NUMERICAL SOLUTION OF THE INTERRELATED DIFFERENTIAL EQUATION OF MOTION IN PHONON ENGINEERING

Year 2017, Volume: 7 Issue: 1, 94 - 100, 01.06.2017

A. Alızadeh H.r. Marası

Abstract

In this work, we study numeric calculations of phonon modes in nanostructures. The motion equation of atoms in a crystal with some simplification, results in a second order ordinary differential equation and two interrelated second order differential equations for 3 polarizations according to 3 dimensions. Although first equation can easily be solved, the next two interrelated equations cannot be solved by usual numerical methods. Based on discretization, a new technique is proposed for studying the motion equations. The results are presented by dispersion curves for shear, dilatational, and flexural modes of phonons.

Keywords

numeric approximation , Eigenvalue problem , dispersion curve.

References

• Alizadeh,A. et al., (2014), Tailoring electronphonon interaction in nanostructures, J. Photonics and Nanostructures - Fundamentals and Applications, 12(2), pp.164172.
• Balandin,A.A., Pokatilov,E.P., and Nika,D.L., (2007), Phonon Engineering in Hetero and Nanostruc- tures, J. Nanoelectronics and Optoelectronics, (2), pp.140170.
• Rostami,A., Alizadeh,A., Baghban H., Alizadeh,T., and Bahar,H.B., (2011), Phonon engineering in nanoscale layered structures, SPIE (7987), 79870F-79871F.
• Pokatilov,E.P., Nika,D.L., and Balandin,A.A., (2003), Phonon spectrum and group velocities in AlN/GaN/AlN and related heterostructures, Superlattices Microstruct., (33), pp.155171.
• Balandin,A.A., Nika,D.L., and Pokatilov,E.P., (2004), Phonon spectrum and group velocities in wurtzite hetero-structures, phys. stat. sol., 1(11), pp.26582661.
• Wright,A.F., (1997), Elastic properties of zinc-blende and wurtzite AlN, GaN, and InN, J. Appl. Phys., 82(2833).
• Marasi,H.R. and Pourmostafa Aqdam,A., (2015), Homotopy analysis method and homotopy Pade approximants for solving the FornbergWhitham equation, Eurasian Math. J., 6(1), pp.65-75.
• Marasi,H.R. and Khezri,E., (2014), Asymptotic distributions of Neumann problem for Sturm-Liouville equation Computational methods for differential equational, 2(1), pp.19-25.
• Mishra,L.N., Agarwal R.P., and Sen,M., (2016), Solvability and asymptotic behavior for some nonlin- ear quadratic integral equation involving Erd´elyi-Kober fractional integrals on the unbounded interval, Progress in Fractional Differentiation and Applications, 2(3), pp.153-168.
• Mishra,L.N., and Agarwal,R.P., (2016), On existence theorems for some nonlinear functional-integral equations, Dynamic Systems and Applications, 25, pp.303-320.
• Mishra,V.N., (2007), Some Problems on Approximations of Functions in Banach Spaces, Ph.D. Thesis, Indian Institute of Technology, Roorkee 247 667, Uttarakhand, India.
• Gandhi,R.B. and Mishra, V.N.,(2016), Local and global results for modified Sz´asz - Mirakjan opera- tors, Math. Method. Appl. Sci. DOI: 10.1002/mma.4171.