^{1,a)}, Peter Offermans

^{2}and Paul M. Koenraad

^{2}

### Abstract

In a previous paper [J. H. Davies, D. M. Bruls, J. W. A. M. Vugs, and P. M. Koenraad, J. Appl. Phys.91, 4171 (2002). Part I.] we compared theory and experiment for the relaxation at a cleaved surface of a strained quantum well of InGaAs in GaAs. The measurements were taken with a scanning tunneling microscope and the analytic calculation used classical elastictheory for a linear, isotropic, homogeneous medium. Qualitative agreement was good but the theory gave only about 80% of the observed displacement. We have therefore extended the calculation to explore the effect of cubic symmetry and the orientation of the cleaved surface. The “strain suppression” method reduces the problem to the response of a half space to traction on its surface. We have calculated this for orthotropic symmetry, which includes the common orientations of orthorhombic, tetragonal, hexagonal, and cubic crystals. Anisotropy has no effect on the shape of the relaxed surface but the magnitude of relaxation changes. For cubic material there is no effect on the strain along the direction of growth if the cleaved surface is a {001} plane and a reduction of a few percent for a {011} plane, which is the case of experimental interest. The outward relaxation is reduced by about 20% due to cubic symmetry for a {001} plane because the shear stiffness of GaAs is higher than in the isotropic model, and is a further 10% smaller for a {011} plane. Thus the results for cubic symmetry lie further from the measurements than those calculated for isotropic material. Interfacial forces may contribute to this discrepancy but we suggest that nonlinear elasticity is probably responsible.

One of the authors (J.H.D.) would like to thank N. W. Ashcroft, S. L. Phoenix, J. P. Sethna, and H. Ustunel for discussions and the Laboratory of Applied and Solid State Physics at Cornell University for their hospitality. He is also grateful for helpful correspondence with D. A. Faux.

I. INTRODUCTION

II. APPROACH

III. SOLUTION OF BOUNDARY-VALUE PROBLEM

A. Airy function

B. Solution with complex variables

C. Boundary conditions

D. Elastic field on surface

IV. COMPLETE SOLUTION FOR RELAXATION OF THE SLAB

V. EFFECT OF SYMMETRY

A. Tetragonal symmetry

B. Hexagonal symmetry and transverse isotropy

C. Cubic rotated

D. Cubic aligned

E. Isotropic symmetry

VI. PERIODIC VARIATION OF COMPOSITION

A. Sinusoidal variation

B. Superlattice

VII. RESULTS

VIII. DISCUSSION

### Key Topics

- Elasticity
- 42.0
- Cladding
- 20.0
- Surface strains
- 16.0
- Lattice constants
- 15.0
- III-V semiconductors
- 12.0

## Figures

(a) Cladding (GaAs) around a slab (quantum well of , with a larger lattice constant), after a surface has been cleaved but before it has relaxed. The sample occupies the half space with along the direction of growth. (b) Sketch of sample after it has relaxed.

(a) Cladding (GaAs) around a slab (quantum well of , with a larger lattice constant), after a surface has been cleaved but before it has relaxed. The sample occupies the half space with along the direction of growth. (b) Sketch of sample after it has relaxed.

(Color online) Components of stress due to relaxation as a function of depth : (a) and (b) in midplane of well, with (c) at edge of well. The results are shown for the isotropic, cubic aligned, and cubic rotated cases. Stress is measured in units of the fictitious traction and distance in units of the half width .

(Color online) Components of stress due to relaxation as a function of depth : (a) and (b) in midplane of well, with (c) at edge of well. The results are shown for the isotropic, cubic aligned, and cubic rotated cases. Stress is measured in units of the fictitious traction and distance in units of the half width .

(Color online) Components of strain due to relaxation as a function of depth : (a) and (b) in midplane of well, with (c) at edge of well. The results are shown for the isotropic, cubic aligned, and cubic rotated cases. Thin lines show strain calculated from the stress for *isotropic* material using the elastic constants for *cubic* material in the two orientations. Strain is measured in units of the mismatch and distance in units of the half width .

(Color online) Components of strain due to relaxation as a function of depth : (a) and (b) in midplane of well, with (c) at edge of well. The results are shown for the isotropic, cubic aligned, and cubic rotated cases. Thin lines show strain calculated from the stress for *isotropic* material using the elastic constants for *cubic* material in the two orientations. Strain is measured in units of the mismatch and distance in units of the half width .

(Color online) Numerical calculations of outward relaxation of surface using ABAQUS to show the effect of cubic symmetry, orientation, and of assuming that the elastic constants in the slab and cladding are identical. The experimental result from I is also shown. Curves have been shifted vertically to coincide near the edge of the plot because the displacement contains an arbitrary constant.

(Color online) Numerical calculations of outward relaxation of surface using ABAQUS to show the effect of cubic symmetry, orientation, and of assuming that the elastic constants in the slab and cladding are identical. The experimental result from I is also shown. Curves have been shifted vertically to coincide near the edge of the plot because the displacement contains an arbitrary constant.

## Tables

Effect of symmetry on elastic constants, strain on surface, and prefactor for outward relaxation of surface; in all cases. For comparison, the final row shows the strains deep within the slab.

Effect of symmetry on elastic constants, strain on surface, and prefactor for outward relaxation of surface; in all cases. For comparison, the final row shows the strains deep within the slab.

Article metrics loading...

Full text loading...

Commenting has been disabled for this content