Bandgap energy bowing parameter of strained and relaxed InGaN layers

This paper focuses on the determination of the bandgap energy bowing parameter of strained and relaxed InxGa1−xN layers. Samples are grown by metal organic vapor phase epitaxy on GaN template substrate for indium compositions in the range of 0<x<0.25. The bangap emission energy is characterized by cathodoluminescence and the indium composition as well as the strain state are deduced from high resolution X-ray diffraction measurements. The experimental variation of the bangap emission energy with indium content can be described by the standard quadratic equation, fitted using a relative least square method and qualified with a chi square test. Our approach leads to values of the bandgap energy bowing parameter equal to 2.87±0.20eV and 1.32±0.28eV for relaxed and strained layers (determined for the first time since the revision of the InN bandgap energy in 2002), respectively. The corresponding modified Vegard’s laws describe accurately the indium content dependence of the bandgap emission energy in InGaN alloy and for the whole range of indium content. Finally, as an example of application, 3D mapping of indium content in a thick InGaN layer is deduced from bandgap energy measurements using cathodoluminescence and a corresponding hyperspectral map.


Introduction
The tunability of the fundamental bandgap of indium gallium nitride (InGaN) across the full visible range spectrum has led to the development of a variety of optoelectronic devices including blue, green, red and white light-emitting diodes [1], blue and green laser diodes [2,3], and solar cells [4][5][6][7]. For the design and the fabrication of such optoelectronic device structures an accurate knowledge of the bandgap energy as a function of the indium content in In x Ga 1−x N alloy for both strained and relaxed layers is required. The composition dependence (x) of the bandgap can be generally described using a modified Vegard's law including, in addition to the linear interpolation, a quadratic term depending on a bowing parameter b: where E GaN NBE and E InN NBE are the values of the emission energy deduced from optical measurements. However, large discrepancies are observed in the published values of b varying in the range 1.4eV-2.8eV [8][9][10][11] since the InN bangap re-evaluation in 2002 [11]. The first reason that could explain such a broad range of values of b is linked to the strain state rate of the studied In x Ga 1−x N layers. Indeed, as shown by Parker et al. [12] and Ponce et al [13] different dependencies of the bandgap emission energy with the indium composition are expected for In x Ga 1−x N layers according to their strain states. It is thus necessary to know the strain states of the In x Ga 1−x N layers and determine the value of the bowing parameter accordingly. The second issue deals with the method used to measure the bandgap energy values. Indeed, the bandgap energy deduced from either photoluminescence (PL) or cathodoluminescence (CL) measurements is Stokes shifted relative to the bandgap energy determined by techniques such as photoreflection, optical absorption or transmission, leading to an underestimation of the bandgap energy for a given indium composition [8,11,13,14]. Last, special attention has to be paid to the measurement temperature since the bandgap energy is expected to vary with temperature according to the well known Varshni's equation. Furthermore, due to indium compositional fluctuation or phase separation, an "S-shape" temperature dependence of the bandgap was reported for InGaN alloy with indium content as low as 3% [15,16]. In this paper, the bandgap energy is characterized using room-temperature CL and PL. High resolution X-ray diffraction (XRD) is used to determine the indium content as well as the strain state rate of the InGaN alloys layers. For both relaxed and strained InGaN layers, the experimental bandgap energy bowing parameters were fitted using a relative least square method from the compositional dependence of the emission bandgap, and qualified by a chi square test [17]. Finally, as an example, CL hyperspectral maps of a thick InGaN layer are plotted in terms of indium content. Results highlight the phase separation issue in such thick layers.

Experimental
Several In x Ga 1−x N samples with a wurtzite structure and different compositions of indium (0<x<0.25) and thicknesses (10-200nm) were grown on c-axis oriented GaN/sapphire template substrates in a T-shape low-pressure metal organic vapor phase epitaxy (MOVPE) reactor [18]. The growth was performed at 800 • C under a reactor pressure of 100 Torr. Nitrogen (N 2 ) was used as carrier gas and trimethylgallium (TMGa), trimethylindium (TMIn) and ammonia (NH 3 ) were employed as precursor sources for Ga, In and elemental N, respectively. The layer thicknesses were deduced from in-situ reflectometry and confirmed by (00. 2) ω-2θ XRD measurements by applying a fitting model to the Pendellösung fringes. Symmetric (00. 4) ω-2θ scans in combination with XRD reciprocal space maps (RSMs) of the asymmetric (11. 4) and (10. 5) planes were used to determine the lattice parameters (a, c) and the degree of relaxation of the InGaN layer. Depending on the strain state, the indium composition can be calculated according to the following procedure. For unstrained layers (fully relaxed films), the composition x is deduced from the well known Vegard's law: c InGaN mes and c GaN mes correspond to the values of the out-of-plane lattice parameter c of InGaN and GaN, respectively, as deduced on our samples, since the compressive strain of the 3.5µm thick GaN on sapphire template is around 1%. For InN, we take the unstrained value of the out-ofplane lattice parameter (c InN 0 ) which is known accurately (within 0.1%) in the literature [19], and equal to 5.705nm. For fully strained layers on GaN template substrates, the out-of-plane and in-plane lattice parameters are related [20] by: The luminescence properties of InGaN structures are investigated by both room-temperature (RT) CL and PL measurements. Experimental conditions are given in [21,22], El Gmili et al. Using a relative least square method, the composition dependences of the emission bandgap energy as deduced from CL measurements are fitted by the standard quadratic Eq. (1). The estimated value of the error in the determination of the bandgap energy is less than or equal to ±1%. This error arises from the determination of the peak wavelength of the luminescence band. It takes into account the accuracy of both the experimental setup and luminescence band mathematical fit. The error in the determination of the indium composition, as deduced from Eqs. (2) and (4), is estimated to be less than or equal to ±3% for both fully relaxed and strained InGaN layers. These errors are calculated by taking into account (i) the error in the determination, from XRD measurements, of the interplanar spacing d 00l of InGaN and GaN layers which can be calculated using the Braggs law differentiation [19]: where θ hkl is the Bragg angle, and Δθ the step size in θ ; (ii) the uncertainty in the knowledge of the literature values of both strain-free out-of-plane lattice constant of InN and Poissons coefficient. This estimation leads to indium composition errors equal to ±1% and ±3% for fully relaxed and strained InGaN layers, respectively. Due to the uncertainty in the knowledge of the literature values of Poissons coefficient, the error is found to be larger in the case of the strained layers. However, in the case of the relaxed layers, the rather scattered experimental InGaN diffraction spots yield an increase of the error in the determination of the values of the lattice parameters. Finally, an error of ±3% is estimated for both relaxed and strained InGaN layers.

Sorting of the InGaN samples according to their strain state rate and indium content
XRD spectra of all the samples were analyzed to determine their indium content, thickness, and strain state rate. Their surface morphology has been also characterized using scanning electron microscopy. Figure 1 summarizes this analysis. According to their morphology, the samples can be sorted in three different sets of samples named A, B, and C. The first set (set A) includes samples exhibiting 2D surface morphology and fully strained InGaN layers, while the third set (set C) comprises samples with 3D surface morphology and fully relaxed InGaN layers. In between these two sets, the set B contains samples with 2D surface morphology, in which 3D indium rich inclusions embedded in the 2D InGaN matrix tend to appear [21], and partially relaxed InGaN layers. Such local indium rich domains contribute to the transition from 2D (set A) to 3D (set C) growth mode observed in samples whose thicknesses are larger than the where B is Burger's vector magnitude, λ the angle between the Burger's vector and the interface, x the Ga content of the In 1−x Ga x N layer, and ν is Poisson's coefficient as discussed above. The values of Burger's vector used to obtain the curve plotted in Fig. 1, are taken from the literature [24] and are equal to 0.324 nm and 0 degrees for the magnitude and the angle respectively.
To support our previous analysis, typical (11. 4) RSMs and (00. 2) ω-2θ scans of samples from set A and C are presented in Fig. 2. Peaks related to fully strained (R=0%) and relaxed (R=100%) films with different indium compositions should lie on the vertical and oblique dashed-line, respectively. For clarity, the strained and fully relaxed diffraction spots are named InGaN#1 and InGaN#2, respectively. In Fig. 2(a), the InGaN layer of set A is shown to be fully strained on GaN, as an InGaN diffraction spot lies on the vertical line corresponding to pseudomorphic InGaN on a GaN template substrate. The indium composition deduced from the (114) RSM and the (004) scan is equal to 10.6%. In addition, the (00. 2) XRD ω-2θ scan revealed several diffraction satellites which fit well with 25nm InGaN layer thickness ( Fig. 2(b)). For  Fig. 2(c). As the strain of the film changes at a fixed composition, the peak should move on isocomposition lines (dashed lines) which have been calculated according to the model proposed in [25], Peireira et al. For InGaN#1, the in-plane peak position occurs very close to the value of the GaN template peak (ΔQ x =-0.0005Å −1 ) indicating that the film is almost fully strained on the substrate. Furthermore, the position of the second InGaN broad diffraction spot corresponding to a fully relaxed layer indicates higher indium content for InGaN#2 than for InGaN#1. In Fig. 2(d), the (002) scan reveals only one InGaN large diffraction peak with no Pendellösung fringes which is consistent with the 3D surface morphology of samples from set C. The InGaN peak corresponds to a 13.8% indium content partially relaxed (around 6%) InGaN sublayer while indium composition is calculated to be equal to 21.3% for the fully relaxed sublayer (In-GaN#2). A structure consisting on a InGaN#2 (R=100%, 120nm) / InGaN#1 (R=6%, 30nm) / GaN template provides a good fit to experimental results, as shown in Fig. 2(d). Since the aim of this paper is the determination of the bandgap bowing parameter in both fully strained and relaxed layers, only results obtained from samples of set A and set C will be considered in the following. Nevertheless, Fig. 3 summarizes RSMs results, showing strain state for all twenty six samples studied in this work.

Determination of the bandgap emission energy
RT depth-resolved CL measurements were performed for electron beam energy varying from 3 to 17keV. For clarity, only the most representative spectra are presented in Fig. 4. Monte-Carlo simulation allows calculation of the overlap shape of the energy loss rate as a function of the layer depth. For an increase of the electron beam energy, the region of maximum loss moves progressively from the near surface to the interface region. As the electron beam energy varies from 3keV to 5keV (12keV), such regions correspond to calculated sample depths of maximum energy loss of 23nm and 46nm (155nm), respectively. Strained films (samples from set A) show two peaks located at around 366nm and 410nm as well as large bands (GaN and InGaN defects transitions [26,27]) whatever the electron beam energy (Fig. 4a). The peaks are attributed to the GaN template substrate and InGaN layer, respectively. The variation in intensity of both peaks with electron beam energy is in agreement with Monte-Carlo simulations. For the samples of set C, CL spectra change drastically with the acceleration voltage, as shown in Fig. 4b. At 3keV, we observe only the InGaN peak at around 510nm and a broad band centered close to 650nm (In- GaN defect band). For an increase of the electron beam energy to 5keV, a second peak appears at lower wavelength (425nm). The splitting of both InGaN CL peaks can be explained in terms of strain relaxation and/or indium composition fluctuation. However, this value is huge (around 438meV) compared to that calculated for both relaxed and strained peaks positions by Pereira et al. [28] for an indium content of 10% and equal to 150meV. Thus, the splitting observed in Fig. 4b cannot be only explained by strained relaxation but also by an increase on the indium composition in InGaN in agreement with the XRD RSM reported in Fig. 2(c). According to such results as well as to Monte-Carlo simulations, the 425nm and 510nm wavelength peak positions are attributed to InGaN#1 (strained InGaN) and InGaN#2 (relaxed InGaN) sublayers, respectively. This interpretation is also confirmed by a nanometer-scale, quantitative composition mapping of InGaN layers obtained from a combination of scanning transmission electron microscopy and energy dispersive x-ray spectroscopy on some of the samples and conducted in a previous study [29,30].

Determination of the bandgap energy bowing parameters
Figures 5a and 5b show the indium content dependencies of the bandgap emission energy as deduced from PL and CL measurements for both fully strained and relaxed InGaN layers, respectively. The value of the bandgap emission energy bowing parameter b can be deduced from these data using a relative least squares method. We have thus conducted such a least squares fit using Eq. (1) and the data of Fig. 5 for both extreme state of strain of the InGaN layers. The GaN bandgap was measured as 3.39eV (RT CL) in agreement with the value reported at 300K by Yam et al. [31] and Islam et al. [8]. That of bulk InN is subject to variations in the literature. We have taken into account values varying between 0.64eV and 0.77eV at 300K [11,32,33], and estimated the bandgap bowing parameter b for relaxed layers within this range of values. Furthermore, in the case of strained layers, to calculate the bandgap of pseudomorphic InN on GaN template we have used the input equation and deformation potentials of references [34,35]. The effect of the strain leads to a theoretical increase of the InN bandgap of 0.1eV. For strained layers, the value of b can thus be fitted for InN bandgaps in the range of 0.74eV to 0.87eV. Conducting fits for many values of the InN bandgap shows that the InN bandgap energy dependences of the bowing parameter b can be described by the following linear equations for strained and relaxed layers, respectively: For both states of strain, we have assumed that, if the InGaN bandgap measurement was done again, there would be 95% chance that the new measurements fall within a 3% maximum distance from the initial measurement. In other words, the bandgap measurement is obtained with a 6% precision at a 5% risk. As no precise statistical method can be found in the literature to take into account the errors on both variables within a statistical fit, we also assume an exact measurement of the composition value. In terms of possible errors on the determination of the values of b, they are to be understood differently for each state of strain. We found that, for the relaxed sample, there was more than 70% chance that the quadratic model is able to correctly describe our measurements. Thus, assuming the quadratic model truthfulness, b is found to fall at a maximum distance of 0.2eV from the values given on the figure with 95% chance. For the strained sample, the model truthfulness probability rises to more than 90%, whereas the maximum error on b, with a 5% risk, is 0.28eV. The possible measurement errors on the bandgap that we have taken into this statistical study are slightly higher than those that have been estimated independently above. This is not surprising and not contradictory since we have not taken into account, in the statistical analysis, the possible errors on the composition, which are in fact relatively greater than that on the bandgap. The linear dependence of b with respect to E InN g (Eq. (7)) allows us to obtain the value of the bandgap bowing parameter from the measured value of the InN bandgap. For instance, a bulk InN bandgap energy of 0.70eV would lead to values of b equal to 2.87±0.2eV and 1.32±0.28eV for relaxed and strained layers, respectively. Table 1 summarizes the values of the bandgap energy bowing parameter b obtained in the frame of this work and compared to values reported by different other authors. To our knowledge, the value of b for strained layers is reported for the first time since the reevaluation of the InN bandgap. For relaxed samples, our value of b is consistent with the results of Islam et al. [8] and Moret et al. [9] and quite different from the data of Kurouchi et al. [10].

Indium content mapping of thick InGaN layers
As an example, we apply the modified Vegard's laws, describing accurately the indium content dependence of the bandgap emission energy in InGaN alloy and for the whole range of indium content, to build 3D mapping of indium content in a thick InGaN layer from bandgap energy measurement using CL and a corresponding hyperspectral map. For this, using the modified Vegard's laws, the change in the wavelength peak (or energy) position can be converted into composition variation for a given state of strain. This is applied to a 130nm thick InGaN layer from the set of samples C. XRD measurements analysis on this sample indicate a structure consisting in a 120nm thick, almost fully relaxed, InGaN#2 layer with 25% of In content, on a 30nm thick, almost fully strained (0.7% of relaxation), InGaN#1 layer with 14.1% of In content. The spatial variation of the luminescence was recorded for different electron beam energy through 400 points over a top sample surface of 5 × 5µm 2 . The corresponding CL hyperspectral mappings obtained at 3keV and 7keV which correspond to the maximum of CL intensity expected for InGaN#2 and InGaN#1 sublayers are presented in Fig. 7a and Fig. 7b, respectively. The plot of the corresponding compositional variation calculated by taking into account the strain state (deduced from XRD RSM measurements) of each sublayer and according to the different Vegard's laws and thus values of the bandgap energy bowing parameter b are shown in Fig. 7c and Fig. 7d. The two sublayers exhibit rather large inhomogeneities. For the near- surface sublayer (InGaN#2) the average indium content is roughly equal to 21% and varies down to 18% at some places. Such compositional inhomogeneity is consistent with the broad InGaN#2 diffraction spot observed on the XRD RSM (not shown here). Near the InGaN/GaN interface, the InGaN#1 sublayer should be more homogeneous. It is not fully the case. This can be explained by the larger electron beam energy required to reach the InGaN#1 sublayer, and thus larger interaction region including the InGaN#2 sublayer. Some indium content fluctuations that appear in Fig. 7b can thus be attributed to the latter. Nevertheless, an average indium