Development of Interatomic Potential of Group IV Alloy Semiconductors for Lattice Dynamics Simulation
Authors: Tomita Motohiro , Ogura Atsushi , Watanabe Takanobu 
Published: February 2020

Introduction
Group IV alloy semiconductor materials are becoming promising for the next-generation devices, e.g., LSI, solar cells, luminous devices, light-emitting/-receiving devices, and thermoelectric generators. One of the most anticipated application is what’s called the Internet of Things, which requires to downsize, large scale integrate, or multi-functionalize. The stress states in the devices will become complicated and inhomogeneous. Thus, there is a great demand to measure the lattice stress in such low-dimensional devices. Raman spectroscopy is one of the powerful methods to evaluate the stress states with high stress sensitivity and non-destruction. However, it has not been clarified well that the atomic concentration dependence of the elastic and phonon properties, including relationship between stress and Raman shift, because it is still difficult to grow a group IV alloy with highly uniform composition and without strain. Therefore, computational simulation approaches, such as molecular dynamics (MD) simulation, are required to predict the atomic concentration dependence of these phonon related properties.
In this study, we investigated and reproduced the lattice constant and phonon frequency in the bulk group IV alloy, especially bulk Si(1-x)Ge
x
, by MD simulation. The phonon related properties were conducted from the Fourier transform of the atom vibration history in the MD data.
Simulation Procedure
The model in MD simulation was Si(1-x)Ge
x
wire with periodic boundary condition. The Ge concentration x in the model were from 0.00 to 1.00. The initial lattice constant ai
was set to 5.431 + 0.2x + 0.027x
2 Å [1]. The initial model length and the one side width of square were 30ai
and 4ai
Å, respectively (see Fig. 1). The employed interatomic potential is the Stillinger-Weber (SW) potential, which is defined by the sum of two- and three-body potential energy terms that depend on local geometric positions. Two types of parameter sets were tested in the present work. One is reported by S. Ethier et al. [2]. The other one is newly determined by us to reproduce the lattice constant and deformation energies of small cluster models by molecular orbital method using HF(Hartree-Fock)/6-311++G** basis set. The MD lattice constant a
MD were obtained by structure optimization. The phonon related properties, including the relative phonon frequency ω/ω
0
Si, were obtained by calculating the dynamical structure factor from MD simulation results at 300 K.
Results and Discussion
Fig. 2 shows the Ge concentration dependence of the MD lattice constant a
MD obtained by structure optimization. In this simulation, the atomic arrangement of Si and Ge were shuffled using random numbers. Five different models for each composition are generated. In Fig. 2, the red plus markers show the MD results using the reported parameter set. The blue cross markers show the results of our parameters. The green solid line shows the experimental data reported by F. Schaffler [1]. The MD results show that the lattice constant is almost reproduced correctly, but the reported parameter set overestimate it at x = 0.45 ~ 0.55. This is improved by our newly designed parameter set.
Fig. 3 shows the Ge concentration dependence of the relative phonon frequency ω/ω
0
Si estimated by the dynamical structure factor simulation. The red plus and the blue cross markers show the MD results and the green solid line shows the experimental data (ω = 520.0 - 0.62x) reported by F. Pezzoli [3]. Both sets of the parameters correctly reproduce the experimental data. Thus, the MD simulation with the SW potential is useful to reproduce the lattice constant and phonon frequency simultaneously even in the Si(1-x)Ge
x
compound. Furthermore, our approach of parametrization based on molecular orbital method is expected to be applicable to other group IV compound systems like SiC and GeSn.
Acknowledgement
This study was partially supported by the Japan Science and Technology Agency’s (JST) CREST program “Development of Silicon-Based Thermoelectric Device Utilizing Computational Phononics” and the Japan Society for the Promotion of Science through a Grant-in-Aid for Scientific Research B (15H03979) and JSPS Fellows (15J07583).
References
[1] F. Schaffler: in Properties of Advanced Semiconductor Materials eds. M. E. Levinshtein et al., Wiley (2001).
[2] S. Ethier et al.: J. Mater. Res., 7 (1992) 2817.
[3] F. Pezzoli et al.: Mater. Sci. Eng. B, 124-125 (2005) 127.

Figure 1