Mechanical Stress on the Nanoscale E-Book

• Herausgeber: Wiley-VCH
• Kategorie: Fachliteratur
• Sprache: Englisch

Seitenzahl: 672

Veröffentlichungsjahr: 2011

Beschreibung

Bringing together experts from the various disciplines involved, this first comprehensive overview of the current level of stress engineering on the nanoscale is unique in combining the theoretical fundamentals with simulation methods, model systems and characterization techniques. Essential reading for researchers in microelectronics, optoelectronics, sensing, and photonics.

Leseprobe

Contents

Cover

Related Titles

Title Page

Copyright

Preface

List of Contributors

Part One: Fundamentals of Stress and Strain on the Nanoscale

Chapter 1: Elastic Strain Relaxation: Thermodynamics and Kinetics

1.1 Basics of Elastic Strain Relaxation

1.2 Elastic Strain Relaxation in Inhomogeneous Substitutional Alloys

1.3 Diffusion

1.4 Strain Relaxation in Homogeneous Mismatched Epitaxial Layers

1.5 Morphological Relaxation of a Solid under Nonhydrostatic Stress

1.6 Elastic Relaxation of 0D and 1D Epitaxial Nanostructures

References

Chapter 2: Fundamentals of Stress and Strain at the Nanoscale Level: Toward Nanoelasticity

2.1 Introduction

2.2 Theoretical Background

2.3 Applications: Size Effects Due to the Surfaces

2.4 Conclusion

Acknowledgment

References

Chapter 3: Onset of Plasticity in Crystalline Nanomaterials

3.1 Introduction

3.2 The Role of Dislocations

3.3 Driving Forces for Dislocations

3.4 Dislocation and Surfaces: Basic Concepts

3.5 Elastic Modeling

3.6 Atomistic Modeling

3.7 Extension to Different Geometries

3.8 Discussion

Acknowledgments

References

Chapter 4: Relaxations on the Nanoscale: An Atomistic View by Numerical Simulations

4.1 Introduction

4.2 Theoretical Models and Numerical Simulations

4.3 Relaxations in Surfaces and Interfaces

4.4 Relaxations in Nanoclusters

4.5 Conclusions

References

Part Two: Model Systems with Stress-Engineered Properties

Chapter 5: Accommodation of Lattice Misfit in Semiconductor Heterostructure Nanowires

5.1 Introduction

5.2 Dislocations in Axial Heterostructure Nanowires

5.3 Dislocations in Core–Shell Heterostructure Nanowires

5.4 Roughening of Core–Shell Heterostructure Nanowires

5.5 Conclusion

Acknowledgements

References

Chapter 6: Strained Silicon Nanodevices

6.1 Introduction

6.2 Impact of Strain on the Electronic Properties of Silicon

6.3 Methods to Generate Strain in Silicon Devices

6.4 Strain Engineering for 22 nm CMOS Technologies and Below

6.5 Conclusions

Acknowledgments

References

Chapter 7: Stress-Driven Nanopatterning in Metallic Systems

7.1 Introduction

7.2 Surface Stress as a Driving Force for Patterning at Nanometer Length Scales

7.3 Nanopatterned Surfaces as Templates for the Ordered Growth of Functionalized Nanostructures

7.4 Stress Relaxation by the Formation of Surface-Confined Alloys

7.5 Conclusion

Acknowledgment

References

Chapter 8: Semiconductor Templates for the Fabrication of Nano-Objects

8.1 Introduction

8.2 Semiconductor Template Fabrication

8.3 Ordered Growth of Nano-Objects

8.4 Conclusions

References

Part Three: Characterization Techniques of Measuring Stresses on the Nanoscale

Chapter 9: Strain Analysis in Transmission Electron Microscopy: How Far Can We Go?

9.1 Introduction: How to Get Quantitative Information on Strain from TEM

9.2 Bending Effects in Nanometric Strained Layers: A Tool for Probing Stress

9.3 Strain Analysis and Surface Relaxation in Electron Diffraction

9.4 Strain Analysis from HREM Image Analysis: Problematic of Very Thin Foils

9.5 Conclusions

Acknowledgment

References

Chapter 10: Determination of Elastic Strains Using Electron Backscatter Diffraction in the Scanning Electron Microscope

10.1 Introduction

10.2 Generation of Electron Backscatter Diffraction Patterns

10.3 Strain Determination Through Lattice Parameter Measurement

10.4 Strain Determination Through Pattern Shift Measurement

10.5 Sampling Strategies: Sources of Errors

10.6 Resolution Considerations

10.7 Illustrative Application

10.8 Conclusions

Acknowledgments

References

Chapter 11: X-Ray Diffraction Analysis of Elastic Strains at the Nanoscale

11.1 Introduction

11.2 Strain Field from Intensity Maps around Bragg Peaks

11.3 Average Strains from Diffraction Peak Shift

11.4 Local Strains Using Submicrometer Beams and Scanning XRD

11.5 Local Strains Derived from the Intensity Distribution in Reciprocal Space

11.6 Phase Retrieval from Strained Crystals

11.7 Conclusions and Perspectives

References

Chapter 12: Diffuse X-Ray Scattering at Low-Dimensional Structures in the System SiGe/Si

12.1 Introduction

12.2 Self-Organized Growth of Mesoscopic Structures

12.3 X-Ray Scattering Techniques

12.4 Data Evaluation

12.5 Results

12.6 Summary

References

Chapter 13: Direct Measurement of Elastic Displacement Modes by Grazing Incidence X-Ray Diffraction

13.1 Introduction

13.2 Elastic Displacement Modes: Analysis and GIXD Observation

13.3 Self-Organized Surfaces

13.4 Vicinal Surfaces

13.5 Conclusion

Acknowledgments

References

Chapter 14: Submicrometer-Scale Characterization of Solar Silicon by Raman Spectroscopy

14.1 Introduction

14.2 Crystal Orientation

14.3 Analysis of Stress and Strain States

14.4 Measurement of Free Carrier Concentrations

14.5 Concluding Remarks

References

Chapter 15: Strain-Induced Nonlinear Optics in Silicon

15.1 Introduction

15.2 Fundamentals of Second Harmonic Generation in Nonlinear Optical Materials

15.3 Second Harmonic Generation and Its Relation to Structural Symmetry

15.4 Strain-Induced Modification of Second-Order Nonlinear Susceptibility in Silicon

15.5 Strained Silicon in Integrated Optics

15.6 Conclusions

References

Index

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Preface

The development of future integrated (“smart”) micro- and nanosystems is generally focusing on further improvements of functionality and performance, enhancement of miniaturization and integration density, and extension into new application fields. In addition to any of these technological developments, reliability, quality, and manufacturing yield are key prerequisites for the development of any complex innovative (“smart”) micro-/nanosystem application. Consequently, new methods, instruments, and tools adjusted to the specific boundary conditions of the miniaturization level down to the nanoscale have to be provided allowing the investigation and understanding of the microstructure, possible failure processes, and reliability risks. In addition, methods and tools allowing the addressing and measurement of locally affected material properties, such as residual stresses, in combination with the microstructure are required. Such instruments and techniques are required to support a focused and rapid technological development and the time-efficient design of components and smart systems.

The particular results of microstructure and stress characterization do not only provide the basis for technological process step improvement but are also required for advanced simulation approaches and models that can be used to consider reliability properties already during the product development stage (“design for reliability” concept). Such concepts gain increasing importance since they allow to reduce time-to-market and development cost.

Present local stress and strain measurements on the nanoscale are based on special transmission electron microscopy techniques such as CBED, HRTEM-GPA, or holographic dark field technology, special scanning electron microscopy techniques such as EBSD or adapted X-ray diffraction techniques such as coherent X-ray diffraction. This book brings together leading groups in these different disciplines to apply these techniques for local strain and stress measurement and its theoretical background.

The book consists of three parts. Part One addresses the fundamentals of stress and strain on the nanoscale including an introduction to thermodynamics, kinetics, and models of elasticity, plasticity, and relaxation. Part Two addresses applications where stress and strain on the nanoscale are relevant such as SiGe devices or nanowires. In Part Three, techniques for measuring stress and strain on the nanoscale are presented such as CBED-TEM, EBSD-REM, different ways to use X-rays, Raman, and nonlinear optical methods.

To our knowledge, it is for the first time that this compendium combines theory, measurement techniques, and applications for stress and strain on the nanoscale. We believe that with increasing complexity of nanoscale devices, the increasing amount of the integration of various technologies, and various aspect ratios, it will be crucial to understand in detail processes and phenomena of nanostress.

This work was stimulated by the cooperation of the Fraunhofer Society, the Max-Planck-Society, the Carnot Association, and the CNRS via the C'Nano-PACA.

This book is dedicated to Prof. Ulrich Gösele, who coinitiated this project.

February 28, 2011                                                                                                                 Ralf Wehrspohn

Halle and Marseille                                                                                                                Margrit Hanbcken

Pierre Müller        

List of Contributors

Michael Becker Max Planck Institute of Microstructure Physics Experimental Department II Weinberg 2 06120 Halle Germany

Sandrine Brochard Institut PPRIME – CNRS UPR 3346 Département de Physique et de Mécanique des Matériaux Espace Phymat, BP 30179 86962 Futuroscope Chasseneuil Cedex France

Christian Bohley Martin-Luther-University Institute of Physics Heinrich-Damerow – Str. 4 06120 Halle Germany

and

Martin-Luther-University Centre for Innovation Competence SiLi-nano Karl-Freiherr-von-Fritsch-Str. 3 06120 Halle (Saale), Germany

Marie-José Casanove CNRS-UPS Centre d'Elaboration de Matériaux et d'Etudes Structurales 29, rue Jeanne Marvig, BP 94347 31055 Toulouse Cedex 4 France

Silke Christiansen Max Planck Institute for the Science of Light Guenther-Scharowsky – Str. 1 91058 Erlangen Germany

Stéphanie Escoubas Aix-Marseille Université IM2NP, Faculté des Sciences et Techniques Campus de Saint-Jérôme Avenue Escadrille Normandie Niemen, Case 142 13397 Marseille Cedex France

and

CNRS, IM2NP (UMR 6242) Faculté des Sciences et Techniques Campus de Saint-Jérôme Avenue Escadrille Normandie Niemen, Case 142 13397 Marseille Cedex France

Joël Eymery CEA/CNRS/Université Joseph Fourier CEA, INAC, SP2M 17 rue des Martyrs 38054 Grenoble Cedex 9 France

Stefan Flachowsky GLOBALFOUNDRIES Fab 1 Wilschdorfer Landstraße 101 01109 Dresden Germany

Christophe Gatel CNRS-UPS Centre d'Elaboration de Matériaux et d'Etudes Structurales 29, rue Jeanne Marvig, BP 94347 31055 Toulouse Cedex 4 France

Frank Glas CNRS Laboratoire de Photonique et de Nanostructures Route de Nozay 91460 Marcoussis France

Julien Godet Institut PPRIME – CNRS UPR 3346 Département de Physique et de Mécanique des Matériaux Espace Phymat, BP 30179 86962 Futuroscope Chasseneuil Cedex France

Ulrich Gösele† Max Planck Institute of of Microstructure Physics Weinberg 2 06120 Halle Germany

Angelika Hähnel Max Planck Institute of Microstructure Physics Weinberg 2 06120 Halle Germany

Margrit Hanbücken CINaM-CNRS Campus de Luminy, Case 913 3288 Marseille Cedex 9 France

Michael Hanke Paul-Drude-Institute for Solid State Electronics Hausvogteiplatz 5-7 10117 Berlin Germany

Jan Hoentschel GLOBALFOUNDRIES Fab 1 Wilschdorfer Landstraße 101 01109 Dresden Germany

Manfred Horstmann GLOBALFOUNDRIES Fab 1 Wilschdorfer Landstraße 101 01109 Dresden Germany

Michael Krause Fraunhofer IWM Walter-Hülse – Str. 1 06120 Halle Germany

Laurence Masson CINaM-CNRS Campus de Luminy, Case 913 3288 Marseille Cedex 9 France

Jean-Sébastien Micha INAC/SPrAM UMR 5819 (CEA-CNRS-UJF) CEA-Grenoble 17 rue des Martyrs 38054 Grenoble Cedex 9 France

Christine Mottet CINaM – CNRS Campus de Luminy, Case 913 13288 Marseille Cedex 9 France

Oussama Moutanabbir Max Planck Institute of of Microstructure Physics Weinberg 2 06120 Halle Germany

Pierre Müller Aix Marseille Université Center Interdisciplinaire de Nanoscience de Marseille UPR CNRS 3118 Campus de Luminy, Case 913 13288 Marseille Cedex 9 France

Shobhana Narasimhan JNCASR Theoretical Sciences Unit Jakkur 560 064 Bangalore India

Olivier Perroud Aix-Marseille Université IM2NP, Faculté des Sciences of et Techniques Campus de Saint-Jérôme Avenue Escadrille Normandie of Niemen, Case 142 13397 Marseille Cedex France

and

CNRS, IM2NP (UMR 6242) Faculté des Sciences et Techniques Campus de Saint-Jérôme Avenue Escadrille Normandie Niemen, Case 142 13397 Marseille Cedex France

Laurent Pizzagalli Institut PPRIME – CNRS UPR 3346 Département de Physique et de Mécanique des Matériaux Espace Phymat, BP 30179 86962 Futuroscope Chasseneuil Cedex France

Anne Ponchet CNRS-UPS Centre d'Elaboration de Matériaux of et d'Etudes Structurales 29, rue Jeanne Marvig, BP 94347 31055 Toulouse Cedex 4 France

Matthias Petzold Fraunhofer Institute for Mechanics of of Materials Halle Walter-Hülse-Str.1 06120 Halle

Geoffroy Prévot Université Pierre et Marie Curie-Paris 6 UMR CNRS 7588, Institut des NanoSciences de Paris Campus Boucicaut, 140 rue de Lourmel 75015 Paris France

Manfred Reiche Max Planck Institute of of Microstructure Physics Weinberg 2 06120 Halle Germany

Vincent Repain CNRS et Université Paris Diderot Matériaux et Phénomènes Quantiques Bâtiment Condorcet – Case 7021 75205 Paris France

Odile Robach CEA-Grenoble INAC/SP2M/NRS 17 rue des Martyrs 38054 Grenoble Cedex 9 France

Christian Roucau CNRS-UPS Centre d'Elaboration de Matériaux et d'Etudes Structurales 29, rue Jeanne Marvig, BP 94347 31055 Toulouse Cedex 4 France

Sylvie Rousset CNRS et Université Paris Diderot Matériaux et Phénomènes Quantiques Bâtiment Condorcet – Case 7021 75205 Paris France

Houda Sahaf CINaM-CNRS Campus de Luminy, Case 913 3288 Marseille Cedex 9 France

George Sarau Max Planck Institute of Microstructure Physics Experimental Department II Weinberg 2 06120 Halle Germany

and

Max Planck Institute for the Science of Light Guenther-Scharowsky – Str. 1 91058 Erlangen Germany

Volker Schmidt Max Planck Institute of Microstructure Physics Experimental Department II Weinberg 2 06120 Halle Germany

Clemens Schriever Martin-Luther-University Institute of Physics Heinrich-Damerow – Str. 4 06120 Halle Germany

and

Martin-Luther-University Centre for Innovation Competence SiLi-nano Karl-Freiherr-von-Fritsch-Str. 3 06120 Halle (Saale), Germany

Olivier Thomas Aix-Marseille Université IM2NP, Faculté des Sciences et Techniques Campus de Saint-Jérôme Avenue Escadrille Normandie Niemen, Case 142 13397 Marseille Cedex France

and

CNRS, IM2NP (UMR 6242) Faculté des Sciences et Techniques Campus de Saint-Jérôme Avenue Escadrille Normandie Niemen, Case 142 13397 Marseille Cedex France

Nicolas Vaxelaire Aix-Marseille Université IM2NP, Faculté des Sciences et Techniques Campus de Saint-Jérôme Avenue Escadrille Normandie Niemen, Case 142 13397 Marseille Cedex France

and

CNRS, IM2NP (UMR 6242) Faculté des Sciences et Techniques Campus de Saint-Jérôme Avenue Escadrille Normandie Niemen, Case 142 13397 Marseille Cedex France

Ralf B. Wehrspohn Martin-Luther-University Institute of Physics Heinrich-Damerow – Str. 4 06120 Halle Germany

and

Fraunhofer Institute for Mechanics of Materials Halle Walter-Hülse-Str. 1 06120 Halle Germany

Joerg V. Wittemann Max Planck Institute of Microstructure Physics Experimental Department II Weinberg 2 06120 Halle Germany

Part One

Fundamentals of Stress and Strain on the Nanoscale

Chapter 1

Elastic Strain Relaxation: Thermodynamics and Kinetics

Frank Glas

1.1 Basics of Elastic Strain Relaxation

1.1.1 Introduction

Although frequently used, the phrase elastic strain relaxation is difficult to define. It usually designates the modification of the strain fields induced in a solid by a transformation of part or whole of this solid. At variance with plastic relaxation, in crystals, elastic relaxation proceeds without the formation of extended defects, thereby preserving lattice coherency in the solid.

Elastic strain relaxation is intimately linked with the notion of instability. Indeed, the transformation considered is often induced by the change of a control parameter (temperature, forces applied, flux of matter, etc.). It may imply atomic rearrangements. Usually the realization of the instability is conditioned by kinetic processes (in particular, diffusion), which themselves depend on the stress state of the system. Elastic relaxation may also occur during the formation of part of a system, for instance, by epitaxial growth. The state with respect to which the relaxation is assessed may then exist not actually, but only virtually, as a term of comparison (e.g., the intrinsic state of a mismatched epitaxial layer grown on a substrate). Moreover, it is often only during growth that the kinetic processes are sufficiently active for the system to reach its optimal configuration.

In the present introductory section, we give a general principle for the calculation of strain relaxation and briefly discuss some analytical and numerical methods. In the next sections, we examine important cases where elastic strain relaxation plays a crucial part. Section 1.2 deals with strain relaxation in substitutional alloys with spatially varying compositions and with the thermodynamics and kinetics of the instability of such alloys against composition modulations. Section 1.3 introduces a kinetic process of major importance, namely, diffusion, and summarizes how it is affected by elastic effects. Section 1.4 treats the case of a homogeneous mismatched layer of uniform thickness grown on a substrate. Section 1.5 shows how a system with a planar free surface submitted to a nonhydrostatic stress is unstable with respect to the development of surface corrugations. Finally, Section 1.6 briefly recalls how the presence of free surfaces in objects of nanometric lateral dimensions, such as quantum dots or nanowires (NWs), permits a much more efficient elastic strain relaxation than in the case of uniformly thick layers.

1.1.2 Principles of Calculation

At given temperature and pressure, any single crystal possesses a reference intrinsic mechanical state in which the strains and stresses are zero, namely, the state defined by the crystal lattice (and the unit cell) of this solid under bulk form. If the crystal experiences a transformation (change of temperature, phase transformation, change of composition, etc.), this intrinsic mechanical state changes to , where again strains and stresses are zero (Figure 1.1 a). The corresponding deformation is the stress-free strain (or eigenstrain) with respect to state ; for instance, for a change of temperature , , where is the thermal dilatation coefficient. If the crystal is mechanically isolated, it simply adopts its new intrinsic state ; it is then free of stresses. This is not the case if the transformation affects only part of the system. We then have two extreme cases. The transformation is incoherent if it does not preserve any continuity between the crystal lattices of the transformed part and of its environment. If, on the contrary, lattice continuity is preserved at the interfaces, the transformation is coherent. This chapter deals with the second case.

Figure 1.1 (a) Stress-free strain relative to the inclusion. (b) The three stages of an Eshelby's process.

Let us call inclusion the volume that is transformed and matrix the untransformed part of the system (indexed by exponents I and M). Coherency is obviously incompatible with the adoption by the inclusion of its stress-free state , the matrix remaining unchanged. The system will thus relax, that is, suffer additional strains, which in general affect both inclusion and matrix. It is a strain relaxation in the following sense: if one imagines the inclusion having been transformed (for instance, heated) but remaining in its original reference mechanical state (which restores coherency, since the matrix has not been transformed from state ), it is subjected to stresses, since forces must be applied at its boundary to bring it from its new intrinsic state back to . With these stresses is associated an elastic energy. The coherent deformation of the whole system constitutes the elastic relaxation.

This suggests a way to calculate relaxation, Eshelby's method (Figure 1.1b) [1]:

1) One applies to the transformed inclusion (state ) the strain , which brings it back to state . This implies exerting on its external surface (whose external normal has components ) the forces per unit area, where is the stress associated1 with the stress-free strain .

2) Having thus restored coherency between inclusion and matrix, one may reinsert the former into the latter. The only change that then occurs is the change of the surface density of forces applied at stage (1) into a body density , since the surface of the inclusion becomes an internal interface.2

3) The resulting state is not a mechanical equilibrium state, since forces must be applied to maintain it. One then lets the system relax by suppressing these forces, that is, by applying forces , while at the same time maintaining coherency everywhere. One thus has to compute the strain field, in the inclusion () and in the matrix (), solution of the elasticity equations for body forces , under the coherency constraint, which amounts to equal displacements at the interface.

We may generalize this approach by not differentiating matrix and inclusion. The whole system experiences a transformation producing an inhomogeneous stress-free strain (defined at any point ) with respect to initial uniform state (perfect crystal). One then applies the body forces producing strain , namely, , where is the stress associated with strain . Finally, one calculates the relaxation field , the solution of the elastic problem with forces that preserves coherency everywhere (displacements must be continuous). It is important to specify the reference state with respect to which one defines the final state of the system. It is often easier to visualize the relaxed state relative to the uniform state ; strain is then simply . If, on the contrary, the elastic energy W stored in the system is to be calculated, we must take as a reference state for each volume element its intrinsic state after transformation (), with respect to which the total strain is . Hence, , where is the strain associated with and where the integral is taken over the whole volume (in the reference state).

In case of an inclusion (Figure 1.1), one may easily show that

(1.1)

This is a fundamental result obtained by Eshelby [1]. In particular, the total elastic energy depends only on the stress in the inclusion.

An example of application to an infinite system with a continuously varying transformation will be given in Section 1.2. Eshelby's method may also be adapted to other problems. In particular, if the interface between matrix and inclusion does not entirely surround the latter (which happens if the inclusion has a free surface), it is not necessary to apply strain to the inclusion at stage 1. It suffices to apply a strain that restores the coherency in the interface, which may make the solution of the problem simpler. An example is given in Section 1.4.

1.1.3 Methods of Calculation: A Brief Overview

The problem thus consists in determining the fields relative to stage 3 of the process. One has to calculate the elastic relaxation of a medium subjected to a given density of body forces. In addition to numerical methods, for instance, those based on finite elements, there exist several analytical methods for solving this problem, in particular, the Green's functions method [2] and the Fourier synthesis method.

In elasticity, Green's function is defined as the component along axis i of displacement at point r caused by a unit body force along j applied at point . For a solid with homogeneous properties, it is a function of the vector joining the two points. One easily shows that for an elastically linear solid (with elastic constants ), the displacement field at stage 3 is

(1.2)

where the integral extends to all points of the volume. Green's functions depend on the elastic characteristics of the medium, but, once determined, any problem relative to this medium is solved by a simple integration. However, if the Green's functions for an infinite and elastically isotropic solid have been known since 1882, only a few cases have been solved exactly. If the medium is not infinite in three dimensions, the Green's functions also depend on its external boundary and on the conditions that are imposed to it. For epitaxy-related problems, the case of the half-space (semi-infinite solid with planar surface) is particularly interesting. These functions have been calculated for the elastically isotropic half-space with a free surface (no external tractions) [3, 4]. Mura's book gives further details [2]. The method also applies to the relaxation of two solids in contact via a planar interface; in this case, this surface is generally not traction-free and the boundary conditions may be on these tractions or on its displacements. Pan has given a general solution in the anisotropic case, valid for all boundary conditions [5].

In the Fourier synthesis method, one decomposes the stress-free strain distribution into its Fourier components: , where is the running wave vector. In linear elasticity, the solution is simply the sum, weighted by the Fourier coefficients , of the solutions relative to each periodic wave of wave vector , which are themselves periodic with the same wave vector. If the system is infinite, the elementary solution is easily determined (see Section 1.2.2). The only nontrivial point is then the integration. This method allows one to treat elegantly the stress-free strain discontinuities, as happening at the interface between a matrix and a misfitting inclusion3 having the same elastic constants.

The number of problems solved by these methods steadily increases. As for inclusions, let us only mention, in addition to Eshelby's pioneering work on the ellipsoidal inclusion [1], the case of parallelepipedic inclusions in an infinite matrix [6] and in a half-space [7, 8] and that of the truncated pyramidal inclusions in an infinite matrix [9] and in a half-space [10]. These are important for being the shapes commonly adopted by semiconducting quantum dots. For a given inclusion, the elastic relaxation may be deeply modified by a free surface, on which the tractions must vanish (Figure 1.2).

Figure 1.2 Comparison of the elastic relaxation of a truncated square-base pyramidal inclusion in an infinite matrix (a) and in a half-space (b). Maps in the symmetry plane of strain component normalized to the intrinsic misfit of the inclusion with respect to the matrix. Analytical calculation by the Fourier method (see also Ref. [10]). Thick blue lines mark the inclusion contour and the free surface, and the intensity scale is the same for (a) and (b).

1.2 Elastic Strain Relaxation in Inhomogeneous Substitutional Alloys

As a first example of strain relaxation, we examine the common case of an alloy whose stress-free state (for instance, its lattice parameter) depends on its composition. If composition variations that preserve lattice coherency develop in an initially homogeneous alloy, internal strains and stresses appear. In this case, elucidating strain relaxation amounts to calculating these fields. We shall see that strains may deeply affect the stability of such an alloy. However, the stability is not determined only by elastic effects. To provide a term of comparison, we shall first consider alloys where compositions variations induce no strain (Section 1.2.1), before turning to the calculation of the elastic fields (Section 1.2.2) and to the way in which they alter the alloy thermodynamics (Section 1.2.3). Finally, Section 1.4 discusses how the presence of a free surface affects strain relaxation and hence stability.

1.2.1 Spinodal Decomposition with No Elastic Effects

Let us consider a bulk binary AB substitutional alloy with atomic concentrations and c in species A and B.4 With each thermodynamic quantity is associated a mixing quantity, the difference of this quantity between the alloy and the same numbers of its atomic constituents taken as pure solids. In the regular solution model, the mixing free enthalpy (per atom) of the disordered alloy at temperature T is , where is an energy, the interaction parameter (here taken per atom), and is the Boltzmann's constant (Figure 1.3 a). If , the alloy tends to decompose, since alloying its constituents produces an energy . However, at high temperatures, entropy may prevent decomposition. The equilibrium state of the system at given temperature and pressure is found by minimizing at constant numbers of atoms.

Figure 1.3 (a) Mixing free enthalpy at two temperatures (full lines), with enthalpic and entropic contributions (dashed lines). (b) High temperature case. (c) Low temperature case; tangent construction, equilibrium compositions , and spinodal compositions , corresponding to inflexion points (circles). (d) Boundaries of miscibility gap (full line) and spinodal gap (dashed lines).

Depending on temperature T, adopts two different forms, with either a single or a double minimum (Figure 1.3 a).5 At high T (Figure 1.3b), the mixing free enthalpy of a homogeneous alloy of any composition is less than that of any mixture of two homogeneous alloys of compositions and into which it might decompose (the latter, , is given by the cord construction), so the alloy is stable. On the contrary, for , with , the tangent construction gives two stable equilibrium compositions , , into which any alloy with intermediate composition tends to decompose (Figure 1.3c). However, two different behaviors are expected. For an alloy of composition intermediate between the compositions and corresponding to the inflexion points of the curve (i.e., such that ), decomposition into two alloys is always energetically favorable, even if these two alloys differ infinitesimally in composition (curve is above its cord). Conversely, for an alloy composition such as , between a stable composition and an inflexion point (), it is only decomposition into two compositions differing by a finite amount that is favored (curve is below its cord). The homogeneous alloy is unstable in the first case and metastable in the second. In the plane (Figure 1.3d), curves , enclose the miscibility gap, inside which the spinodal curve, locus of points , , separates the metastable (M) and unstable (or spinodal, I) domains. The process through which an alloy quenched below starts decomposing via composition variations of arbitrary small amplitude is spinodal decomposition.

The same results are obtained by linear stability analysis [11]. This general method consists in studying the instability of a system not against an arbitrary perturbation but against a Fourier component of the latter. If the problem is linear (a linear combination of solutions is a solution, the boundary conditions being themselves linear), global instability is equivalent to the existence of at least one unstable elementary perturbation. Here, we study the instability of an alloy of average composition against a sinusoidal composition modulation of amplitude along the direction : . The mixing free enthalpy of the modulated alloy, found by averaging over a modulation wavelength , is at order 2 in . The excess of mixing enthalpy due to modulation is . We thus recover the previous result, namely, an alloy of average composition is unstable () at temperature against a composition modulation of vanishing amplitude if .6

The foregoing analysis assumes that fractioning the alloy into two phases or modulating its composition produces no excess energy.7 It is however not generally the case, and this can be taken into account by adding to the energy density of the composition modulation a Landau-type phenomenological gradient term that opposes abrupt composition variations [12]. This produces a critical decomposition wavelength, lower wavelengths being energetically unfavorable. The gradient term may be of purely chemical origin, but, in addition, in alloys with size effects that decompose coherently, stresses appear between regions having different compositions and generate additional elastic energy. The next section describes the modifications of the previous results due to these stresses and to the way in which they relax.

1.2.2 Elastic Strain Relaxation in an Alloy with Modulated Composition

Let us assume, for simplicity, that the alloy is of cubic structure or elastically isotropic and that its intrinsic lattice parameter varies linearly with composition (Vegard's law) with and , the relative lattice mismatch between the pure constituents and . If the alloy composition lies within the metastable domain M, one may expect the nucleation of finite volumes of phases with distinct compositions, hence with different lattice parameters, and decomposi-tion may be coherent or not, depending on the size and composition contrast. In the unstable domain I, the transformation into the stable phases may however occur via the amplification of composition modulations starting with zero amplitude (Section 1.2.1). One then expects the decomposition to be coherent, at least at the start of the process. This is the case we treat here. Two major questions then arise. What are the elastic strain fields induced by a given composition distribution? How does the associated elastic energy modify the stability of the alloy?

To answer these questions, let us consider again a sinusoidal composition modulation in direction . From Vegard's law, this induces a spatial modulation of intrinsic parameter with amplitude . As regards elasticity (Section 1.1.2), this corresponds to a stress-free strain modulation with respect to the homogeneous state ().8 From Section 1.1.2, the elastic relaxation with respect to the latter is obtained by imposing a density of body forces . If the alloy is elastically isotropic,9. We look for a relaxation displacement field that is continuous (coherency condition) and has the same wavelength as the perturbation: . The relaxation strain is then 10 and the associated stress . From the equilibrium between stresses and forces , one gets and . The relaxation field is thus a tetragonal strain modulation along in phase with the intrinsic modulation but amplified by factor [12]. The relaxed lattice parameter remains equal to in the directions normal to . The elastic energy is that of the total strain fields () (Section 1.1.2). Its density is easily found to be , where is the Young's modulus normalized by Avogadro's number in order to obtain a density per atom.

What is the meaning of relaxation here? With respect to the intrinsic (reference) state of each volume element of the modulated alloy, in which elastic energy is zero by definition, there is an increase of elastic energy due to the coherency between these elements. On the contrary, the energy is lower than that in the state where all elements (with different compositions) would adopt the average lattice parameter (virtual reference state at the end of stage 2; Figure 1.4). If, at least during the early stages of decomposition, the system prefers continuous composition variations and coherent relaxation, it is because the formation of finite domains with distinct compositions, which would all adopt their intrinsic lattice parameter, would induce interfacial defects whose cost in energy would be even higher (see also Section 1.4).

1.2.3 Strain Stabilization and the Effect of Elastic Anisotropy

The total excess of mixing free enthalpy due to modulation is found by adding the elastic relaxation energy to the term calculated in Section 1.2.1 without taking stresses into account. One finds . The condition of instability of alloy with composition against modulations of vanishing amplitude thus changes from to

(1.3)

Since , the extra term is positive; so the condition for instability is more restrictive in terms of composition. If the mixing enthalpy at low is a double-well curve (Figure 1.3c), we see that the domain of unstable compositions is reduced and the critical temperature lowered, an effect often called stress stabilization. In the regular solution model, the critical temperature decreases from to . This reduction may reach several hundreds of kelvins in some metallic alloys [12]. The elastic contribution may even be large enough to render negative the calculated , which means that the instability is then totally suppressed (no composition satisfies Eq. (1.3)).

The previous calculations, based on isotropic elasticity, do not specify any favored direction of modulation. To extend the calculations to cubic crystals [13], it suffices to replace term in Eq. (1.3) by a modulus depending on modulation direction . In particular, if the elastic constants satisfy to . Modulations then tend to form in the soft directions of type . These considerations are corroborated by the observation in spinodally decomposed alloys of a characteristic microstructure, manifested in the transmission electron microscopy images by a modulated contrast in the soft directions [11, 14]. More generally, in elastically anisotropic materials, the shape of inclusions and their relative disposition tend to be determined by elastic relaxation [11].

1.2.4 Elastic Relaxation in the Presence of a Free Surface

The presence of a free surface may also deeply affect strain relaxation. In the infinite solid considered so far, the elastic energy of any composition distribution is the sum of those of its Fourier components, which do not interact since if . This does not necessary hold in a solid bounded by a surface (see Section 1.5). However, this is still true for a planar half-space. Moreover, for a modulation with a wave vector parallel to the surface, the elastic energy may be considerably reduced with respect to the infinite solid. This reduction stems from the extra stress relaxation permitted by the free surface. The optimal modulations have an amplitude that is exponentially attenuated in the direction normal to the surface. Strain stabilization is thus less pronounced than in the bulk and the critical temperature increases accordingly [8]. In the regular solution model, one finds a new critical temperature , such that [15, 16].11 Composition modulations in directions parallel to the substrate have indeed been observed in epitaxial layers of semiconducting alloys [17, 18].

More generally, the strain relaxation of misfitting inclusions may be strongly affected if they lie close to a free surface, typically within a distance on the order of their dimensions (Figure 1.2) [10].

1.3 Diffusion

Many transformations (such as considered in Section 1.1.1) require a redistribution of matter in the system to become effective. Hence, the realization of the instability is conditioned by kinetic processes, in particular, diffusion. In this sense, the elastic relaxation that accompanies the transformation is also conditioned by diffusion. However, one usually considers that the timescale for diffusion is much longer than that for the mechanical adjustment (relaxation) of the system to the instantaneous distribution of atoms, so the system is continuously in a state of mechanical equilibrium (but of course not in global thermodynamic equilibrium) during the transformation (quasistatic approximation). This is not to say that diffusion and strain relaxation are independent: we shall see in Section 1.3.2 that diffusion is affected by the elastic strain and stress fields. Before this, we briefly consider diffusion without elastic effects.

1.3.1 Diffusion without Elastic Effects

As an example of how diffusion conditions the realization of an instability, consider a bulk alloy AB subject to decomposition (as in Section 1.2), and first ignore the elastic effects. If the homogeneous alloy is quenched below its critical temperature , it becomes unstable and tends to decompose (Section 1.2.1). The concentration of, say, component B becomes nonuniform. According to Fick's law, this induces a diffusive flux of B atoms, , where is the diffusion coefficient of B [19] and is the atomic volume in the homogeneous reference state of the alloy.12 Setting , we have . Since , this flux tends to smooth the concentration gradient, which should inhibit the formation of the composition modulation. It is because the B flux is not limited to Fick's diffusive term that the modulation may actually develop.

Indeed, generally, to the diffusive flux must be added a transport flux depending on the force exerted on each B atom [19]. If this force derives from a potential (), the Nernst–Einstein equation (demonstrated by canceling the total flux at equilibrium and by using the appropriate statistics) leads to . For instance, in the case of charged particles in an electric field, the transport flux is directly related to the electrostatic force exerted on each particle. Now, if the alloy is ideal, the chemical potential of species B at concentration is , where is the equilibrium concentration at temperature , so the diffusive flux becomes and the total flux is , where is a generalized potential, including external forces.

In the case of the alloy, a similar approach applies, but the definition of the appropriate potential and the generalization of Fick's law require some care. If the alloy is substitutional, A and B share the same crystal lattice and, in the absence of vacancies, any B atom leaving a site must be replaced by an A atom so that the fluxes of A and B are opposite: . The B flux then becomes , where is a phenomenological diffusion mobility, and are the local and concentration-dependent values of the chemical potentials of A and B, and is the diffusion potential that replaces and generalizes the chemical potential. Since is the change of free enthalpy when an A atom is replaced by a B atom at the point considered, we have , where is the free enthalpy (per atom) of the alloy at composition c. Moreover, by definition of the free enthalpy of mixing introduced in Section 1.2, we have , where , are the position-independent chemical potentials of the pure elements. Hence, . It now becomes clear that in the spinodal domain, which is precisely defined by (Section 1.2.1), the diffusion flux of species B does not oppose the gradient (as follows from the sole consideration of Fick's diffusive flux) but actually tends to amplify it. In other words, as expected in this domain, the alloy is unstable against composition modulations of vanishing amplitude.

This calculation of the diffusion flux is actually much richer than simply confirming the static analysis carried out in Section 1.2.1. It opens the way to a study of the kinetics of decomposition (provided is known), since the evolution of the composition profile with time t obeys the usual conservation equation . The interested reader is referred to, for example, Cahn's publications [14]. This general approach also permits to take into account any effect that modifies the energy of the system (e.g., electric or magnetic fields) by simply adding its contribution to the free enthalpy g. In the next section, we show how the effect of elastic strain fields on diffusion can be calculated in this way.

1.3.2 Diffusion under Stress in an Alloy

The elastic contribution to the free enthalpy per atom is simply , where is the strain corresponding to the transformation of a given volume element from its intrinsic state to its final relaxed state (Figure 1.1b and 1.4) and is the associated stress (see Section 1.1.2). Since is defined only formally ( being the strain between the independent stress-free volume elements and the final relaxed coherent state), it is preferable to express the density of elastic energy as a function of the elastic fields describing the transformation of the homogeneous alloy with composition into the modulated and relaxed alloy (the extreme stages of Figure 1.4), namely, . Elasticity introduces a complication, namely, nonlocal effects, since the stress and strain at a given point depend not only on local composition (as was the case with the chemical potentials) but also on the whole concentration distribution. Nevertheless, it is possible to express the effect of elasticity as an additional term in the diffusion potential [20], so that the diffusion equation becomes , with

Figure 1.4 Eshelby process for a bulk alloy with modulated composition.

(1.4)

where are the elastic compliances. If the latter do not depend on composition, the third term of Eq. (1.4) disappears. If this is the case and if the material is cubic or elastically isotropic, then , where is the relative lattice mismatch between the pure constituents introduced in Section 1.2.2; so , where is simply the local dilatation. Note that only nonhomogeneous stresses affect diffusion in this way, although, in addition, stress may affect the elastic constants and the diffusion coefficients. For more details on diffusion under stress, including surface diffusion, see Refs [21–23].

Spinodal decomposition may be studied in this way, considering a possible gradient energy of chemical origin (Section 1.2.1) plus the elastic energy. This leads to different growth rates for perturbations of different wavelengths, with a rather narrow peak centered around a fastest developing wavelength. Experimentally, spinodally decomposed alloys indeed tend to exhibit a microstructure with a fairly well-defined decomposition periodicity [14].

1.4 Strain Relaxation in Homogeneous Mismatched Epitaxial Layers

1.4.1 Introduction

Knowing which amount of a given material can be deposited coherently on a mismatched substrate is of great importance in the field of epitaxy. For semiconducting materials in particular, the extended defects that form when plastic relaxation occurs often affect deleteriously the electrical and optical properties of heterostructure-based devices, and great care is usually taken to remain in the coherency domain during growth. In a simple equilibrium picture, the transition between elastic and plastic relaxation is governed by the total energy of the system. It is thus important to analyze elastic strain relaxation in the coherent case. In the present section, we consider the standard case of a layer of uniform thickness, before turning to possible thickness variations in Section 1.5.

We consider mismatched heterostructures formed by depositing a homogeneous layer L of uniform thickness t () onto a semi-infinite planar substrate S (), both being single crystals. In the spirit of Section 1.1.2, one may consider that in its intrinsic stress-free state, L results from an elastic distortion of S, described by the stress-free strain that transforms the intrinsic lattice of S into that of L [8, 24]. The methods of Section 1.1 may then be applied. We further assume that S and L have the same structure and differ only by the magnitude of their lattice parameters, so the stress-free strain is a pure isotropic dilatation, equal to the relative difference between the lattice parameters of the layer and of the substrate (or lattice mismatch), (which is positive if the layer is compressed by the substrate).13

If the deposit L extends infinitely parallel to the planar interface (plane ), the lattice mismatch makes it impossible for S and L to retain their intrinsic bulk stress-free states if the interface is coherent, simply because the spacings of the lattice planes that cross the interface are different for S and L. Hence the necessity of an accommodation of the lattice mismatch, which can take two extreme forms. If the coherency at the interface is preserved, thanks to a deformation of one or both materials, the accommodation is purely elastic. Conversely, accommodation may be realized plastically, via the formation of a network of misfit dislocations at the interface, which thereby becomes incoherent (section 1.1.1). Leaving aside plastic relaxation, the detailed discussion of which falls outside the scope of the present chapter, we shall briefly show how the methods of Section 1.1 give the solution of the elastic problem.

1.4.2 Elastic Strain Relaxation

If the deposit is much thinner than the substrate, one can safely consider (ignoring possible curvature effects) that only the former is strained while the latter retains its bulk lattice. Moreover, the layer has a free surface , so the substrate/layer interface does not entirely surround the layer. As mentioned in Section 1.1.2, it is then not necessary to apply strain at stage 1 of the Eshelby process. It suffices to apply a strain that restores lattice continuity across the S/L interface, that is, such that . This may be achieved by applying forces along x and y on the elementary cubes that compose the layer, but not along z, so at this (modified) stage 1, (Figure 1.5 a). Considering, to simplify, an elastically isotropic medium, the equations of elasticity (see footnote 9) indicate that one then has and (all nondiagonal strains and stresses are zero). Because the forces corresponding to , are uniform, they cancel mutually when the layer elements are reassembled at stage 2, so the corresponding density of body forces is zero. Moreover, since there are no forces along z, the layer is entirely free of applied forces at the end of stage (2), so that considering stage (3) becomes irrelevant. This adaptation of Eshelby's method thus offers a particularly simple solution of the problem.

Figure 1.5 (a) Tetragonal strain of a unit cell of parameter subjected to biaxial stress (here taken negative, corresponding to compression); . (b) Planar (dashed line) and corrugated (full line) surfaces. (c) Schematics of strain at crests (c) and valleys (v) with respect to intrinsic (i) and tetragonally strained (q) unit cells.

To summarize, elastic relaxation affects only the thin layer, which adapts its parameter to that of the substrate in the interface plane and strains tetragonally (extending or contracting, depending on the sign of ) in the normal (z) direction, with an amplification factor with respect to intrinsic strain , since the dilatation along z with respect to the substrate is . The corresponding elastic energy per unit volume of layer is readily found to be .

1.4.3 Critical Thickness

One of the major issues regarding strain accommodation in heterostructures is to find out which factors determine the mode of relaxation (elastic or plastic). It is usually observed that dislocations do not form until the growing layer reaches some critical thickness. Basically, such a critical thickness exists because the energies stored in the system per unit area scale differently with layer thickness h. In the coherent state (elastic relaxation), we have just seen that the energy is uniformly distributed in the layer, so the energy per unit area scales with h. On the contrary, in the plastically relaxed state, the density of dislocations of a given type that accommodates a given mismatch (ensuring that the material passes from the stress-free lattice parameter of the substrate to that of the layer across their interface) is fixed and inversely proportional to the relative mismatch [25]. In this case, as a first approximation, the layer is strained in the vicinity of the interface (because of the nonuniform strain fields of the dislocations) but quickly recover its stress-free parameter away from it, so the elastic energy per unit area does not depend on h. Actually, since the dislocations have a long-range strain field, the elastic energy increases with h, but only logarithmically, much more slowly than in the elastic case (it even saturates when the layer thickness becomes larger the dislocation spacing). In addition, the dislocation cores contribute a constant term to the energy. Hence, the energy stored is larger in the plastic case at low layer thicknesses and in the elastic case at high thicknesses.

These considerations are at the origin of the most widely used criterion for calculating for a given couple of materials, which consists in comparing the total energies of a given heterostructure in the coherent and plastically relaxed states as a function of thickness and in defining as that thickness at which the energy in the former becomes larger than in the latter. This is an equilibrium criterion, equivalent to finding the thickness at which the misfit-induced force acting on a preexisting dislocation tends to pull and extend it into the S/L interface [26–28]. Other criteria are of a kinetic nature and deal with the nucleation of the misfit dislocations or with their motion toward the interface. The critical thickness decreases rapidly when the S/L mismatch increases [26]. In practice, for typical semiconductor materials, for and for .

1.5 Morphological Relaxation of a Solid under Nonhydrostatic Stress

1.5.1 Introduction

Consider a homogeneous half-space , subjected to a uniform biaxial stress (exponent q) in plane and with a planar traction-free surface (). Assuming, for the sake of simplicity, that the medium is elastically isotropic, its response (relaxation) to this stress has been calculated in another context in Section 1.4.2:14 it is a uniform tetragonal strain , with ; the nondiagonal terms are zero (Figure 1.5 a).

It has been known for a few decades that if one abandons the arbitrary constraint that the free surface be planar (Figure 1.5b), the system can relax even more and adopt a different state, with lower elastic energy [29–31].15,16 Why such a morphological (planar → nonplanar) transformation might reduce the elastic energy is easily understood: at the crests of the surface, the system may deform not only in direction z but also laterally Figure 1.5c), since there is no matter to prevent it from doing so [31]. To confirm the decrease of the total energy, we must however also examine the strain in the valleys and allow for the fact that the area of the corrugated surface is larger than that of the planar one.

1.5.2 Calculation of the Elastic Relaxation Fields

In the present case, elastic relaxation stands for the modification of the strain fields accompanying the planar → nonplanar transformation. Since the latter implies an actual change of shape following a redistribution of matter, Eshelby's method (which deals with the change of intrinsic state of a given volume) is not adapted. Instead, we directly solve the elastic problem, the boundary condition being that the corrugated surface remains traction-free. Since this problem has no exact solution for an arbitrary surface profile, we study the elastic response of the system to an elementary perturbation (in the spirit of the linear stability analysis, Section 1.2.1), namely, a sinusoidal modulation (hereafter, undulation) along x of the position h of the surface along z, , measured with respect to the planar state (Figure 1.5b).17,18 Let us look for the fields , that have to be added to the “q” fields to obtain the total (equilibrium) field. Given the symmetry of the problem, no quantity depends on y; hence . For the same reason, , so for . The solution of such a plane strain problem is known to derive from an Airy function, solution of differential equation , via relations , , and [33]. Setting , one finds that must satisfy differential equation , the general solution of which is , with A, B, C, and D constants. Since the fields must remain finite for , one has . Finally,

(1.5)

since and hence . A shear strain thus appears in plane . The undulated surface with normal remains traction-free under total field . Hence,

(1.6)

the stress being calculated in . We now assume that the amplitude of the undulation is small compared to its wavelength (i.e., ), and compute the fields at first order in (even in the case of a sinusoidal perturbation, there is no exact solution at finite amplitude). Then, , . From Eqs (1.6) and (1.5), one gets . By using Eq. (1.5), one obtains the stresses, from which the relaxation strains derive via the appropriate relations (see footnote 9). The nonzero components of this strain field are as the following:

(1.7)

(1.8)

(1.9)

1.5.3 ATG Instability

The elastic energy W of the solid with undulated surface, per unit area of planar surface (reference state), is easily calculated from the total fields: . It follows that the variation of elastic energy per unit area when the system transforms from planar to undulated state is

(1.10)

the second equality being valid at lowest (second) order in . Replacing in Eq. (1.10) the “s” fields by their expressions calculated from Eqs (1.7)–(1.9) via the linear elasticity formulas, we get

(1.11)

Since , any surface undulation reduces the elastic energy of the system.19 This fundamental result specifies the driving force for the instability.

However, an undulation also increases the effective area of the free surface and induces a surface strain. Here, we only consider the first effect, which translates into an excess energy per unit reference area, equal to , where is the surface free energy (assumed to be independent of the slight orientation changes of the surface) and is the relative variation of area. The latter equals , that is, at order 2 in . The total energy variation per unit area due to the undulation is thus

(1.12)

The planar surface is unstable with respect to an undulation if . This is equivalent to the undulation wavenumber being less than the critical value . In other words, there exists a critical wavelength , such that the planar free surface of the biaxially stressed half-space is unstable with respect to any undulation with wavelength larger than , with

(1.13)

The existence of a critical wavelength is due to the fact that the undulation-induced fields (which reduce elastic energy) penetrate the solid over a depth of the order of (Eqs (1.5) and (1.7)–(1.9)), whereas the excess energy due to the increased area is independent of . Equation (1.12) indicates that an undulation with a given wavelength is all the more easy to create that is high and all the more difficult that surface energy is high.

This analysis can be generalized. The point is that the stress must be nonhydrostatic. The instability of a solid subjected to such stresses with respect to morphological perturbations of its surface is often called ATG (Asaro–Tiller–Grinfel'd) [29, 30]. It has been observed (with millimetric wavelengths) at the surface of crystals under uniaxial stress [34]. However, the ATG instability is particularly important for epitaxy. One indeed attributes to it the often-observed formation of undulations at the surface or at the interfaces of semiconducting layers mismatched with respect to their substrates [35, 36]. Indeed, as seen in Section 1.4, if the mismatch is , the semi-infinite substrate exerts biaxial strain .20 Hence, the free surface of the layer is unstable against surface undulations with wavelengths larger than the critical value given by Eq. (1.13). For typical strains on the order of a percent, this wavelength is only on the order of a few tens to a few hundreds of nanometers. This is a mode of strain relaxation that differs from the usual tetragonal distortion of uniformly thick layers (Section 1.4) by its morphology and its elastic fields and also from their plastic relaxation, since relaxation remains elastic (no extended defect appears). However, at the atomic scale, the surface undulation of a low-index surface corresponds to the modulation of the spacing of preexisting or newly created surface steps. One may indeed recover the instability by considering, for instance, a vicinal surface, slightly misoriented with respect to a high-symmetry orientation, and hence composed of facets separated by steps. It has been shown that under nonhydrostatic stress, the steps interact attractively and thus tend to accumulate in bunches [37].

1.5.4 Kinetics of the ATG Instability

Here also kinetics matter, since they condition the actual formation of the undulation. Srolovitz has treated in a simple fashion two mechanisms whereby a planar surface may undulate, namely, surface diffusion and evaporation/condensation [31]. In both cases, one obtains a mode that develops more rapidly than the others, with a wavelength on the order of (as in the case of spinodal decomposition; see Section 1.3.2). This explains simply why if Eq. (1.12) leads to a semi-infinite band of unstable wavelengths, the experiments show a rather well-defined wavelength. In the case of epitaxy, one must also take into account the influx of matter from the fluid phase (molecular beams, gas or liquid) in addition to the transport of matter along the surface. This has been done by Spencer et al. [38] who, in addition to substrate rigidity, identify two kinetic factors that tend to inhibit the formation of the undulation, namely, a low temperature (which reduces surface diffusion) and a high growth rate (which buries the undulations before they can develop).

1.5.5 Coupling between the Morphological and Compositional Instabilities

Let us consider a half-space of a regular solution alloy (A, B) with (Section 1.2.1). We know that if its free surface remains planar, it is unstable for against composition modulations with arbitrary wavelengths, if gradient energy is ignored (Section 1.2.4). We also know that if it remains homogeneous, its planar surface is unstable against undulations with wave vectors , at any (Section 1.5.3). In both cases, elastic relaxation is of primary importance: it determines and it is the driving force for undulation. In this section, we consider briefly how a possible coupling between the two instabilities affects their respective domains of existence.21 To answer this question, we calculate the elastic relaxation of a layer of average mismatch with a composition modulation along a direction parallel to the substrate/layer interface, but allowing a z-dependent amplitude; without loss of generality, we then write the modulation with , so that its stress free strain with respect to the substrate is . Assume that its free surface is undulated: . As in the case of the purely morphological perturbation (Section 1.5.2), the elastic relaxation fields cannot be computed exactly; we limit ourselves to the first order in both at [24]. The excess of elastic energy (per unit area) of the undulated/modulated state with respect to the planar/homogeneous state is then quadratic in these two variables:

(1.14)

where . In Eq. (1.14), the terms in and correspond respectively to a pure undulation (Eq. (1.12)) and to a composition modulation in a planar half-space [24]. Whatever the sign of product , the cross-product becomes negative with an appropriate choice of the sign of (the phase of the modulation). The elastic energy of the mixed perturbation is then less than the sum of the two perturbations taken separately. If we assume that and that and have the same sign, then for , the layer is more mismatched than on average, and it is interesting to have a larger relaxation there, hence also a crest (). The coupling vanishes if the layer is on average lattice matched to its substrate [15]. To obtain the total excess of free energy, we add to the excess free enthalpy of mixing and the excess surface energy, as done respectively in Sections 1.2.3 and 1.5.3 [24]. The conclusion is that not only the instability domains but also the very nature of the instability are modified. We must consider together the two parameters that in the case of uncoupled disturbances (Figure 1.6 a) have critical values separating stable and unstable domains, namely, T and k for the compositional (C) and morphological (M) instabilities, respectively. With coupling (Figure 1.6b), one finds in the plane an extended domain of morphocompositional (MC) instability. To each wavenumber k corresponds a critical temperature

Figure 1.6 Half-space under biaxial stress. (a) Instability domains with respect to separate lateral composition modulation (C) or surface undulation (M), in plane (wave vector, temperature) (S: stability). (b) Domain of joint morphocompositional instability (MC).

(1.15)

where is the bulk critical temperature (Section 1.2.3).

As usual, this thermodynamical analysis must be completed by a kinetic analysis that will decide if the instability will actually develop, depending on the matter transport mechanisms available [39–41].

1.6 Elastic Relaxation of 0D and 1D Epitaxial Nanostructures

We have seen in Section 1.4 that one way to prevent the formation of dislocations during the growth of a mismatched epitaxial layer on a substrate is to keep the layer thickness below its critical value for plastic relaxation. This becomes impractical at high lattice mismatch , since the critical thickness decreases rapidly when increases (Section 1.4.3). In such cases, one may play on the dimensionality and dimensions of the deposit (and sometimes of the substrate) to prevent or hinder dislocation formation. Indeed, when the constraint of infinite lateral extension is lifted, the deposit may recover its intrinsic (stress-free) state even if the interface remains coherent. The lattice planes may then deform continuously from the spacing of the substrate toward the intrinsic spacing of the deposit over some distance from the interface. This is realized in quantum dots and nanowires.

1.6.1 Quantum Dots

Section 1.5 indicates that a mismatched epitaxial layer may reduce its total energy by developing a surface undulation. The same driving force leads to the nucleation of coherent islands (instead of a uniformly thick 2D layer) of a strongly mismatched epitaxial deposit in the Volmer–Weber (VW) or Stranski–Krastanov (SK) growth modes.22 Observed for more than 25 years [42], the SK growth of semiconductors spurred the spectacular development of quantum dot nanostructures.23