^{1,a)}, J. L. Hollander

^{1}, C. McAleese

^{1}, M. J. Kappers

^{1}, M. A. Moram

^{1}and C. J. Humphreys

^{1}

### Abstract

There is increasing interest in III-nitride films and multiple quantum well structures grown in non-polar or semi-polar orientations for application in light-emitting devices. We describe a method to obtain the compositions and the thicknesses of layers within III-nitride quantum well or superlattice structures grown in non-polar or semi-polar orientations, based on X-ray scattering. For each new crystallographic orientation considered, new axes were obtained and both the lengths and angles of these new axes calculated relative to the original conventional reference axes. These angles provide the coefficients of the matrix to transform the elastic constants published in the conventional setting (as used for polar *c*-plane oriented III-nitrides) into the appropriate new values. The new characteristic lengths and new elastic constants are then put into the general equation that relates the composition of a fully strained layer to the experimentally measured out-of-plane alloy *d*-spacing. Thus we have (a) determined the alloy composition from the difference between the experimentally measured alloy *d*-spacing and that of the substrate and (b) calculated the strained *d*-spacing for a given alloy composition for input to simple kinematical simulation software. In addition for quantum well structures the thickness ratio of well-to-barrier must be determined. Here we use the minima in the low angle reflectivity data. The repeat thickness and thus the thicknesses of the well and barrier layers, can be obtained from either the low or the high-angle data. We then cross-check by comparing the experimental and the simulated high-angle diffraction data. This method has been applied successfully to heteroepitaxial non-polar and semi-polar GaN/AlGaN and InGaN/GaN multiple quantum well structures and may also be used to find the composition of epilayers. The method works even in the presence of tilt between the superlattice and the GaN “template’, although in this case additional high-angle diffraction data at different settings must be collected.

The authors thank EPSRC Grant No. EP/EC35167/1 for funding and also Gates-Cambridge Trust and Winston Churchill Foundation for funding JLH.

I. INTRODUCTION

II. PRINCIPLES OF THE METHOD

A. Composition and strain relationships

B. Transformation to new axes

C. Application

D. Multiple quantum wells

E. Simulated diffraction data

III. EXPERIMENTAL

A. Film growth

B. X-ray scattering

IV. RESULTS

A. Non-polar () GaN/AlGaN MQWs – (note, crystallographically equivalent to ())

B. Semi-polar () InGaN/GaN MQWs

C. Semi-polar () GaN/AlGaN MQWs

V. DISCUSSION

VI. CONCLUSIONS

A.

1. Note added in proof

### Key Topics

- Multiple quantum wells
- 51.0
- Elasticity
- 38.0
- Reflectivity
- 26.0
- X-ray diffraction
- 22.0
- Thin film growth
- 16.0

## Figures

Illustration of the axes *x* _{1}, *x* _{2}, and *x* _{3} for three orientations: polar *c*-plane, non-polar *a*-plane and semi-polar and definition of the reciprocal space vectors *S* _{x}, *S* _{y}, and *S* _{z} (*S* = 2 sinθ/λ).

Illustration of the axes *x* _{1}, *x* _{2}, and *x* _{3} for three orientations: polar *c*-plane, non-polar *a*-plane and semi-polar and definition of the reciprocal space vectors *S* _{x}, *S* _{y}, and *S* _{z} (*S* = 2 sinθ/λ).

(Color online) X-ray reflectivity data for a 10-repeat non-polar GaN/AlGaN MQW structure (Sample A). From the bottom of the figure to the top, the data sets show the experimental data, then simulated data calculated from the best-fit model (*t* _{AlGaN} = 109 Å, *t* _{GaN} = 70 Å), then from a model with a higher well-to-barrier thickness ratio (*t* _{AlGaN} = 106 Å, *t* _{GaN} = 73 Å) and finally from a model with a lower well-to barrier thickness ratio (*t* _{AlGaN} = 112 Å, *t* _{GaN} = 67 Å). The calculated data are offset vertically for clarity.

(Color online) X-ray reflectivity data for a 10-repeat non-polar GaN/AlGaN MQW structure (Sample A). From the bottom of the figure to the top, the data sets show the experimental data, then simulated data calculated from the best-fit model (*t* _{AlGaN} = 109 Å, *t* _{GaN} = 70 Å), then from a model with a higher well-to-barrier thickness ratio (*t* _{AlGaN} = 106 Å, *t* _{GaN} = 73 Å) and finally from a model with a lower well-to barrier thickness ratio (*t* _{AlGaN} = 112 Å, *t* _{GaN} = 67 Å). The calculated data are offset vertically for clarity.

(Color online) ω/2θ scans of the reflection for a 10-repeat non-polar GaN/AlGaN MQW structure (Sample A). From the bottom of the figure to the top, the data sets show the experimental data collected using an analyzer crystal, then simulated data calculated using the best-fit model (Al_{0.169}Ga_{0.831}N thickness 109 Å and GaN thickness 70 Å), then experimental data obtained using an open detector, then simulated data from a model with a higher Al content and a higher well-to-barrier thickness ratio (Al_{0.173}Ga_{0.727}N thickness 106 Å and GaN thickness 73 Å) and finally from a model with a lower Al content and a lower well-to-barrier thickness ratio (Al_{0.164} Ga_{0.836}N thickness 112 Å and GaN thickness 67 Å). The calculated data are offset vertically for clarity.

(Color online) ω/2θ scans of the reflection for a 10-repeat non-polar GaN/AlGaN MQW structure (Sample A). From the bottom of the figure to the top, the data sets show the experimental data collected using an analyzer crystal, then simulated data calculated using the best-fit model (Al_{0.169}Ga_{0.831}N thickness 109 Å and GaN thickness 70 Å), then experimental data obtained using an open detector, then simulated data from a model with a higher Al content and a higher well-to-barrier thickness ratio (Al_{0.173}Ga_{0.727}N thickness 106 Å and GaN thickness 73 Å) and finally from a model with a lower Al content and a lower well-to-barrier thickness ratio (Al_{0.164} Ga_{0.836}N thickness 112 Å and GaN thickness 67 Å). The calculated data are offset vertically for clarity.

(Color online) X-ray reflectivity data for a 10-repeat semi-polar InGaN/GaN MQW structure (Sample B). From the bottom of the figure to the top, the data sets show the experimental data, then simulated data calculated from the best-fit model (*t* _{InGaN} = 33 Å, *t* _{GaN} = 81 Å), then from a model with a higher well-to-barrier thickness ratio (*t* _{InGaN} = 36 Å, *t* _{GaN} = 78 Å) and finally from a model with a lower well-to-barrier thickness ratio (*t* _{InGaN} = 30 Å, *t* _{GaN} = 84 Å). The calculated data are offset vertically for clarity.

(Color online) X-ray reflectivity data for a 10-repeat semi-polar InGaN/GaN MQW structure (Sample B). From the bottom of the figure to the top, the data sets show the experimental data, then simulated data calculated from the best-fit model (*t* _{InGaN} = 33 Å, *t* _{GaN} = 81 Å), then from a model with a higher well-to-barrier thickness ratio (*t* _{InGaN} = 36 Å, *t* _{GaN} = 78 Å) and finally from a model with a lower well-to-barrier thickness ratio (*t* _{InGaN} = 30 Å, *t* _{GaN} = 84 Å). The calculated data are offset vertically for clarity.

A wide-angle reciprocal space map showing the measurable reflections that can be accessed in (a) the plane containing the 112 reflection, in which theanddirections both lie in-plane (φ = 0) and (b) the plane containing the 112 and 006 reflections and the *c*-axis (φ = 90). The spot size is approximately indicative of the relative intensity expected for each reflection.

A wide-angle reciprocal space map showing the measurable reflections that can be accessed in (a) the plane containing the 112 reflection, in which theanddirections both lie in-plane (φ = 0) and (b) the plane containing the 112 and 006 reflections and the *c*-axis (φ = 90). The spot size is approximately indicative of the relative intensity expected for each reflection.

(Color online) Reciprocal space maps of the and reflections for a 10-repeat semi-polar InGaN/GaN MQW structure (Sample B), taken with the sample set at φ = 90.

(Color online) Reciprocal space maps of the and reflections for a 10-repeat semi-polar InGaN/GaN MQW structure (Sample B), taken with the sample set at φ = 90.

(Color online) ω/2θ scans of the reflection taken from a 10-repeat semi-polar InGaN/GaN MQW structure (Sample B). From the bottom of the figure to the top, the data sets show the experimental data collected using an analyzer crystal, then simulated data calculated using the best-fit model (In_{0.12}Ga_{0.82}N thickness 33 Å and GaN thickness 81 Å), then experimental data obtained using an open detector with a slit in front of it, then simulated data from a model with a lower In content and a higher well-to-barrier thickness ratio (In_{0.11}Ga_{0.89}N thickness 36 Å and GaN thickness 78 Å) and finally from a model with a higher In content and a lower well-to-barrier thickness ratio (In_{0.13}Ga_{0.87}N thickness 30 Å and GaN thickness 84 Å). The calculated data are offset vertically for clarity.

(Color online) ω/2θ scans of the reflection taken from a 10-repeat semi-polar InGaN/GaN MQW structure (Sample B). From the bottom of the figure to the top, the data sets show the experimental data collected using an analyzer crystal, then simulated data calculated using the best-fit model (In_{0.12}Ga_{0.82}N thickness 33 Å and GaN thickness 81 Å), then experimental data obtained using an open detector with a slit in front of it, then simulated data from a model with a lower In content and a higher well-to-barrier thickness ratio (In_{0.11}Ga_{0.89}N thickness 36 Å and GaN thickness 78 Å) and finally from a model with a higher In content and a lower well-to-barrier thickness ratio (In_{0.13}Ga_{0.87}N thickness 30 Å and GaN thickness 84 Å). The calculated data are offset vertically for clarity.

(Color online) X-ray reflectivity data from a 10-repeat semi-polar GaN/AlGaN MQW structure (Sample C). From the bottom of the figure to the top, the data sets show the experimental data, then simulated data calculated from the best-fit model ((*t* _{AlGaN} = 105 Å, *t* _{GaN} = 69 Å), then from a model with a higher well-to-barrier thickness ratio (*t* _{AlGaN} = 102 Å, *t* _{GaN} = 72 Å) and finally from a model with a lower well-to barrier thickness ratio (*t* _{AlGaN} 108 Å, *t* _{GaN} 66 Å). The calculated data are offset vertically for clarity.

(Color online) X-ray reflectivity data from a 10-repeat semi-polar GaN/AlGaN MQW structure (Sample C). From the bottom of the figure to the top, the data sets show the experimental data, then simulated data calculated from the best-fit model ((*t* _{AlGaN} = 105 Å, *t* _{GaN} = 69 Å), then from a model with a higher well-to-barrier thickness ratio (*t* _{AlGaN} = 102 Å, *t* _{GaN} = 72 Å) and finally from a model with a lower well-to barrier thickness ratio (*t* _{AlGaN} 108 Å, *t* _{GaN} 66 Å). The calculated data are offset vertically for clarity.

(Color online) Reciprocal space maps of the reflection for a 10-repeat semi-polar GaN/AlGaN MQW structure (Sample C) taken with the sample set at different in-plane rotational positions, (a) φ = - 90, (b) φ = 0 and (c) φ = 90.

(Color online) Reciprocal space maps of the reflection for a 10-repeat semi-polar GaN/AlGaN MQW structure (Sample C) taken with the sample set at different in-plane rotational positions, (a) φ = - 90, (b) φ = 0 and (c) φ = 90.

(Color online) Reciprocal space maps of the , , , and 0006 reflections for a 10-repeat semi-polar GaN/AlGaN MQW structure (Sample C).

(Color online) Reciprocal space maps of the , , , and 0006 reflections for a 10-repeat semi-polar GaN/AlGaN MQW structure (Sample C).

(Color online) ω/2θ scans of the reflection taken from a 10-repeat semi-polar GaN/AlGaN MQW structure (Sample C). From the bottom of the figure to the top, the data sets show the experimental data collected using an analyzer crystal, then simulated data calculated using the best-fit model (Al_{0.126}Ga_{0.874}N thickness 105 Å and GaN thickness 69 Å), then experimental data obtained using an open detector, then simulated data from a model with a lower Al content and a lower well-to-barrier thickness ratio (Al_{0.122}Ga_{0.878}N thickness 108 Å and GaN thickness 66 Å) and finally from a model with a higher Al content and a lower well-to-barrier thickness ratio (Al_{0.129}Ga_{0.871}N thickness 102 Å and GaN thickness 72 Å). The calculated data are offset vertically for clarity.

(Color online) ω/2θ scans of the reflection taken from a 10-repeat semi-polar GaN/AlGaN MQW structure (Sample C). From the bottom of the figure to the top, the data sets show the experimental data collected using an analyzer crystal, then simulated data calculated using the best-fit model (Al_{0.126}Ga_{0.874}N thickness 105 Å and GaN thickness 69 Å), then experimental data obtained using an open detector, then simulated data from a model with a lower Al content and a lower well-to-barrier thickness ratio (Al_{0.122}Ga_{0.878}N thickness 108 Å and GaN thickness 66 Å) and finally from a model with a higher Al content and a lower well-to-barrier thickness ratio (Al_{0.129}Ga_{0.871}N thickness 102 Å and GaN thickness 72 Å). The calculated data are offset vertically for clarity.

## Tables

Summary of the transformed elastic constants appropriate for commonly used semi-polar and non-polar orientations.

Summary of the transformed elastic constants appropriate for commonly used semi-polar and non-polar orientations.

Lattice parameters used to calculate lengths in Table I.

Lattice parameters used to calculate lengths in Table I.

Example of GaN described in a normalized cubic base with conversion to the semi-polar () setting.

Example of GaN described in a normalized cubic base with conversion to the semi-polar () setting.

Summary of X-ray results.

Summary of X-ray results.

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