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Surface Science

Volume 488, Issues 1-2
1 August 2001
Pages 52-72

DOI: 10.1016/S0039-6028(01)01081-0
PII: S0039-6028(01)01081-0
Copyright © 2001 Elsevier Science B.V. All rights reserved.

Elastic relaxation during 2D epitaxial growth: a study of in-plane lattice spacing oscillations

P. Müller, , a, P. Turbanb, L. Lapenaa and S. Andrieub

a Centre de Recherche sur les Mécanismes de la Croissance Cristalline,1 CRMC2-CNRS, Campus de Luminy, Case 913, 13288 Marseille Cedex 9, France
b Laboratoire de Physique des Matériaux, UMR7556, Univ. H. Poincaré, F-54506 Vandoeuvre, France

Received 9 February 2001;  accepted 6 April 2001.  Available online 18 July 2001.

Abstract

The purpose of this paper is to report some new experimental and theoretical results about the analysis of in-plane lattice spacing oscillations during two-dimensional (2D) homo and hetero epitaxial growth. The physical origin of these oscillations comes from the finite size of the strained islands. The 2D islands may thus relax by their edges, leading to in-plane lattice spacing oscillations during the birth and spread of these islands. On the one hand, we formulate the problem of elastic relaxation of a coherent 2D epitaxial deposits by using the concept of point forces and demonstrate that the mean deformation in the islands exhibits an oscillatory behaviour. On the other hand, we calculate the intensity diffracted by such coherently deposited 2D islands by using a mean model of a pile-up of weakly deformed layers. The amplitude of in-plane lattice spacing oscillations is found to depend linearly on the misfit and roughly linearly on the nucleation density. We show that the nucleation density may be approximated from the full-width at half maximum of the diffracted rods at half coverages. The predicted dependence of the in-plane lattice spacing oscillations amplitude with the nucleation density is thus experimentally verified on V/Fe(0 0 1), Mn/Fe(0 0 1), Ni/Fe(0 0 1), Co/Cu(0 0 1) and V/V(0 0 1).

Author Keywords: Growth; Molecular beam epitaxy; Reflection high-energy electron diffraction (RHEED); Surface stress

Article Outline

1. Introduction
2. Model
2.1. Epitaxial misfit
2.2. Equilibrium strain in deposited islands
2.2.1. Displacements and deformation fields
2.2.2. Discussion
2.3. Diffracted intensity
2.3.1. Diffraction by a weakly deformed layer
2.3.2. Diffraction of the whole system: an interference average model
2.3.3. Comments about FWHM oscillations
2.4. Conclusion
3. Comparison with experimental data
3.1. Description of experiments
3.2. FWHM versus nucleation density
3.3. <a||> oscillations amplitude versus nucleation density
3.4. Elastic relaxation and effective misfit
4. Conclusion and perspectives
Acknowledgements
Appendix A
Appendix B
Appendix C
References


 (4K)
Fig. 1. Sketch of the model: set of periodic (L apart) infinite ribbons (width ) of material A deposited onto a mismatched substrate B.

(11K)
Fig. 2. Normalised displacement fields uxi(x)/(Kma/) where u0x(x), (x)uxInt(x) and uxB(x)=ux0(x)+uxInt(x), are the displacement field in absence of elastic relaxation, the elastic relaxation contribution and the total field, respectively (see Eq. (9) and Table 1). (a)¯(c) correspond to a coverage =0.1, 0.5 and 0.99, respectively and are calculated for a coalescence size L=100. Owing to the elastic relaxation, the greater the coverage is, the closer the border of the islands and the weaker the displacement field.

(10K)
Fig. 3. (a) Normalised displacement uxi/(Kma/) calculated at the island edge x=/2 as a function of coverage for L=100. ux0, uxInt and uxB=ux0+uxInt, obtained from Eq. (9) and Table 1, are the displacement field in absence of elastic relaxation, the elastic relaxation contribution and the total field, respectively. Notice that uxB reaches a maximum at half coverage. (b) Normalised mean deformation <>/(Kma/) as a function of coverage for various values of the coalescence size L.

(4K)
Fig. 4. Model used to calculate the diffracted intensities: the first growing layer is considered as a full weakly deformed layer whose inner potential varies with coverage, the underlying layers being considered as full nondeformed layers (see the end of Section 2.3.1). The electron beams diffracted by each layer interfere each other.

(9K)
Fig. 5. Diffracted intensity calculated from Eq. (15a) as a function of q for various values of coverage =0.1, 0.2, 0.4, 0.5, 0.6, 0.8 (with =1.4°, n=5; =0.5 and V=10 V). Notice the intensity oscillation and the position oscillation as well.

(7K)
Fig. 6. Shift of the diffracted intensity maximum versus coverage calculated from Eq. (15a) with =0.5 for m=5%. () L=100, () L=20.

(7K)
Fig. 7. (a) Shift of the diffracted intensity maximum versus misfit calculated for L=20. (b) Shift of the diffracted intensity maximum versus nucleation density calculated for m=5%. In a large domain of nucleation density the variation is linear. (The diffracted intensity maximum are calculated for =0.5.) In both cases the fat line corresponds to the overestimated shift calculated by Eq. (15c) whereas the squares linked by the thin line corresponds to the true shift calculated from Eq. (15a) (see text).

(6K)
Fig. 8. Variation of the nucleation density (calculated from the RHEED peak FWHM) with the substrate temperature, and comparison to the nucleation density obtained by Stroscio et al. [27,28] using STM.

(39K)
Fig. 9. FWHM and in-plane lattice spacing oscillations (IPLOSs) obtained without (bottom) and with initial oxygen surface concentration (top) during the growth of Mn on (0 0 1) Fe (left), V on (0 0 1) Fe (middle), and Co on (0 0 1) Cu (right). The FWHM is increased in presence of oxygen due to the increase of nucleation centres. As a consequence, the 2D islands size decreases and the relaxation effect occurring at the islands edge is easier to detect.

(7K)
Fig. 10. Maximum variation of the detected IPLSOSs versus FWHM for Mn, Ni and V on Fe(0 0 1), Co on Cu(0 0 1) and V on V(0 0 1).

Table 1. Analytical expressions of the displacements fields at the surface z=0 (<1K)
The term ux0(x) is the elastic field in absence of any elastic interaction and the term uxB(x)=ux0(x)+uxInt(x) the total elastic field where uxInt(x) is the contribution of the elastic interactions. Notice that these expressions are valid for -<x< expected at x=±/2 where some local divergences occur because of the use of point forces.

Table 2. Analytical expressions of the mean deformations inside and outside the islands, in absence of any elastic interaction (subscript 0), only due to interactions (subscript int) and their sum (<1K)

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1 Associated to the Universities of Aix-Marseille II and III.

Corresponding author. Fax: +91-41-8916; email: muller@crmc2.univ-mrs.fr
This document
SummaryPlus
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Journal Format-PDF (514 K)
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Surface Science
Volume 488, Issues 1-2
1 August 2001
Pages 52-72


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