Jan H. van der Merwe

Misfit dislocation generation in epitaxial layers

Abstract The misfit between an epilayer and a substrate may be accommodated either by misfit strain or by misfit dislocations, or by both jointly. For an epitaxial monolayer (ML) a critical misfit, depending primarily on bonding, exists below which it is stable when in registry with the substrate. When growth continues with the formation of a multilayer, misfit dislocations will enter at some critical thickness. The main objectives of this article are to critically review theoretical work aimed at explaining (1) the conditions under which an epilayer will grow in a ML-by-ML fashion to yield a uniform film and (2) the reasons why observed critical thicknesses and residual strains are often significantly in excess of the predicted ones, in terms of equilibrium and non-equilibrium concepts. Both (1) and (2) are of great fundamental and technological interest.

The role of structural ledges as misfit-compensating defects: fcc-bcc interphase boundaries

An energetic justification is given for structural ledges as interfacial misfit compensating defects, based on elastic considerations. We detail the NW-x configuration in which the orientation is imposed by close matching along the (21 l)fcc and (110)bcc directions and (111) fcc and (110) bcc planes, respectively. Special prominence is given to the terrace between structural ledges, where it is shown that an elastic relaxation obtains at specific terrace lengths based on atomic matching. It is this relaxation that makes a stepped interface more stable than a planar one at all meaningful values of misfit.

Structural ledges in interphase boundaries

This article addresses the properties of stepped misfitting interfaces and their energetic preference to planar misfitting interfaces. It highlights: (a) the purely geometrical or rigidlike, (b) the rigid (unrelaxed) energetic, and (c) the relaxed energetic properties of stepped interfaces. In (a), we address (1) the accommodation of misfit by the step or ledge mode through the cancellation of the mismatch, that builds up along a terrace, by the forwardpattern advance effected by a step,i.e., the relative displacement of atomic patterns on either side of the interface as observed in crossing a structural ledge along the interface, (2) the sideways (shear) pattern advance which seems to be energetically undesirable, (3) the need for tilt-type misfit dislocations to accommodate the misfit normal to the interface, and (4) the fact that at {III}fcc(face-centered cubic)/{110}bcc(body-centered cubic) interfaces with rhombic symmetries, the misfits, as well as the pattern advances, are interrelated through the ratior = b/a of nearest-neighbor distances in the crystals. In (b), we exploit the rigid model approach that (1) yields ideality criteria for minimum energy and provides energetic justification for the step mode of misfit accommodation, (2) confirms that the average terrace widthl[inx defined by this mode also meets the condition for positive energy gain, and (3) defines the upper and lower energy bounds to provide a perspective of the system energetics. In (c), the foregoing considerations are refined by a transition to the harmonic (elastic) model to yield (1) the dependence of the mean energy per atom of a stepped interface on interfacial misfit and pattern advance, as well as the dependence of the mean energy per atom of a planar interface on misfit, (2) expressions for the stresses related to the atomic interaction between opposing terraces, (3) atomic displacements that might be probed by modern analytical techniques, and (4) resolved shear stresses and normal stresses that may facilitate the formation of glide dislocations in the presence of applied stresses. The boundary in a two-dimensional space—spanned by misfit and pattern advance—between regions where stepped interfaces are more stable than planar ones has been determined, suggesting that a critical misfit exists above which only planar interfaces are stable. Whereas the resolved shear stress related to the formation of structural ledges may facilitate the formation of dislocations in the presence of a subcritical applied stress, the corresponding displacements (bending) of atomic planes are probably observable only with strain contrast electron microscopy techniques.

Misfit dislocation energy in epitaxial overgrowths of finite thickness

Abstract For the purpose of predicting the structure of an epitaxial bicrystal, which has been grown under equilibrium conditions and in which one of the members is of finite thickness, one needs the dependence on misfit, bonding and thickness of the energy E d associated with the misfit dislocations required to accommodate the interfacial misfit. In the present paper approximate expressions are deduced for E d . These are compared with each other and with existing ones. It is shown that the existing expressions each have only a very narrow range of applicability. Of the two new approaches presented, the one introduces a refinement of the well-known concept of the shielding of the field of a dislocation by others and by free surfaces. The other one uses the exact solution of the governing equation of a simple case as a first approximation for the more complicated governing equation of the present problem. The former is shown to be rather inaccurate for thin films and thus calls for caution in the application of the shielding principle. The latter is inferred to be the best overall approximation.

Theoretical considerations in growing uniform epilayers

We briefly describe, theoretical considerations in the quest to grow crystalline films with perfection in uniformity of thickness. The endeavor of perfect crystallinity is best served by growing epitaxially. Thickness uniformity is facilitated by growing in a monolayer-by-monolayer mode-so called Frank-van der Merwe (FM) mode. The island (Volmer-Weber=VW) mode is undesirable; it promotes roughening and the creation of crystalline imperfections. Equilibrium criteria, proposed by Bauer, on the basis of shape equilibrium, predict that FM growth is favored by strong substrate bonding. The dependence of shape equilibration on adatom migration barriers, substrate temperature, and deposition rate is discussed. Modification of the system to improve adverse conditions by intermixing, alloying, and the use of surfactants is outlined. VW growth may also commence after completion of one or a few monolayers (MLs) in FM mode. This form of island growth, also known as Stranski-Krastanov (SK) growth, is mainly due to electronic effects, but may also be triggered by epitaxial strain relief. The tailoring of growth into FM-like growth, when VW or SK conditions exist-by an appropriate increase of supersaturation usually effected by controlling the substrate temperature and deposition rate, is briefly discussed.

Strain relaxation in epitaxial overlayers

The misfit between an epilayer and a substrate may be accommodated by misfit dislocations (MDs) or misfit strain (MS) or both. In a small misfit system the misfit is accommodated by MS alone up to a critical thickness whereafter the residual MS decreases as the thickness increases. The main objective of this paper is to review theoretical work aimed at understanding MS relief in a growing epilayer by the introduction of MDs, with the view of resolving the discrepancies between predicted and observed critical thickness and residual MS after onset of MS relief. Since the predictions are based on equilibrium principles, equilibrium theories for monolayers (MLs) in the Frenkel-Kontorowa model, and for thickening epilayers (growing ML-by-ML) in the Volterra model, are briefly summarized, including some consideration of the conditions for ML-by-ML growth. Since equilibration can be drastically retarded by barriers to nucleation and motion of dislocations the observed quantities are most often non-equilibrium values. Calculations show (i) that MD sources are normally needed to the onset of MS relief, (ii) that threading dislocations are the “softest” sources, (iii) that even with threading dislocation sources MS relief may lag behind equilibrium predictions and (iv) that Peierls friction may lead to an infinitely large critical thickness.

Role of misfit strain and proximity in epigrowth modes I: Strong epilayer-substrate interaction

Abstract The objectives of this study are to assess the effects of misfit strain and substrate proximity on the evolution of growth modes — emphasizing the transition to Stranski-Krastanov (SK) growth — during thickening of a growing epilayer. The study comprises a generalization of Bauers equilibrium criteria and their application to the growth of Cu and Ni on W(110) — a strong interacting substrate — using embedded-atom methods (EAM) calculations. It is suggested that proximity is propagated by epilayer-substrate bond strength and interlayer relaxation, and that the proximity range is strongly influenced by cutoffs which model the limited range of electronic induced atom-atom interaction. The study shows that the contribution of misfit strain energy to shaping growth modes is nonlinear and indeed vanishes outside the proximity range. The predictions are in agreement with the observed Frank-van der Merwe (FM) growth of the first monolayer (ML) and a SK transition thereafter. EAM calculations also confirm the anticipated relative strengths of bonding. Calculated activation energies of surface migration suggest that adatoms are significantly more mobile on top of thin epilayers than on the bare strong interacting substrate and that this will also strongly influence equilibration kinetics.

The elastostatics of atomic steps on crystal surfaces : I. A pair of steps of opposite sign enclosing a lower terrace

Abstract This paper emanates from the fact that the “free” equilibrium atomic spacings of a surface monolayer (ML) differ from those of the crystal interior. The mechanical effects of this “surface misfit” f in the presence of a pair of monatomic steps enclosing a lower terrace have been analysed using a simple phenomenological model in which the analysis reduces to a boundary value problem in isotropic linear elasticity. The model predicts: (i) that the induced strains, displacements, stresses, energies and forces between steps are respectively proportional to f, af, μf, μa2f2 and μaf2, where a and μ are, respectively, the cubic lattice parameter and shear modulus; (ii) that the misfit normal to the surface contributes less than 1% to the interaction energy of the steps; (iii) that the leading terms in the asymptotic series of the energy and force of interaction of steps are inverse squares and inverse cubes of the step-step separation, respectively; (iv) that the interaction between the two steps (of opposite sign) is attractive; (v) that the predicted effects are significant near the steps; (vi) that the maximum stresses are large; and (vii) that the predicted quantities, though not quantitatively accurate, provide good guidelines for continued theoretical and experimental research.

Elastic constants of fcc 〈111〉 films 1. Free monolayers

Abstract The aim of the investigation is to calculate the properties of free-standing strained fcc monolayers (MLs): their equilibrium lattice constants, their anisotropic cubic stiffness constants and the influence of anharmonocity. Embedded atom methods (EAMs), employing bulk-fitting parameters, were used to model the atomic interactions. The calculations, which are conducted for films of Ni and Cu bounded by close-packed (cp) 〈111〉 planes, and constrained to remain plane, predict (a) that EAMs yield — at least for Ni and Cu — fairly accurate values of anisotropic cubic stiffness constants for the bulk and hopefully also for thin films, including MLs, (b) that the equilibrium lattice constants of free-standing MLs are less than those of the bulk by a few percent, (c) that the stiffness constants of such MLs at the ML equilibrium configuration are appreciably different from those of the bulk, (d) that the strain energy per atom at 25% strain for a ML is about twice that for the bulk and (e) that anharmonicity reduces the strain energy per atom at 25% strain by about a factor of two, as compared with the value based on the harmonic model. The present results are in good agreement with comparable previous results for Cu.

The formation of misfit dislocations by climb in pseudomorphic monolayers

Abstract We have used a hybrid Lennard-Jones/Frank-van der Merwe (LJ-FM) model and an embedded-atom-method (EAM) to analyse the energetics of an isolated adatom A, which is adsorbed at a second-layer lattice point above a pseudomorphic (ps) monolayer (ML) of Ni or Co on a Mo{110} substrate, and is allowed to climb down quasi-statically into the ps ML where it constitutes a critical nucleus for the formation of a misfit dislocation (MD). In the LJ-FM approach misfit could be introduced in a somewhat artificial, but adequate way to predict tendencies. The LJ-FM analysis showed (i) that a Ni adatom has a well defined stable second-layer position when Mo is replaced by bcc Ni, (ii) that the equilibrium position is less well defined with Mo substrate even with imposed zero misfit and (iii) that as the mistfit with Mo becomes more negative, approaching the Ni and Co misfits of about −20%, a flat energy trough develops which allows spontaneous access to the interior of the ps ML. In the EAM approach only Co reaches, more or less, the interior of the ps ML; in the case of Ni, and Cu which was also considered, strong core repulsion of substrate adatoms limits the downward climb to only 1 A. Clearly, negative (positive) misfit of large magnitude, is a prerequisite to the nucleation of heavy (light) MDs by climb in a process which is facilitated by strong (weak) absorbate-substrate bonding.