Editing Analytical Model for the Optical Functions of Indium Gallium Nitride with Application to Thin Film Solar Photovoltaic Cells

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{{Source data
| type = Paper
| cite-as = Dirk V. P. McLaughlin and J.M. Pearce, [http://dx.doi.org/10.1016/j.mseb.2011.12.008 "Analytical Model for the Optical Functions of Indium Gallium Nitride with Application to Thin Film Solar Photovoltaic Cells"], Materials Science and Engineering: B, 177, 239-244 (2012). [http://mtu.academia.edu/JoshuaPearce/Papers/1540242/Analytical_Model_for_the_Optical_Functions_of_Indium_Gallium_Nitride_with_Application_to_Thin_Film_Solar_Photovoltaic_Cells open access]
}}

This paper presents the preliminary results of optical characterization using spectroscopic ellipsometry of wurtzite indium gallium nitride (In<sub>x</sub>Ga<sub>1−x</sub>N) thin films with medium indium content (0.38 < x < 0.68) that were deposited on silicon dioxide using plasma-enhanced evaporation. A Kramers–Kronig consistent parametric analytical model using Gaussian oscillators to describe the absorption spectra has been developed to extract the real and imaginary components of the dielectric function (ɛ1, ɛ2) of In<sub>x</sub>Ga<sub>1−x</sub>N films. Scanning electron microscope (SEM) images are presented to examine film microstructure and verify film thicknesses determined from ellipsometry modeling. This fitting procedure, model, and parameters can be employed in the future to extract physical parameters from ellipsometric data from other In<sub>x</sub>Ga<sub>1−x</sub>N films.

=== Background ===

In<sub>x</sub>Ga<sub>1−x</sub>N is a III-V [[semiconductor]] alloy with optoelectronic properties that are well-suited for application in solar [[photovoltaic]] (PV) cells. By altering the indium content in the alloy, the band gap of In<sub>x</sub>Ga<sub>1−x</sub>N can be tuned from 0.7 eV to 3.4 eV spanning nearly the entire solar spectrum. However, due to a variety of factors such as atomic lattice mismatch between InN and GaN, it is difficult to grow high-quality In<sub>x</sub>Ga<sub>1−x</sub>N films with a medium-high indium content using conventional deposition techniques. The In<sub>x</sub>Ga<sub>1−x</sub>N thin films characterized in this paper could be deposited on an inexpensive silicon dioxide (SiO<sub>2</sub>) substrate with a thin GaN buffer layer using a novel plasma-enhanced evaporation deposition system. This paper presents the results of optical characterization using spectroscopic ellipsometry of wurtzite In<sub>x</sub>Ga<sub>1−x</sub>N thin films with medium indium content (0.38<x<0.68) and proposes a Kramers-Kronig consistent parametric model for the optical functions of In<sub>x</sub>Ga<sub>1−x</sub>N.

=== Experimental ===

The In<sub>x</sub>Ga<sub>1-x</sub>N films were characterized via spectroscopic ellipsometry using the lab's [[QAS Ellipsometer|J.A. Woollam Co. vertical-variable angle spectroscopic ellipsometer]] (V-VASE) with a photon range of 0.8 eV to 4.5 eV.

In order to extract useful information about the thin films using ellipsometry, a Kramers-Kronig consistent parametric model was developed to fit the raw ellipsometric through a regression-based data analysis. In building the parametric model, each unique layer of material in the sample must be represented: Si wafer, SiO<sub>2</sub> substrate (grown on the Si wafer), In<sub>x</sub>Ga<sub>1-x</sub>N layer and a surface roughness layer (treated as a Bruggeman Effective Medium Approximation consisting of a 50/50 mixture of In<sub>x</sub>Ga<sub>1-x</sub>N film and void space. In order to represent the unique absorbing In<sub>x</sub>Ga<sub>1-x</sub>N layer, parametric dispersion relationships were used. A Cauchy layer with Urbach absorption was first used over the non-absorbing (transparent) regions of the ellipsometric data output. In order to produce a real physical shape for dispersion over the entire energy range (absorbing regions now included), the Cauchy layer was converted into a general oscillator layer using 2-3 Gaussian oscillators to enforce Kramers-Kronig consistency.

Top-down and cross-sectional SEM images were taken using a [[Field emission scanning electron microscope lab protocol at Queen's University|field-emission scanning electron microscope (SEM)]] in the Queen's Physics Department.

=== Results ===

For this paper, four In<sub>x</sub>Ga<sub>1−x</sub>N thin films with estimated indium contents of x=0.38,0.54,0.64 and 0.68 were characterized. The top-down SEM images of these films are shown below:

[[File:Dirk Ellips Paper Figure 1.jpeg|thumb|500px|center|Top-down SEM images of In<sub>x</sub>Ga<sub>1−x</sub>N films increasing in indium content from a-d]]

The In<sub>x</sub>Ga<sub>1−x</sub>N films display a clear relationship of larger grain or nanocolumn diameters with increasing indium content, which has been reported [[Effects of Substrate Temperature on Indium Gallium Nitride Nanocolumn Crystal Growth|previously]]. Additionally, the two highest indium content films show not only thicker much better defined and isolated platelets/nanocolumns.

The following image shows the cross-sectional view of the In<sub>0.68</sub>Ga<sub>0.32</sub>N thin film:

[[File:Dirk Ellips Paper Figure 2.jpeg|thumb|400px|center|Cross-sectional SEM image of a 221 nm-thick In<sub>0.68</sub>Ga<sub>0.32</sub>N film]]

This image reveals the partially coalesced nanocolumnar microstructure of the high indium content films and verifies the accuracy of the film thickness (221 nm) determined from the ellipsometric model.

The Kramers-Kronig consistent parametric dispersion model described earlier was found to fit the experimental data (Ψ and Δ) very well with mean-squared errors (MSE) under 16. This, in addition to film thicknesses confirmed by SEM imaging, allowed the conclusion that the model developed is both accurate and physically meaningful. Using these models for the In<sub>x</sub>Ga<sub>1−x</sub>N films, film thicknesses and optical properties were obtained. The following graphs show the real and imaginary parts of the dielectric function, ε<sub>1</sub> and ε<sub>2</sub>, as a function of photon energy for the analyzed In<sub>x</sub>Ga<sub>1−x</sub>N films:

[[File:Dirk Ellips Paper Figure 3.jpeg|thumb|600px|center|Real and imaginary parts of the dielectric function of In<sub>x</sub>Ga<sub>1−x</sub>N films increasing in indium content from a-d]]

The In<sub>x</sub>Ga<sub>1−x</sub>N films show similar trends for the real and imaginary parts of the dielectric function. As expected, the ε<sub>2</sub> curves indicate strong absorption at the higher photon energies above the band gaps. However, there is also evidence of non-zero absorption at the low energy end which could indicate sub-gap absorption due to a high density of absorbing free electrons, dislocations and defects. The rather broad curves of the dielectric functions and index of refraction curves can an be largely attributed to the large lattice mismatch the In-rich In<sub>x</sub>Ga<sub>1−x</sub>N films presented here suffer from which tend to cause structural disorder and slight fluctuations in composition. Absorption coefficients extracted from the model were found to be on the order of 10<sup>5</sup> cm<sup>-1</sup> at the absorption edges and this strong absorption, despite imperfect crystallinity, illustrates the potential of the In<sub>x</sub>Ga<sub>1−x</sub>N for photovoltaic applications.

The paper also presents the parameters of the Gaussian oscillators used to fit the ellipsometric data. The parameters presented can be used to help describe the optical functions and absorption of similar wurtzite In-rich In<sub>x</sub>Ga<sub>1−x</sub>N films. This is particularly important for the complex nature of proposed multi-junction solar photovoltaic devices, which rely on multiple layers of In<sub>x</sub>Ga<sub>1−x</sub>N films with varying indium content.

=== Conclusions ===

A Kramers-Kronig consistent parametric model has been developed for the optical functions of wurtzite InxGa1-xN alloy films of medium indium contents (0.38<x<0.68) deposited by a novel plasma-enhanced evaporation deposition system. This model employing simple Gaussian oscillators was used to fit spectroscopic ellipsometric data over the 0.8 eV to 4.5 eV range to obtain film thicknesses, dielectric functions and absorption coefficients. Using analytical expressions to accurately describe the optical functions of InxGa1-xN films is an extremely important step in understanding the semiconductor and its utilization in high-efficiency solar photovoltaic cells. The optical characterization methods employed and the model developed can be used as a basis for the optical characterization of similar InxGa1-xN thin films. 

=== Citation List ===

The list of sources used in the paper can be found [[Analytical Model for the Optical Functions of Indium Gallium Nitride with Application to Thin Film Solar Photovoltaic Cells - references|here]] which drew papers from the more extensive [[InGaN Solar Cells Literature Review|InGaN literature review]]

=== Contact ===

The author can be contacted at pvdirk@gmail.com

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