structural ledges at phase boundaries—II. F.C.C.-B.C.C. interfaces in Nishiyama-Wasserman orientation

Abstract The ledge mode of misfit accommodation was first postulated for boundaries between b.c.c. and f.c.c. metal phases; the interfaces being {111} f.c.c.-{110} b.c.c. planes and relative orientations varying from Nishiyama-Wassermann (NW) to Kurdjumov-Sachs (KS). Here the geometrical quantities are uniquely related by the misfit ratio r of atomic diameters. We consider the so-called NW-x configuration in which the orientation is imposed by close matching along the 〈 2 11〉 ; f.c.c. and 〈 1 10〉 b.c.c. (taken as x-) directions. From the fact that no net shear pattern displacements are present with x-ledges (ledges normal to the x-direction) it is concluded that they are energetically preferable to y-ledges and justifies the approach of an energetic comparison between stepped interfaces with x-ledges and a planar interface containing conventional misfit dislocations (MDs). The NW-x stepped configuration is at first subjected to a rigid model analysis, i.e. a model with rigid crystals and periodic (truncated Fourier representation) interfacial interaction. This analysis provides (i) energetic justification for a relation between terrace periodicity and misfit cancellation, (ii) values of upper and lower average energy bounds, (iii) a method for estimating interfacial shear moduli, and (iv) a motivation for the suggestion that a relative rigid translation of the crystals is needed for energy minimization. The average energy per atom of a stepped interface ( ϵ s ) provides for misfit at the terrace patch and riser interface, as well as for misfit normal to the interface. The planar interface energy ϵ p accounts for interfacial misfit only. From a display of the excess energy Δ ϵ = ϵ s − ϵ p plotted against r it is concluded that stepped interfaces are more stable than planar interfaces for all meaningful values of the misfit ratio r, i.e. within the range where the NW-x orientation is viable.

Interfacial energy: bicrystals of semi-infinite crystals

Abstract Crystalline materials are normally bounded by surfaces and interconnected by interfaces – defects. These defects greatly influence the properties of structural materials. Their influence is normally quantified in terms of defect energies – the surface and interfacial free energies – that play, e.g., a decisive role in shaping the thickness uniformity of epilayers, a feature of great importance in the fabrication of devices. The quantification of these defect energies and their dependence on the interfacial misfit between the crystals are thus highly desirable objectives. Embedded atom method (EAM) interaction potentials are used for purposes of quantification. Their advantages and disadvantages are noted. The interpretation of the results is greatly facilitated by application of optimum Fourier coefficients. Predictions of significance, that can be inferred from sample calculations, are: (i) that the interfacial energy is normally not small compared to the surface energies and (ii) that the variation of interfacial energy with misfit is significant and can be understood in terms of the optimum Fourier coefficients.

Elastic and structural properties of f.c.c. {111} thin films part 2. A monolayer on a {110} b.c.c. substrate☆

Abstract Simple qualitative considerations suggest that the inherent mechanical properties—equilibrium structure and elastic constants—of thin epilayers are influenced by proximity effects. The computational effort to calculate these properties, using any form of atomic interaction, becomes enormous when epilayer and substrate are discommensurate and the repeat period is large—an important feature of misfit strain relief in epilayers. A procedure is proposed by which this problem can be overcome for an epimonolayer, and can be extended to multilayers. The procedure involves the following assumptions: (i) the mechanical behavior of the monolayer (ML) is governed by the principle of minimum energy, the average energy per ML atom being minimized in this case, (ii) the field of interaction emanating from the substrate is periodic (with the periodicity of the substrate surface) within the plane of the ML, (iii) the substrate field can be mapped by calculation involving the translation of the ML—as if “rigid”, having registry dimensions and allowing for height equilibration—on the substrate surface, (iv) the ML may be constrained to its average (constant) equilibrium height with negligible discrepancies in the energy. The procedure is demonstrated, and numerically justified, by its application to {111} MLs of Ni and Cu in Nishiyama-Wassermann orientation on W {110}, using embedded-atom-method potentials. The calculations produce convincing evidence to substantiate the validity of the procedure, showing that the contribution of the substrate to the embedding energy of ML atoms can be fairly accurately described in terms of its average electron density in the ML plane, the effect of the periodic oscillations in the electron density being negligible. A similar procedure is valid for the embedding of substrate atoms. The application to Ni and Cu on W shows that proximity effects are drastic: the in-plane elastic constants of a supported ML, an ML in the crystal interior and a free standing ML are respectively and crudely in the ratio 1:1.5:2.5. Proximity effects are likewise important in anharmonicity.

Interplay between misfit strain relief and Stranski–Krastanov growth in fcc (111)/bcc (110) ultrathin film epitaxy: Part 1. Analytical approach

Abstract The main objective of this investigation is to study the impact of misfit (homogeneous) strain (MS) relief by misfit dislocations (MDs) — or a misfit vernier where the relaxation into oscillatory strains characteristic of MDs becomes insignificant — on the Stranski–Krastanov (SK) growth mode, the perception being that the driving force for the transition is dominated by short range proximity effects of the vacuum and a strongly bonding substrate. The influence of misfit strain relief with increasing thickness on the transition from two-dimensional (2D) to three-dimensional (3D) growth, characteristic for SK growth, is investigated. Here, the dominating parameters are identified and the mathematical formulae for the governing relations, needed for quantification in Part 2, using embedded atom methods (EAM) potentials, are derived: (i) the dependence of substrate coverage on epilayer misfit ( f ) and deformation ( ē ); (ii) the structure of Fourier series to model fcc (111) epilayer–bcc (110) substrate interaction potentials V ( x ,xa0 y ) for a Kurdjumov–Sachs (KS) orientation, including optimum Fourier coefficients V hk for low order truncations; (iii) the harmonic intra-epilayer interaction suitable to describe MDs and to model misfit vernier accommodation of mismatch related to homogeneous Poisson strain; (iv) the average in-plane strain energy suitable (a) to handle the large repeat periods involved in homogenous epilayer deformation within the short wavelength periodic epilayer–substrate interaction potentials and (b) to define stiffness constants for epilayer deformation within such a field; (v) stability criteria for MS relief of 2D coherent and one-dimensional (1D) KS coherent — registry of two opposing sets of parallel closest packed atom rows on either side of the interface — epilayers in Nishiyama–Wassermann and KS orientations, respectively, applying continuum and discrete approaches; and (vi) criteria for growth mode realization, emphasizing transitions to the SK mode and its dependence on proximity effects. The analytical considerations reveal (a) that the transition from a 2D coherent misfit accommodation mode to a 1D KS mode is favored by a large excess strain energy e 2D − e 1D and large substrate surface free energy, but opposed by strong epilayer–substrate bonding and (b) the MS energy e makes no contribution to the growth mode discriminant if proximity effects of the interface and free surface are absent.

An assessment of the roles of climb and glide in misfit strain relief

In order to assess the relative contributions of glide and climb processes in the relaxation of misfit strain in heteroepitaxial layers, the glide and climb velocities of dislocations are compared at several temperatures. It is shown that the glide velocity is much greater than the climb velocity under normal conditions. For copper and silicon, about four orders of magnitude can exist at 3/4 of the melting temperature with lower temperatures leading to larger differences. One therefore expects relaxation of misfit strain to proceed primarily by glide mechanisms. Two cases are addressed here in which climb processes can be important: (i) by‐passing of an obstacle and (ii) redistribution of an irregular array of interfacial misfit dislocations into a regular array of lower energy.