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Optical Materials Express

Optical Materials Express

  • Editor: David Hagan
  • Vol. 4, Iss. 5 — May. 1, 2014
  • pp: 1030–1041
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Bandgap energy bowing parameter of strained and relaxed InGaN layers

G. Orsal, Y. El Gmili, N. Fressengeas, J. Streque, R. Djerboub, T. Moudakir, S. Sundaram, A. Ougazzaden, and J.P. Salvestrini  »View Author Affiliations


Optical Materials Express, Vol. 4, Issue 5, pp. 1030-1041 (2014)
http://dx.doi.org/10.1364/OME.4.001030


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Abstract

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.

© 2014 Optical Society of America

1. Introduction

2. Experimental

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 d00l of InGaN and GaN layers which can be calculated using the Braggs law differentiation [19

19. M. A. Moram and M. E. Vickers, “X-ray diffraction of III-nitrides,” Rep. Prog. Phys. , 72, 036502(2009). [CrossRef]

]:
Δdhkldhkl~cot(Δθhkl)×Δθ
(5)
where θhkl 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.

3. Results and discussion

3.1. 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

21. Y. El Gmili, G. Orsal, K. Pantzas, A. Ahaitouf, T. Moudakir, S. Gautier, G. Patriarche, D. Troadec, J. P. Salvestrini, and A. Ougazzaden, “Characteristics of the surface microstructures in thick InGaN layers on GaN,” Opt. Mat. Exp. 3(8), 1111–1118 (2013). [CrossRef]

], 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 critical layer thickness. To highlight that point, the critical layer thickness for plastic relaxation computed from the model reported by Fischer et al. [23

23. A. Fischer, H. Kuhne, and H. Richter, “New Approach in Equilibrium Theory for Strained Layer Relaxation,” Phys. Rev. Lett. , 73, 2712–2715 (1994). [CrossRef] [PubMed]

] is also reported (dashed line) in Fig. 1. According to that model, the critical layer thickness hc is:
hclnhcB=B×cosλ0.0836×x×(1+1ν44π×cos2λ×(1+ν))
(6)
where B is Burger’s vector magnitude, λ the angle between the Burger’s vector and the interface, x the Ga content of the In1−xGaxN 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

24. M. Leyer, J. Stellmach, Ch. Meissner, M. Pristovsek, and M. Kneissl, “The critical thickness of InGaN on (0001) GaN,” J. Cryst. Growth , 310, 4913–4915 (2008). [CrossRef]

] and are equal to 0.324 nm and 0 degrees for the magnitude and the angle respectively.

Fig. 1: InGaN surface morphology versus indium composition and epilayer thickness. The theoretical critical layer thickness as deduced from the model of Fisher et al. [23] is also reported (dashed line).

Fig. 2: Examples of typical (114) RSM and ω − 2θ XRD scans obtained in the case of (a–b) a sample from set A (2D morphology and fully strained InGaN layers) and (c–d) a sample from set C (3D morphology and fully relaxed InGaN layers), respectively.
Fig. 3: Summary of RSMs results, showing strain state (for samples of set C only the relaxed sublayer is reported) for all twenty six samples studied in this work.

3.2. Determination of the bandgap emission energy

Fig. 4: Typical room temperature CL spectra obtained at different electron beam energy for samples from set A (a) and set C (b).

3.3. 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

31. F. K. Yam and Z. Hassan, “InGaN: An overview of the growth kinetics, physical properties and emission mechanisms,” Superlattices and Microstructures 43, 1–23 (2008). [CrossRef]

] and Islam et al. [8

8. M. R. Islam, M. R. Kaysir, M. J. Islam, A. Hashimoto, and A. Yamamoto, “MOVPE Growth of Inx Ga1−xN (x ≈ 0.4) and Fabrication of Homo-junction Solar Cells,” J. Mater. Sci. Technol. , 29(2), 128–136 (2013). [CrossRef]

]. 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

11. J. Wu, W. Walukiewicz, K. M. Yu, J. W. Ager, E. E. Haller, H. Lu, and W. J. Schaff, “Small bandgap bowing in In1−x GaxN alloys,” Appl. Phys. Lett. , 80, 4741–4743 (2002). [CrossRef]

, 32

32. W. Walukiewicz, “Narrow bandgap group III-nitride alloys,” Physica E 20, 300–307 (2004). [CrossRef]

, 33

33. T. Matsuoka, H. Okamoto, M. Nakao, H. Harima, and E. Kurimoto, “Optical bandgap energy of wurtzite InN,” Appl. Phys. Lett. 81, 1246–1248 (2002). [CrossRef]

], 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

34. Q. Yan, P. Rinke, M. Scheffler, and C. G. Van, “Strain effects in group-III nitrides: Deformation potentials for AlN, GaN, and InN,” Appl. Phys. Lett 95, 121111 (2009). [CrossRef]

,35

35. H. Y. Peng, M. M. McCluskey, Y. M. Gupta, M. Kneissl, and N. M. Johnson, “Shock-induced band-gap shift in GaN: Anisotropy of the deformation potentials,” Phys. Rev. B 71, 115207 (2005). [CrossRef]

]. 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:
{b(strained)=1.154×EgInN+0.396b(relaxed)=1.230×EgInN+2.010
(7)
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 EgInN (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

8. M. R. Islam, M. R. Kaysir, M. J. Islam, A. Hashimoto, and A. Yamamoto, “MOVPE Growth of Inx Ga1−xN (x ≈ 0.4) and Fabrication of Homo-junction Solar Cells,” J. Mater. Sci. Technol. , 29(2), 128–136 (2013). [CrossRef]

] and Moret et al. [9

9. M. Moret, B. Gil, S. Ruffenach, O. Briot, Ch. Giesen, M. Heuken, S. Rushworth, T. Leese, and M. Succi, “Optical, structural investigations and band-gap bowing parameter of GaInN alloys,” J. Cryst. Growth 311, 2795–2797 (2009). [CrossRef]

] and quite different from the data of Kurouchi et al. [10

10. M. Kurouchi, T. Araki, H. Naoi, T. Yamaguchi, A. Suzuki, and Y. Nanish, “Growth and properties of In-rich InGaN films grown on (0001) sapphire by RF-MBE,” Phys. Stat. Sol. (B) , 241, 2843–2848 (2004). [CrossRef]

].

Fig. 5: Low In content dependence of the bandgap emission energy in fully strained (a) and relaxed (b) InGaN layers. The solid lines represent the best relative least squares method fit according to Eq. (1). For an InN bandgap energy of 0.7eV, the values of the bowing parameter are equal to 1.32±0.28eV and 2.87±0.2eV for fully strained and relaxed InGaN layers, respectively.

Table 1:. Room temperature values of the emission bandgap energy bowing parameter b obtained in the frame of this work and compared to values reported by different other authors since the InN bandgap re-evaluation. Note that the data of Moret et al. [9] are obtained at 10K.

table-icon
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Using the values of the bowing parameter b and Eq. (1), it is possible to extrapolate the indium content dependence of the bandgap energy of the InGaN alloys to the whole indium content range and for the different InN bandgap values. Figures 6a and 6b show values of the bandgap energy measured within this work and by other authors, as well as their extrapolations for both strained and relaxed InGaN layers. As can be seen, for both strain state of the layers, our results are in good agreement with data of references [8

8. M. R. Islam, M. R. Kaysir, M. J. Islam, A. Hashimoto, and A. Yamamoto, “MOVPE Growth of Inx Ga1−xN (x ≈ 0.4) and Fabrication of Homo-junction Solar Cells,” J. Mater. Sci. Technol. , 29(2), 128–136 (2013). [CrossRef]

, 12

12. C. A. Parker, J. C. Roberts, S. M. Bedair, M. J. Reed, S. X. Liu, and N. A. El- Masry, “Determination of the critical layer thickness in the InGaN/GaN Heterostructures,” Appl. Phys. Lett. , 75(18), 2776–2778 (1999). [CrossRef]

, 13

13. P. A. Ponce, S. Srinivasan, A. Bell, L. Geng, R. Liu, M. Stevens, J. Cai, H. Omiya, H. Marui, and S. Tanaka, “Microstructure and electronic properties of InGaN alloys,” Phys. Stat. Sol. B 240, 273–284 (2003). [CrossRef]

]. Data of Moret et al. [9

9. M. Moret, B. Gil, S. Ruffenach, O. Briot, Ch. Giesen, M. Heuken, S. Rushworth, T. Leese, and M. Succi, “Optical, structural investigations and band-gap bowing parameter of GaInN alloys,” J. Cryst. Growth 311, 2795–2797 (2009). [CrossRef]

], which have been obtained for composition close to the two binary endpoints and at low temperature, are slightly larger than our experimental values. Such a difference is consistent with the temperature bandgap variation described by the Varshni’s equation. For indium-rich InGaN alloys, the data of [32

32. W. Walukiewicz, “Narrow bandgap group III-nitride alloys,” Physica E 20, 300–307 (2004). [CrossRef]

], Walukiewicz et al., and [10

10. M. Kurouchi, T. Araki, H. Naoi, T. Yamaguchi, A. Suzuki, and Y. Nanish, “Growth and properties of In-rich InGaN films grown on (0001) sapphire by RF-MBE,” Phys. Stat. Sol. (B) , 241, 2843–2848 (2004). [CrossRef]

], Kurouchi et al., seem to be in a better agreement with the fits whose InN bandgap energy and bowing parameters values are equal to 0.77eV and 2.96eV, respectively.

Fig. 6: Extrapolation of the In content dependence of the bandgap energy of InGaN alloys to the whole indium content range according to the different InN bandgap values, for a) strained and b) relaxed layers. Data obtained in this work and by other authors are shown for comparison.

3.4. 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 × 5μm2. 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 content of 14.5% with some variations down to 13% can be seen in the InGaN#1 sublayer.

Fig. 7: CL hyperspectral maps (a–b) and corresponding indium content maps (c–d) of a 130nm thick InGaN layer from the set of samples C obtained for electron beam energy of 3keV (b–d, InGaN#2 sublayer) and 7keV (a–c, InGaN#1 sublayer).

4. Conclusion

We have determined the bandgap energy bowing parameter of both strained and relaxed InGaN layers. The indium content bandgap energy dependence was fitted using a least square method leading to a range of values of the bowing parameter according to the value of the InN bandgap energy. Our results are demonstrated to be significant for CL measurements and especially to map the spatial compositional variation of a given sample with different state of strain. This work also provides interesting tools for the analysis and characterization of InGaN-based nanostructures and heterostructures.

Acknowledgments

This study has been funded by the French National Research Agency (ANR) under the NOVA-GAINS (Grant no ANR-12-PRGE-0014-02) ANR project and GANEX Laboratory of Excellence (Labex) project.

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T. Matsuoka, H. Okamoto, M. Nakao, H. Harima, and E. Kurimoto, “Optical bandgap energy of wurtzite InN,” Appl. Phys. Lett. 81, 1246–1248 (2002). [CrossRef]

34.

Q. Yan, P. Rinke, M. Scheffler, and C. G. Van, “Strain effects in group-III nitrides: Deformation potentials for AlN, GaN, and InN,” Appl. Phys. Lett 95, 121111 (2009). [CrossRef]

35.

H. Y. Peng, M. M. McCluskey, Y. M. Gupta, M. Kneissl, and N. M. Johnson, “Shock-induced band-gap shift in GaN: Anisotropy of the deformation potentials,” Phys. Rev. B 71, 115207 (2005). [CrossRef]

OCIS Codes
(120.0120) Instrumentation, measurement, and metrology : Instrumentation, measurement, and metrology
(310.6860) Thin films : Thin films, optical properties

ToC Category:
Semiconductors

History
Original Manuscript: February 18, 2014
Revised Manuscript: April 4, 2014
Manuscript Accepted: April 5, 2014
Published: April 29, 2014

Citation
G. Orsal, Y. El Gmili, N. Fressengeas, J. Streque, R. Djerboub, T. Moudakir, S. Sundaram, A. Ougazzaden, and J.P. Salvestrini, "Bandgap energy bowing parameter of strained and relaxed InGaN layers," Opt. Mater. Express 4, 1030-1041 (2014)
http://www.opticsinfobase.org/ome/abstract.cfm?URI=ome-4-5-1030


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