Nanoscale Structuring by Misfit Dislocations in Si1-xGex/Si Epitaxial Systems

Similar documents
Uniform and ordered self-assembled Ge dots on patterned Si substrates with selectively epitaxial growth technique

Surface composition of ordered alloys: An off-stoichiometric effect

The dipole moment of a wall-charged void in a bulk dielectric

Crack Tip Flipping: A New Phenomenon yet to be Resolved in Ductile Plate Tearing

Bending of nanoscale ultrathin substrates by growth of strained thin films and islands

Ordering of Nanostructures in a Si/Ge 0.3 Si 0.7 /Ge System during Molecular Beam Epitaxy

Sound absorption properties of a perforated plate and membrane ceiling element Nilsson, Anders C.; Rasmussen, Birgit

A Note on Powers in Finite Fields

Kommerciel arbejde med WAsP

Modelling Salmonella transfer during grinding of meat

Reducing Uncertainty of Near-shore wind resource Estimates (RUNE) using wind lidars and mesoscale models

Steady State Comparisons HAWC2 v12.2 vs HAWCStab2 v2.12

Morphological evolution of single-crystal ultrathin solid films

Exposure Buildup Factors for Heavy Metal Oxide Glass: A Radiation Shield

Determination of packaging induced 3D stress utilizing a piezocoefficient mapping device

Numerical modeling of the conduction and radiation heating in precision glass moulding

Realtime 3D stress measurement in curing epoxy packaging

Ion exchange model for phase proton exchange waveguide in LiNbO3.

Validation of Boundary Layer Winds from WRF Mesoscale Forecasts over Denmark

Optimizing Reliability using BECAS - an Open-Source Cross Section Analysis Tool

Spontaneous lateral composition modulation in InAlAs and InGaAs short-period superlattices

Radioactivity in the Risø District January-June 2014

An Equivalent Source Method for Modelling the Lithospheric Magnetic Field Using Satellite and Airborne Magnetic Data

Determination of the minimum size of a statistical representative volume element from a fibre-reinforced composite based on point pattern statistics

Reflection and absorption coefficients for use in room acoustic simulations

Design possibilities for impact noise insulation in lightweight floors A parameter study

Feasibility of non-linear simulation for Field II using an angular spectrum approach

Radioactivity in the Risø District July-December 2013

Chapter 3. Step Structures and Epitaxy on Semiconductor Surfaces

Multiple Scenario Inversion of Reflection Seismic Prestack Data

Lidar calibration What s the problem?

Optimal acid digestion for multi-element analysis of different waste matrices

Spontaneous Lateral Composition Modulation in InAlAs and InGaAs Short-Period Superlattices

On the Einstein-Stern model of rotational heat capacities

Potential solution for rain erosion of wind turbine blades

Diffusion of sulfuric acid in protective organic coatings

Plan for Lectures #4, 5, & 6. Theme Of Lectures: Nano-Fabrication

Luminescence of laterally ordered Ge islands along Š100 directions

Dynamic Effects of Diabatization in Distillation Columns

Relaxation of a Strained Elastic Film on a Viscous Layer

A. Portavoce, P.Gas. L2MP-CNRS, Univ. Aix-Marseille 3, Campus St Jérôme, Marseille Cedex 20, France. I. Berbezier, A. Ronda,

A comparison study of the two-bladed partial pitch turbine during normal operation and an extreme gust conditions

Reconstruction and intermixing in thin Ge layers on Si 001

Measurements of the angular distribution of diffuse irradiance

Matrix representation of a Neural Network

Plane Wave Medical Ultrasound Imaging Using Adaptive Beamforming

Global Wind Atlas validation and uncertainty

Lecture contents. Stress and strain Deformation potential. NNSE 618 Lecture #23

Reflection of sound from finite-size plane and curved surfaces

Space charge fields in DC cables

Physics and Material Science of Semiconductor Nanostructures

Accelerating interior point methods with GPUs for smart grid systems

Solar radiation and thermal performance of solar collectors for Denmark

Optical properties of InAlGaAs quantum wells: Influence of segregation and band bowing

Order- N Green's Function Technique for Local Environment Effects in Alloys

Quantum-well-induced ferromagnetism in thin films

Strongly 2-connected orientations of graphs

Nanostructure Fabrication Using Selective Growth on Nanosize Patterns Drawn by a Scanning Probe Microscope

Vortex statistics for turbulence in a container with rigid boundaries Clercx, H.J.H.; Nielsen, A.H.

A numerical study of vortex-induced vibrations (viv) in an elastic cantilever

Single Shooting and ESDIRK Methods for adjoint-based optimization of an oil reservoir

Validation and comparison of numerical wind atlas methods: the South African example

Supplementary Figure 1. Planar-view annular dark-field scanning transmission electron microscopy (ADF-STEM) and energy-dispersive X-ray (EDX) images

Design of Robust AMB Controllers for Rotors Subjected to Varying and Uncertain Seal Forces

On numerical evaluation of two-dimensional phase integrals

Nanoprobing of semiconductor heterointerfaces: quantum dots, alloys and diffusion

Kinetic Monte Carlo simulation of nucleation on patterned substrates

Vectorial analysis of dielectric photonic crystal VCSEL

Application of mesoscale models with wind farm parametrisations in EERA-DTOC

Raman spectroscopy of self-assembled InAs quantum dots in wide-bandgap matrices of AlAs and aluminium oxide

Geometry explains the large difference in the elastic properties of fcc and hcp crystals of hard spheres Sushko, N.; van der Schoot, P.P.A.M.

Blueshift of the silver plasmon band using controlled nanoparticle dissolution in aqueous solution

University of Bath. Publication date: Document Version Early version, also known as pre-print. Link to publication

The colpitts oscillator family

Validity of PEC Approximation for On-Body Propagation

Impact of the interfaces for wind and wave modeling - interpretation using COAWST, SAR and point measurements

Observation of Bulk Defects by Scanning Tunneling Microscopy and Spectroscopy: Arsenic Antisite Defects in GaAs

Comparisons of winds from satellite SAR, WRF and SCADA to characterize coastal gradients and wind farm wake effects at Anholt wind farm

Crystalline Surfaces for Laser Metrology

Self-excitation of space charge waves

The Wien Bridge Oscillator Family

Generating the optimal magnetic field for magnetic refrigeration

Geometry of power flows and convex-relaxed power flows in distribution networks with high penetration of renewables

A zero-free interval for chromatic polynomials of graphs with 3-leaf spanning trees

Molecular orientation via a dynamically induced pulse-train: Wave packet dynamics of NaI in a static electric field

Analysis of Biaxially Stressed Bridge Deck Plates

Scaling during shadowing growth of isolated nanocolumns

Characterization of thick graded Si 1 x Ge x /Si layers grown by low energy plasma enhanced chemical vapour deposition

Spontaneous Pattern Formation from Focused and Unfocused Ion Beam Irradiation

Compositional variations in strain-compensated InGaAsP/InAsP superlattices studied by scanning tunneling microscopy

Unknown input observer based scheme for detecting faults in a wind turbine converter Odgaard, Peter Fogh; Stoustrup, Jakob

Document Version Publisher s PDF, also known as Version of Record (includes final page, issue and volume numbers)

Precise control of size and density of self-assembled Ge dot on Si(1 0 0) by carbon-induced strain-engineering

Prediction of airfoil performance at high Reynolds numbers.

A note on non-periodic tilings of the plane

Effect of steady-state hydrogen coverage on the evolution of crosshatch morphology during Si 1Àx Ge x ÕSi 001 growth from hydride precursors

Kinetic Monte Carlo simulation of semiconductor quantum dot growth

Modification of the surface electronic and chemical properties of Pt(111) by subsurface 3d transition metals

A Trust-region-based Sequential Quadratic Programming Algorithm

Benchmarking of optimization methods for topology optimization problems

Transcription:

Downloaded from orbit.dtu.dk on: Dec 17, 2017 Nanoscale Structuring by Misfit Dislocations in Si1-xGex/Si Epitaxial Systems Shiryaev, S.Y.; Jensen, Flemming; Hansen, J. Lundsgaard; Petersen, Jon Wulff; Larsen, Arne Nylandsted Published in: Physical Review Letters Link to article, DOI: 10.1103/PhysRevLett.78.503 Publication date: 1997 Document Version Publisher's PDF, also known as Version of record Link back to DTU Orbit Citation (APA): Shiryaev, S. Y., Jensen, F., Hansen, J. L., Petersen, J. W., & Larsen, A. N. (1997). Nanoscale Structuring by Misfit Dislocations in Si1-xGex/Si Epitaxial Systems. Physical Review Letters, 78(3), 503-506. DOI: 10.1103/PhysRevLett.78.503 General rights Copyright and moral rights for the publications made accessible in the public portal are retained by the authors and/or other copyright owners and it is a condition of accessing publications that users recognise and abide by the legal requirements associated with these rights. Users may download and print one copy of any publication from the public portal for the purpose of private study or research. You may not further distribute the material or use it for any profit-making activity or commercial gain You may freely distribute the URL identifying the publication in the public portal If you believe that this document breaches copyright please contact us providing details, and we will remove access to the work immediately and investigate your claim.

Nanoscale Structuring by Misfit Dislocations in Si 12x Ge x Si Epitaxial Systems S. Yu. Shiryaev, 1 F. Jensen, 2 J. Lundsgaard Hansen, 1 J. Wulff Petersen, 2 and A. Nylandsted Larsen 1 1 Institute of Physics and Astronomy, University of Aarhus, DK-8000 Aarhus C, Denmark 2 Mikroelektronik Centret, Building 345 East, The Technical University of Denmark, DK-2800 Lyngby, Denmark (Received 22 March 1996) New capabilities of misfit dislocations for spatial manipulation of islands in Si 12x Ge x Si heteroepitaxial systems have been elucidated. Formation of highly ordered Ge-island patterns on substrates prestructured by slip bands of misfit dislocations is revealed. The major sources leading to the ordering are identified to be dislocation strain fields at the surface and modifications of the nearsurface-layer morphology induced by dislocation slip. [S0031-9007(96)01997-7] PACS numbers: 68.55. a, 61.16.Bg, 61.16.Ch, 61.72.Lk The interplay between surface morphology and misfit dislocations in heteroepitaxial semiconductor systems has recently been a subject of numerous studies [1]. It manifests itself in a diverse range of phenomena, such as formation of gradual surface undulations (cross-hatch patterns) arising from the surface mass transport driven by the strain fields of the misfit dislocations [2 4], elastic [5] and plastic [6] displacements of the surface by misfit dislocations, and dislocation nucleation at surfaceripple structures in continuous heteroepitaxial layers [7 10]. Alignment of the surface-ripple domains along dislocation lines was observed in Si 12x Ge x Si heteroepitaxial systems and attributed to ripple-dislocation interactions mediated by strain [7,10]. An ordering of InP islands on a strained InGaP buffer was found in Ref. [11] and explained by a preferential island nucleation driven by the misfit-dislocation strain. In this Letter we report on a new phenomenon formation of highly ordered patterns of isolated Ge islands on Si Si 12x Ge x Si substrates prestructured by the introduction of misfit dislocations. An outstanding feature of these patterns is that they are almost entirely composed of well-separated rows of sharply aligned Ge islands and mimic the underlying dislocation-slip-band morphology. The basic idea of our experiments is illustrated in Fig. 1. A heteroepitaxial system is grown consisting of a compositionally graded Si 12x Ge x layer on a (001) Si substrate and on top of that a uniform Si 12x Ge x layer comprising a quantum well (QW) used as a marker layer, and finally a silicon cap layer. The structure is metastable and has been shown to relax during postgrowth annealing through a self-organized process leading to the formation of very narrow, shear bands of misfit dislocations in the graded layer [12]. This has dramatic consequences for the initial structure, such as the following: (i) formation of shear steps both on the surface and at the heterointerfaces in the bulk of the epitaxial system and (ii) appearance of a surface, elastic-strain pattern associated with the highly nonuniform distribution of the misfit dislocations in the graded layer. After relaxation, Ge is deposited on the surface to study the effect of the surface morphology and surface strain on the growth of three-dimensional (3D) Ge islands on Si. The samples were grown at 530 ± C by molecular-beam epitaxy (MBE) and included graded layers identical to those used in our previous work [13]. The top uniform Si 0.8 Ge 0.2 layer included a 5 nm thick, Si 0.6 Ge 0.4 QW and a 10 nm thick Si cap. The relaxation of the mismatch strain in the samples was achieved directly in the MBE chamber by flashing the substrate temperature up to 800 ± C for 15 s. After the flash the temperature was reduced to 740 ± C and a Ge film with a nominal thickness of 1.2 nm was deposited on top of the relaxed structure, at a deposition rate of 0.02 nm s. For comparison, the same amount of Ge was also deposited on the surface of a virgin Si substrate and on the surface of the above heterostructure without a Si cap. The atomic force microscopy (AFM) images displayed in Fig. 2 illustrate the results of the Ge deposition on the surface of a virgin (001)Si wafer (a) and of prestructured Si 12x Ge x Si (b) and Si-cap Si 12x Ge x Si substrates (c). Well-defined Ge islands are resolved in all the images indicating 3D growth in all the cases. The islands are randomly distributed on the surface of the Si substrate [Fig. 2(a)] in agreement with previous observations of Ge islanding on (001) Si (see, for example, Ref. [14]). In contrast to this structure, ordered Ge-island patterns are FIG. 1. Schematic diagram illustrating the structuring of Si Si 12x Ge x Si substrate with dislocations in a compositionally graded layer. The dashed lines illustrate the regions of dislocation slip. C and QW denote a Si cap and a quantum well, respectively. 0031-9007 97 78(3) 503(4)$10.00 1997 The American Physical Society 503

FIG. 2. AFM image of a 7.7 3 7.7 mm 2 surface area showing the result of deposition of a 1.2 nm thick Ge film on the surface of a bare 001 Si substrate (a) and patterned Si 12x Ge x Si (b) and Si Si 12x Ge x Si (c) systems. The height scale (black to white) is (a) 41 nm, (b) 53 nm band, and (c) 63 nm, respectively. A height profile (d) through an island in (c) along the line P-Q is shown. FIG. 3. Cross-sectional TEM images of the sample shown in Fig. 2(c). (a) Rows (a, b, and c) of Ge islands are aligned along the 110 surface direction. The sample is tilted off the 110 zone axis to visualize the (001) surface thereby displaying the rows of Ge islands. (b) Enlarged TEM image of one island (F and G mark the steps produced by a shear band on, respectively, the Si 0.6 Ge 0.4 marker, and the interface between the Si cap and Si 0.8 Ge 0.2 ). The dashed lines indicate 111 slip planes corresponding to particular bands in the graded layer. found on the surfaces modified by the misfit dislocations [Figs. 2(b) and 2(c)]. The islands formed on the surface of bare Si 0.8 Ge 0.2 [Fig. 2(b)] are distributed dominantly in relatively wide straps oriented along 110 surface directions. Although a weak geometrical correlation can be seen on this surface (a part of the islands accumulates around the transition regions between elevated and depressed areas of the surface), the islands do not preferentially populate the depressions or elevations. A spectacular effect is observed on the surface of the heterostructure with the Si cap. The island pattern found in this case [Fig. 2(c)] mimics the slip morphology of the surface and almost all (about 90%) Ge islands are aligned in sharp rows parallel to the 110 surface directions. Height profiles through the islands [see Fig. 2(d)] reveal that the islands are located at the edges of the shear steps. Furthermore, in case (a) the fraction of the surface area covered with the islands (18%) is much higher than that in case (c) (8%). Note also that the island rows in case (c) are separated by large rectangular areas completely free of islands. These observations suggest that Ge from these areas redistributes to the islands in the rows [15] leaving the Ge-film thickness in these areas in the range of the wetting-layer thickness ( 3 ML [16]). The spatial relationship between the Ge islands and the misfit dislocations in the sample shown in Fig. 2(c) becomes apparent in cross-sectional TEM images (Fig. 3). The Ge islands are now seen to be localized in the vicinity of the slip regions on the surface (the so called 504 band-yield regions). In Fig. 3(a) island rows a, b, and c propagate along the 110 -surface directions perpendicular to the cross-sectional plane and are positioned at the yield regions of, respectively, bands A, B, and C composed of dislocations gliding on the (111) (A) and 111 (B and C) planes perpendicular to the cross-sectional plane. Figure 3(b) is a magnified image of one island and the underlying structure. The yield region of the slip band corresponding to this island is defined in Fig. 3(b) by the (111) plane intersecting slip step F on the Si 0.6 Ge 0.4 marker and slip step G at the interface between the Si cap and the underlying Si 0.8 Ge 0.2 layer. It can be recognized that the position of the Ge island is displaced from the band-yield region to the elevated part of the step structure. Detailed examination of cross-sectional images shows that this spatial relationship is characteristic of islands aligned along the band-yield regions. This observation is consistent with the AFM images where such a relationship is found for a number of Ge rows [marked by arrows in Fig. 2(c)]. On the other hand, inspection of the crosssectional images of the noncapped samples revealed no such spatial correlation between the islands and slip-band regions; the islands in the straps were found at both sides of these regions. In addition, a heavy strain contrast was found in both the islands and the substrate just below the islands for all the samples shown in Fig. 2. Hence, we conclude that they belong to the category of islands which are partially relaxed through elastic displacements in both the island and the substrate [14].

A characteristic feature of the capped samples is that the Si-cap thickness is locally decreased at the elevated sides of the bulk slip steps [G in Fig. 3(b)] and locally increased at the opposite sides. In the areas remote from the bandyield regions this thickness reaches its nominal value of 10 nm. We found that this modification of the Si-cap layer occurs during annealing in the MBE chamber before the Ge deposition, and attribute this effect to a surface redistribution of Si which results in a smoothing of the sharp slip-step bunches highly energetically unfavorable on the free surface. In general, there are several sources which may provide energetically favorable sites for island location on the surface. They include surface steps [17,18], local deviations of the surface from planar orientation [19], and the nonuniform strain at the surface which provides the sites with a reduced mismatch between Ge and Si [10,20,21]. The island positions found in our case do not coincide with the slip steps on the surface either for noncapped or capped samples. Therefore, we conclude that the island ordering observed in the present work is not mediated by the slip steps. Furthermore, the islands in the noncapped samples do not preferentially populate the elevations or depressions on the surface. Hence, the only source which can induce the highly nonuniform island distribution in this case is the nonuniform elastic strain from the dislocation bands. This conclusion is supported by the results of recent experimental and theoretical studies of the vertical self-alignment of islands in multilayered heteroepitaxial systems [20,21] and alignment of InP islands on strained InGaP buffers along dislocation bunches [11]. Although these studies focus on the different aspects of the island formation process (either island-nucleation events [20] or surface diffusion leading to a nonuniform thickness of the wetting layer [21]), they imply a common driving force for the island alignment which is nonuniform strain at the surface. The above results show that the presence of the Si cap has a dramatic influence on the island pattern which manifests itself in the extremely sharp island alignment along the band-yield regions. There is also a significant difference in the island-size distributions for the capped and noncapped samples [see Figs. 2(b) and 2(c)]. In this work we focus on the most striking effect which is the sharp alignment. To explain this effect we shall consider the near-surface layer structure which includes the Si cap and the interface between the cap and the Si 0.8 Ge 0.2 layer. There are two features of this structure that may contribute to such an alignment. The first feature is the sharp steps at the interface between the tensile-strained Si cap and the relaxed Si 0.8 Ge 0.2 layer [G in Fig. 3(b)]. Because these slip steps perturb the biaxial strain between the Si cap and Si 0.8 Ge 0.2, and lie very close to the surface, they induce local surface-strain variations and, thus, may provide low-energy sites for the islands. The second feature is the local variation in the Si cap thickness around the band-yield regions. It appears that the positions of the islands correspond to those sides of the regions where the thickness of the Si cap film is smaller [Fig. 3(b)]. This correlation suggests that the substrate sites with a thinner Si cap may be energetically more favorable for the islands than the sites with a thicker Si cap. A reason for this may be that the substrate in these regions is softer, i.e., it allows larger elastic deflections of the Ge Si interface from the planar orientation providing a higher degree of the elastic relaxation in the islands. This suggestion is based on recent calculations which show that the elastic displacement fields originating from islands penetrate to substrate depths larger than the island heights [22]. Therefore, since the Si cap thickness in our case is smaller than the island heights, both the Si cap and underlying Si 0.8 Ge 0.2 alloy volumes are involved in the relaxation. Note that the substrate regions under the islands where the cap is thinner include smaller Si volumes per unit surface area. Hence, these regions are softer because Si has larger elastic constants than Si 0.8 Ge 0.2 [23]. Finally, the present results do not allow for a decoupling of the effects of the near-surface steps and the Si-cap thickness. Thus we cannot make any clear conclusions concerning their relative role in the sharp island confinement. A separation of these effects would, however, be of great interest not only from a scientific but also a technological point of view since it may provide new routes for the alignment of nanodot structures. In summary, new capabilities of misfit dislocations for spatial structuring of heteroepitaxial semiconductor systems at the nanometer length scale have been elucidated. Formation of ordered Ge-island patterns on Si 12x Ge x Si and Si-cap Si 12x Ge x Si substrates prestructured by dislocation bands is revealed. The most spectacular ordering was observed on surfaces of samples with Si caps where the islands appear to be confined to one-dimensional rows aligned along slip regions on the surface. It was demonstrated that the ordering of the islands is not mediated by the slip-step bunches on the surface. The results suggest that the ordering on the noncapped substrates originates from the nonuniform dislocation strain fields and that the sharp lateral confinement of the islands on the capped substrates is mediated by the near-surface layer structure modified by the dislocation slip. Although the analysis done in this work is qualitative and omits aspects of island nucleation and growth [8,18,24], it captures, in our opinion, the major sources which may lead to the sharp lateral island confinement, and thus provides a framework for further, detailed studies. This work was supported by the Danish Minister of Industry and Research in the framework of CNAST. Special thanks to Pia Bomholt for the help in the preparation of the cross-sectional TEM specimens. [1] For a review, see articles in Mater. Res. Soc. Bull. 21, No. 4 (1996). [2] J. W. P. Hsu, E. A. Fitzgerald, Y.-H. Xie, P. J. Silverman, and M. J. Cardillo, Appl. Phys. Lett. 61, 1293 (1993). 505

[3] M. Albrecht, S. Christiansen, J. Michler, W. Dorsch, H. P. Strunk, P. O. Hansson, and Bauser, Appl. Phys. Lett. 67, 1232 (1995). [4] F. Jonsdottir and L. B. Freund, Mater. Res. Soc. Symp. Proc. 317, 309 (1994). [5] M. A. Lutz, R. M. Feenstra, F. K. LeGoues, P. M. Mooney, and J. O. Chu, Appl. Phys. Lett. 66, 724 (1995). [6] S. Yu. Shiryaev, F. Jensen, and J. Wulff Petersen, Appl. Phys. Lett. 64, 3305 (1994). [7] D. E. Jesson, K. M. Chen, S. J. Pennycook, T. Thundat, and R. J. Warmack, Science 268, 1161 (1995). [8] D. E. Jesson, K. M. Chen, and S. J. Pennycook, Mater. Res. Soc. Bull. 21, 31 (1996). [9] A. G. Cullis, A. J. Pidduk, and M. T. Emeny, Phys. Rev. Lett. 75, 2368 (1995). [10] A. G. Cullis, D. J. Robbins, S. J. Barnett, and A. J. Pidduk, J. Vac. Sci. Technol. A 12, 1924 (1994). [11] K. Häusler, F. Noll, and K. Eberl, Solid State Electron. 40, 803 (1996). [12] S. Yu. Shiryaev, J. Lundsgaard Hansen, A. Nylandsted Larsen, F. Jensen, and J. Wulff Petersen, Phys. Rev. B 52, 15 881 (1995). [13] S. Yu. Shiryaev, F. Jensen, J. Wulff Petersen, J. Lundsgaard Hansen, and A. Nylandsted Larsen, J. Crystal Growth 157, 132 (1995). [14] D. J. Eaglesham and M. Cerullo, Phys. Rev. Lett. 64, 1943 (1990). [15] To estimate the total amount of Ge atoms in the islands in the sample shown in Fig. 3(c), we used the average island dimensions (diameter equal to 100 nm and height equal to 21 nm) determined from plan-view and cross-sectional TEM images. The fraction of the surface covered with islands was determined from large-scan-area AFM images. We found that the Ge content of the islands amounts to 2 ML which means that roughly 50% of the total amount of the deposited Ge 4 ML is redistributed to the islands. [16] H. Sunamura, S. Fukatsu, N. Usami, and Y. Shiraki, J. Crystal Growth 157, 265 (1995). [17] K. M. Chen, D. E. Jesson, S. J. Pennycook, M. Mostoller, T. Kaplan, T. Thundat, and R. J. Warmack, Phys. Rev. Lett. 75, 1582 (1995). [18] Y.-W. Mo, D. E. Savage, B. S. Swartzentruber, and M. G. Lagally, Phys. Rev. Lett. 65, 1020 (1990). [19] Recent calculations by L. J. Gray, M. F. Chisholm, and T. Kaplan [Appl. Phys. Lett. 66, 1924 (1995)] show that strain energies in the Ge Si system are very sensitive to relatively small deviations of the interface from the planar orientation. [20] J. Tersoff, C. Teichert, and M. G. Lagally, Phys. Rev. Lett. 76, 1675 (1996). [21] Q. Xie, A. Madhukar, P. Chen, and N. P. Kobayashi, Phys. Rev. Lett. 75, 2542 (1995). [22] S. Christiansen, M. Albrecht, H. P. Strunk, and H. J. Maier, Appl. Phys. Lett. 64, 3617 (1994). [23] S. P. Baker and E. Arzt, in Properties of Strained and Relaxed Silicon Germanium, edited by E. Kasper, EMIS Data Reviews Series No. 12 (INSPEC, IEE, London, 1995), p. 67. [24] J. Tersoff and F. K. LeGoues, Phys. Rev. Lett. 72, 3570 (1994). 506