Journal of Computational Electronics

, Volume 9, Issue 3–4, pp 160–172 | Cite as

Modified valence force field approach for phonon dispersion: from zinc-blende bulk to nanowires

Methodology and computational details
Article

Abstract

The correct estimation of the thermal properties of ultra-scaled CMOS and thermoelectric semiconductor devices demands for accurate phonon modeling in such structures. This work provides a detailed description of the modified valence force field (MVFF) method to obtain the phonon dispersion in zinc-blende semiconductors. The model is extended from bulk to nanowires after incorporating proper boundary conditions. The computational demands by the phonon calculation increase rapidly as the wire cross-section size increases. It is shown that nanowire phonon spectra differ considerably from the bulk dispersions. This manifests itself in the form of different physical and thermal properties in these wires. We believe that this model and approach will prove beneficial in the understanding of the lattice dynamics in the next generation ultra-scaled semiconductor devices.

Keywords

Dynamical matrix Nanowire Phonons Valence Force Field 

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References

  1. 1.
    Buin, A., Verma, A., Anantram, M.: Carrier-phonon interaction in small cross-sectional silicon nanowires. J. Appl. Phys. 104, 053716 (2008) CrossRefGoogle Scholar
  2. 2.
    Buin, A.K., Verma, A., Svizhenko, A., Anantram, M.P.: Significant enhancement of hole mobility in [110] silicon nanowires compared to electrons and Bulk silicon. Nano Lett. 8(2), 760–765 (2008) pMID: 18205425 [online]. Available: http://pubs.acs.org/doi/abs/10.1021/nl0727314 CrossRefGoogle Scholar
  3. 3.
    Mingo, N., Yang, L.: Phonon transport in nanowires coated with an amorphous material: an atomistic Green’s function approach. Phys. Rev. B 68(24), 245406 (2003) CrossRefGoogle Scholar
  4. 4.
    Mingo, N., Yang, L., Li, D., Majumdar, A.: Predicting the thermal conductivity of Si and Ge nanowires. Nano Lett. 3(12), 1713–1716 (2003) CrossRefGoogle Scholar
  5. 5.
    Wang, J., Wang, J.-S.: Dimensional crossover of thermal conductance in nanowires. Appl. Phys. Lett. 90(24), 241908 (2007) CrossRefGoogle Scholar
  6. 6.
    Peelaers, H., Partoens, B., Peeters, F.M.: Phonon band structure of Si nanowires: a stability analysis. Nano Lett. 9(1), 107–111 (2009) CrossRefGoogle Scholar
  7. 7.
    Keating, P.N.: Effect of invariance requirements on the elastic strain energy of crystals with application to the diamond structure. Phys. Rev. 145(2), 637–645 (1966) CrossRefGoogle Scholar
  8. 8.
    Sui, Z., Herman, I.P.: Effect of strain on phonons in Si, Ge, and Si/Ge heterostructures. Phys. Rev. B 48(24), 17938–17953 (1993) CrossRefGoogle Scholar
  9. 9.
    Fu, H., Ozolins, V., Alex, Z.: Phonons in GaP quantum dots. Phys. Rev. B 59(4), 2881–2887 (1999) CrossRefGoogle Scholar
  10. 10.
    McMurry, H., Solbrig, A. Jr., Boyter, J.: The use of valence force potentials in calculating crystal vibrations. J. Phys. Chem. Solids 28(12), 2359–2368 (1967) CrossRefGoogle Scholar
  11. 11.
    Weber, W.: Adiabatic bond charge model for the phonons in diamond, Si, Ge, and α-Sn. Phys. Rev. B 15(10), 4789–4803 (1977) CrossRefGoogle Scholar
  12. 12.
    Rustagi, K., Weber, W.: Adiabatic bond charge model for the phonons in A(III)B(V) semiconductors. Solid State Commun. 18, 673–675 (1976) CrossRefGoogle Scholar
  13. 13.
    Markussen, T., Jauho, A.-P., Brandbyge, M.: Heat conductance is strongly anisotropic for pristine silicon nanowires. Nano Lett. 8(11), 3771–3775 (2008) CrossRefGoogle Scholar
  14. 14.
    McMurry, H.L., Solbrig, A.W., Boyter, J.K., Noble, C.: The use of valence force potentials in calculating crystal vibrations. J. Phys. Chem. Solids 28, 2359–2368 (1967) CrossRefGoogle Scholar
  15. 15.
    Zou, J., Balandin, A.: Phonon heat conduction in a semiconductor nanowire. J. Appl. Phys. 89(5), 2932–2938 (2001) CrossRefGoogle Scholar
  16. 16.
    Zhang, Y., Cao, J.X., Xiao, Y., Yan, X.H.: Phonon spectrum and specific heat of silicon nanowires. J. Appl. Phys. 102(10), 104303 (2007) CrossRefGoogle Scholar
  17. 17.
    Li, X., Maute, K., Dunn, M.L., Yang, R.: Strain effects on the thermal conductivity of nanostructures. Phys. Rev. B 81(24), 245318 (2010) CrossRefGoogle Scholar
  18. 18.
    Thonhauser, T., Mahan, G.D.: Phonon modes in Si [111] nanowires. Phys. Rev. B 69(7), 075213 (2004) CrossRefGoogle Scholar
  19. 19.
    Zhao, H., Tang, Z., Li, G., Aluru, N.R.: Quasiharmonic models for the calculation of thermodynamic properties of crystalline silicon under strain. J. Appl. Phys. 99(6), 064314 (2006) CrossRefGoogle Scholar
  20. 20.
    Lazarenkova, O.L., von Allmen, P., Oyafuso, F., Lee, S., Klimeck, G.: Effect of anharmonicity of the strain energy on band offsets in semiconductor nanostructures. Appl. Phys. Lett. 85(18), 4193–4195 (2004) CrossRefGoogle Scholar
  21. 21.
    Hendrikse, Z.W., Elout, M.O., Maaskant, W.J.A.: Computation of the independent elements of the dynamical matrix. Comput. Phys. Commun. 86(3), 297–311 (1995) CrossRefGoogle Scholar
  22. 22.
    Landauer, R.: Spatial variation of currents and fields due to localized scatterers in metallic conduction. IBM J. Res. Dev. 1(3), 223–231 (1957) CrossRefMathSciNetGoogle Scholar
  23. 23.
    Wallace, D.C.: Thermodynamics of Crystals. Dover, New York (1998) Google Scholar
  24. 24.
    Weinstein, B.A., Piermarini, G.J.: Raman scattering and phonon dispersion in Si and GaP at very high pressure. Phys. Rev. B 12(4), 1172–1186 (1975) CrossRefGoogle Scholar
  25. 25.
    Dongarra, J.: Survey of sparse matrix storage formats, (1995) [online]. Available: http://www.netlib.org/linalg/html_templates/node90.html
  26. 26.
    Dongarra, J.: Mathworks, Matlab eig reference (2010) [online]. Available: http://www.mathworks.com/help/techdoc/ref/eig.html
  27. 27.
    Dongarra, J.: Matlab eig reference (2010) [online]. Available: http://www.mathworks.com/help/techdoc/ref/eigs.html
  28. 28.
    Klimeck, G., Oyafuso, F., Boykin, T.B., Bowen, R.C., von Allmen, P.: Development of a nanoelectronic 3-D (NEMO 3-D) simulator for multimillion atom simulations and its application to alloyed quantum dots. Comput. Model. Eng. Sci. (CMES) 3(5), 601–642 (2002) MATHGoogle Scholar
  29. 29.
    Paul A., Luisier, M., Neophytou, N., Kim, R., Geng, J., McLennan, M., Lundstrom, M., Klimeck, G.: Band Structure Lab, May 2006 [online]. Available: http://nanohub.org/resources/1308
  30. 30.
    Nilsson, G., Nelin, G.: Study of the homology between silicon and germanium by thermal neutron spectrometry. Phys. Rev. B 6(10), 3777–3786 (1972) CrossRefGoogle Scholar
  31. 31.
    Electronic archive, new semiconductor materials—characteristics and properties, Ioffe Physico-Technical Institute Website, 2001, http://www.ioffe.ru/SVA/NSM/Semicond/
  32. 32.
    Madelung, O.: Semiconductors—HandBook, 3rd edn. Springer, Berlin (2004) Google Scholar
  33. 33.
    de Gironcoli, S.: Phonons in Si-Ge systems: An ab initio interatomic-force-constant approach. Phys. Rev. B 46(4), 2412–2419 (1992) CrossRefGoogle Scholar
  34. 34.
    Eryiğit, R., Herman, I.P.: Lattice properties of strained GaAs, Si, and Ge using a modified bond-charge model. Phys. Rev. B 53(12), 7775–7784 (1996) CrossRefGoogle Scholar
  35. 35.
    Lazarenkova, O.L., von Allmen, P., Oyafuso, F., Lee, S., Klimeck, G.: An atomistic model for the simulation of acoustic phonons, strain distribution, and Grüneisen coefficients in zinc-blende semiconductors. Superlattices Microst. 34(3–6), 553–556 (2003) CrossRefGoogle Scholar
  36. 36.
    Blackford, L.S., Choi, J., Cleary, A., D’Azevedo, E., Demmel, J., Dhillon, I., Dongarra, J., Hammarling, S., Henry, G., Petitet, A., Stanley, K., Walker, D., Whaley, R.C.: ScaLAPACK Users’ Guide. Society for Industrial and Applied Mathematics, Philadelphia (1997) MATHGoogle Scholar

Copyright information

© Springer Science+Business Media LLC 2010

Authors and Affiliations

  • Abhijeet Paul
    • 1
  • Mathieu Luisier
    • 1
  • Gerhard Klimeck
    • 1
  1. 1.School of Electrical and Computer Engineering and Network for Computational NanotechnologyPurdue UniversityWest LafayetteUSA

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