^{1}, X. Weng

^{1}, W. Ye

^{1}, D. Dettling

^{1}, S. Hanson

^{1}, G. Obeidi

^{1}and R. S. Goldman

^{1,a)}

### Abstract

We have investigated stress evolution in dilute nitride alloy filmsgrown by plasma-assisted molecular-beam epitaxy. For coherently strained films, a comparison of stresses measured via *in situ* wafer curvature measurements, with those determined from x-ray rocking curves using a linear interpolation of lattice parameter and elastic constants, suggests significant bowing of the elastic properties of GaAsN. The observed stress differences are used to quantify the composition-dependent elastic constant bowing parameters. For films with , *in situ* wafer curvature measurements reveal a signature for stress relaxation. Atomic force microscopy and transmission electron microscopymeasurements indicate that stress relaxation occurs by a combination of elasticrelaxation via island formation and plastic relaxation associated with the formation of stacking faults.

This work was supported in part by the National Science Foundation Early Faculty CAREER, Instrumentation for Materials Research, and Nanoscale Exploratory Research Programs (Grant Nos. 9773707, 9975701, and 0210714); the DoD Multidisciplinary University Research Initiative administered by the Air Force Office of Scientific Research under Grant No. F49620-00-1-0328; the Department of Energy, through the National Renewable Energy Laboratory Photovoltaics Beyond the Horizon Program under Contract No. ACQ-1-30619-14; the TRW Foundation; and k-Space Associates, Inc. The authors thank S.-H. Wei, P. Carrier, and M. D. Thouless for useful discussions and acknowledge the assistance of the staff of the Electron Microbeam Analysis Laboratory (EMAL) at the University of Michigan. The JEOL 2010 FX analytical electron microscope at EMAL is supported by NSF under Grant No. 9871177.

I. INTRODUCTION

II. EXPERIMENTAL PROCEDURES

III. COHERENTLY STRAINED GaAsNFILMS

IV. STRESS RELAXATION IN GaAsNFILMS

V. DEVIATION OF LATTICE PARAMETER FROM VEGARD’S LAW

VI. BOWING OF GaAsNELASTIC CONSTANTS

VII. CONCLUSIONS

### Key Topics

- III-V semiconductors
- 54.0
- Elasticity
- 31.0
- Thin films
- 28.0
- Lattice constants
- 25.0
- Stress relaxation
- 24.0

## Figures

(Color online) Atomic force microscopy (AFM) images of the surfaces of 100-nm-thick, coherently strained GaAsN films grown with =(a) 7, (b) 18, and (c) 29. A 250-nm-thick, partially-relaxed GaAsN film is shown in (d). The gray-scale ranges displayed are (a) 100, (b) 5, (c) 5, and (d) 70 nm. Cuts of the tip height defined by the arrows are shown below each AFM image.

(Color online) Atomic force microscopy (AFM) images of the surfaces of 100-nm-thick, coherently strained GaAsN films grown with =(a) 7, (b) 18, and (c) 29. A 250-nm-thick, partially-relaxed GaAsN film is shown in (d). The gray-scale ranges displayed are (a) 100, (b) 5, (c) 5, and (d) 70 nm. Cuts of the tip height defined by the arrows are shown below each AFM image.

Stress-thickness product vs thickness for (a) 500-nm-thick films and (b) the first 250 nm of films. In (a), the slope is approximately constant for all films, indicating negligible stress relaxation. In (b), the slope is approximately constant for films with , while for the slope begins to decrease, indicating stress relaxation. Inset: (004) XRC for the same films. For , there is one distinct epilayer peak, while for there is a split epilayer peak.

Stress-thickness product vs thickness for (a) 500-nm-thick films and (b) the first 250 nm of films. In (a), the slope is approximately constant for all films, indicating negligible stress relaxation. In (b), the slope is approximately constant for films with , while for the slope begins to decrease, indicating stress relaxation. Inset: (004) XRC for the same films. For , there is one distinct epilayer peak, while for there is a split epilayer peak.

(a) Stress measured by *in situ* multibeam optical stress sensor (MOSS) (triangles and squares) measurements and calculated from *ex situ* x-ray rocking curve (XRC) (diamonds and circles) measurements using a linear interpolation of lattice parameter and elastic constants. For samples with N composition greater than , indicated by the dotted line, the apparent stress difference is significantly greater than the error bars of the measurements. All samples were grown with a high ratio , unless otherwise noted. N compositions were determined from XRCs using a linear interpolation of lattice parameters and elastic constants. (b) Composition dependence of the bowing parameter, , for which the MOSS and XRC stresses are equal, shown as open diamonds and solid circles for and 15, respectively. Weighted linear fits to the data points for and 15 are indicated by the dashed and solid lines, respectively. N compositions were determined from XRCs, with bowing of the elastic constants included. (c) Composition dependence of and using for which the MOSS and XRC stresses are equal, with (solid lines) and (dashed lines).

(a) Stress measured by *in situ* multibeam optical stress sensor (MOSS) (triangles and squares) measurements and calculated from *ex situ* x-ray rocking curve (XRC) (diamonds and circles) measurements using a linear interpolation of lattice parameter and elastic constants. For samples with N composition greater than , indicated by the dotted line, the apparent stress difference is significantly greater than the error bars of the measurements. All samples were grown with a high ratio , unless otherwise noted. N compositions were determined from XRCs using a linear interpolation of lattice parameters and elastic constants. (b) Composition dependence of the bowing parameter, , for which the MOSS and XRC stresses are equal, shown as open diamonds and solid circles for and 15, respectively. Weighted linear fits to the data points for and 15 are indicated by the dashed and solid lines, respectively. N compositions were determined from XRCs, with bowing of the elastic constants included. (c) Composition dependence of and using for which the MOSS and XRC stresses are equal, with (solid lines) and (dashed lines).

Dark-field cross-sectional transmission electron microscopy (TEM) images for films with =(a) 2.4% and (b) 3.0%, collected with a [002] two-beam condition. In addition, bright-field cross-sectional TEM images collected with a [004] two-beam condition are shown for =(c) 2.4% and (d) 3.0%. Finally, cross-sectional high-resolution TEM images are shown for =(e) 2.4% and (f) 3.0%. In both cases, the zone axis is .

Dark-field cross-sectional transmission electron microscopy (TEM) images for films with =(a) 2.4% and (b) 3.0%, collected with a [002] two-beam condition. In addition, bright-field cross-sectional TEM images collected with a [004] two-beam condition are shown for =(c) 2.4% and (d) 3.0%. Finally, cross-sectional high-resolution TEM images are shown for =(e) 2.4% and (f) 3.0%. In both cases, the zone axis is .

GaAsN alloy lattice parameter vs nitrogen concentration, predicted by various lattice-parameter models, as well as observed experimentally by x-ray rocking curves ( axis) and nuclear reaction analysis (Ref. 29) ( axis). The model assumes a linear interpolation of the binary lattice parameters (i.e., Vegard’s law), the model assumes a linear interpolation of the binary unit-cell volumes, the model assumes a quadratic interpolation of the binary unit-cell volumes, the model uses Murnaghan’s equation of state to minimize the strain energy, and in the model, the linearly interpolated lattice parameter is modified to include the effect of lattice expansion due to the incorporation of N–N split interstitials. The gray line indicates the least-squares fit to the experimental data.

GaAsN alloy lattice parameter vs nitrogen concentration, predicted by various lattice-parameter models, as well as observed experimentally by x-ray rocking curves ( axis) and nuclear reaction analysis (Ref. 29) ( axis). The model assumes a linear interpolation of the binary lattice parameters (i.e., Vegard’s law), the model assumes a linear interpolation of the binary unit-cell volumes, the model assumes a quadratic interpolation of the binary unit-cell volumes, the model uses Murnaghan’s equation of state to minimize the strain energy, and in the model, the linearly interpolated lattice parameter is modified to include the effect of lattice expansion due to the incorporation of N–N split interstitials. The gray line indicates the least-squares fit to the experimental data.

## Tables

Elastic constants and lattice parameters for GaAs and GaN used for x-ray rocking curve analysis.

Elastic constants and lattice parameters for GaAs and GaN used for x-ray rocking curve analysis.

Stress measured by multibeam optical stress sensor (MOSS) and determined from x-ray rocking curve (XRC) measurements assuming different models for the film lattice parameter, for six samples with various N compositions. The model assumes a linear interpolation of the binary lattice parameters (i.e., Vegard’s law). The model accounts for deviation from a linear interpolation due to lattice expansion caused by the incorporation of N–N split interstitials.

Stress measured by multibeam optical stress sensor (MOSS) and determined from x-ray rocking curve (XRC) measurements assuming different models for the film lattice parameter, for six samples with various N compositions. The model assumes a linear interpolation of the binary lattice parameters (i.e., Vegard’s law). The model accounts for deviation from a linear interpolation due to lattice expansion caused by the incorporation of N–N split interstitials.

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