EELS
simulations in
III
‐Nitride ternary alloys by
DFT
Authors: Ascunce Alberto Eljarrat , Sastre Xavier , Estradé Sònia , Peiró Francesca 
Published: December 2016

III‐V nitride ternary alloys, composed of two different third‐column metals,
e.g
. Al, Ga, In… , and nitrogen, as in the case of Al
x
Ga
1‐x
N, are semiconductor materials that for the last years have been playing a crucial role in the development of novel applications. They are of foremost importance for the optoelectronic industry, for instance for the recent development of blue laser applications. Often in these devices, the desirable reduction of the typical integrated circuit dimensions is translated in increasing challenges to the growth and characterization techniques employed. Among the later, analytical transmission electron microscope (TEM) is an invaluable for its ability to obtain structural and chemical information about the structures and materials at the nanometer scale. For instance, electron energy‐loss spectroscopy (EELS), a technique that is available in most modern TEM machines, allows the measurement of important valence properties by probing the low‐loss region of the spectrum, containing signals from inter‐band transitions and plasmon excitation.

We present a theoretical study of low‐loss EELS using super‐cell models for different concentrations of the metals, x, that allow to systematically study the whole compositional range, 0<x<1, with Δx = 0.125 resolution [1]. This study is carried out for the three foremost III‐nitride semiconductor ternary alloys, Al
x
Ga
1‐x
N, In
x
Al
1‐x
N and In
x
Ga
1‐x
N. In order to do this, automated DFT simulations have been carried out using Wien2k software and home‐made scripts. Additionally to the typical DFT simulation scheme, we have corrected our calculations using the modified Becke‐Johnson (mBJ) exchange‐correlation potential. This correction represents a critical improvement over the former calculation, using generalized gradient approximation (GGA), which predicted wrong band‐gap values.

For each concentration, x, of the ternary nitride compounds, A
x
B
1‐x
N, with elements A and B combinations of Al, Ga and In, we obtain from our DFT simulations the complex dielectric function (CDF), ε(E) = ε
1
+i·ε
2
, where E is the energy‐loss. Energy‐loss spectra are proportional to the imaginary part of the inverse CDF, Im[‐1/ε], also called the energy‐loss function (ELF, see Figs. 1 and 2). Figure 1 contains the ELF‐series obtained for the three studied ternary nitrides. In these series, the composition‐related behavior of the most intense peak in EELS, the plasmon, is depicted. It is generally accepted that the observed features in ternary nitride EELS, like band gap and plasmon onset energy, are related to the features observed in the pure binaries through a Vegard law of the form,

Where Ei is the observed energy for a feature i, and b is called the bowing parameter. We have used this formula to analyze both band gap and plasmon energy, Egap and Ep, respectively. In this sense, Egap is directly measured in the calculated band structures and density of states. Conversely, Ep is retrieved from a model‐based fit of the ELF series. For this purpose, we chose the Drude model of quasi‐free electron gas, which is a typical approach in experimental EELS. Figure 3 contains the results from these analyses, where the Egap and Ep appear in red and green color, respectively. The results show a somewhat inconsistent behavior of these two parameters in terms of slope and bowing of the derived Vegard laws (solid lines). This problem is especially poignant in In‐rich compounds, in which the band gap offset is greater and also interband transitions are more important.

Because of this inconsistency, we have developed an alternative method to locate the plasmon energy, following the zero of the real part of the CDF (Fig. 2); Ecut, such that ε
1
(Ecut) = 0. A more reasonable agreement with the theoretical band gap as well as the experimentally measured Vegard law is obtained from this parameter. Finally, the role played by interband transitions in the calculations of In‐rich compounds is also addressed.