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Gaussian free field in the background of correlated random clusters, formed by metallic nanoparticles

  • Jafar Cheraghalizadeh
  • Morteza N. Najafi
  • Hossein Mohammadzadeh
Regular Article
  • 42 Downloads

Abstract

The effect of metallic nano-particles (MNPs) on the electrostatic potential of a disordered 2D dielectric media is considered. The disorder in the media is assumed to be white-noise Coulomb impurities with normal distribution. To realize the correlations between the MNPs we have used the Ising model with an artificial temperature T that controls the number of MNPs as well as their correlations. In the T → 0 limit, one retrieves the Gaussian free field (GFF), and in the finite temperature the problem is equivalent to a GFF in iso-potential islands. The problem is argued to be equivalent to a scale-invariant random surface with some critical exponents which vary with T and correspondingly are correlation-dependent. Two type of observables have been considered: local and global quantities. We have observed that the MNPs soften the random potential and reduce its statistical fluctuations. This softening is observed in the local as well as the geometrical quantities. The correlation function of the electrostatic and its total variance are observed to be logarithmic just like the GFF, i.e. the roughness exponent remains zero for all temperatures, whereas the proportionality constants scale with TT c . The fractal dimension of iso-potential lines (D f ), the exponent of the distribution function of the gyration radius (τ r ), and the loop lengths (τ l ), and also the exponent of the loop Green function x l change in terms of TT c in a power-law fashion, with some critical exponents reported in the text. Importantly we have observed that D f (T) − D f (T c ) ~ 1/√ξ(T), in which ξ(T) is the spin correlation length in the Ising model.

Keywords

Solid State and Materials 

References

  1. 1.
    A. Moisala, A.G. Nasibulin, E.I. Kauppinen, J. Phys. Condens. Matter 15, S3011 (2003) ADSCrossRefGoogle Scholar
  2. 2.
    K. Subrahmanyam, A.K. Manna, S.K. Pati, C. Rao, Chem. Phys. Lett. 497, 70 (2010) ADSCrossRefGoogle Scholar
  3. 3.
    X. Zhou, X. Huang, X. Qi, S. Wu, C. Xue, F.Y. Boey, Q. Yan, P. Chen, H. Zhang, J. Phys. Chem. C 113, 10842 (2009) CrossRefGoogle Scholar
  4. 4.
    P.V. Kamat, J. Phys. Chem. Lett. 1, 520 (2009) CrossRefGoogle Scholar
  5. 5.
    A. Miller, K. Welford, B. Daino, in Nonlinear optical materials and devices for applications in information technology (Springer Science & Business Media, New York, 2013), Vol. 289 Google Scholar
  6. 6.
    R. Haglund, R. Magruder, K. Becker, R. Zuhr, J. Wittig, L. Yang, Opt. Lett. 18, 373 (1993) ADSCrossRefGoogle Scholar
  7. 7.
    V.M. Shalaev, A.K. Sarychev, Phys. Rev. B 57, 13265 (1998) ADSCrossRefGoogle Scholar
  8. 8.
    W. Cai, D.A. Genov, V.M. Shalaev, Phys. Rev. B 72, 193101 (2005) ADSCrossRefGoogle Scholar
  9. 9.
    V. Kravets, S. Neubeck, A. Grigorenko, A. Kravets, Phys. Rev. B 81, 165401 (2010) ADSCrossRefGoogle Scholar
  10. 10.
    K.H. Kim, S.H. Choe, Plasmonics 12, 855 (2017) CrossRefGoogle Scholar
  11. 11.
    K.H. Kim, Plasmonics (2018), DOI: https://doi.org/10.1007/s11468-017-0687-x
  12. 12.
    K.H. Kim, Laser Phys. 23, 115401 (2013) ADSCrossRefGoogle Scholar
  13. 13.
    X. Meng, K. Fujita, Y. Moriguchi, Y. Zong, K. Tanaka, Adv. Opt. Mater. 1, 573 (2013) CrossRefGoogle Scholar
  14. 14.
    V. Subramanian, E. Wolf, P.V. Kamat, J. Phys. Chem. B 105, 11439 (2001) CrossRefGoogle Scholar
  15. 15.
    M.E. Franke, T.J. Koplin, U. Simon, Small 2, 36 (2006) CrossRefGoogle Scholar
  16. 16.
    Y.F. Huang et al., RSC Adv. 3, 16080 (2013) CrossRefGoogle Scholar
  17. 17.
    P. Yu, Y. Yao, J. Wu, X. Niu, A.L. Rogach, Z. Wang, Sci. Rep. 7, 7696 (2017) ADSCrossRefGoogle Scholar
  18. 18.
    X. Li, O. Niitsoo, A. Couzis, J. Colloid Interface Sci. 465, 333 (2016) ADSCrossRefGoogle Scholar
  19. 19.
    K. Arya, Z. Su, J.L. Birman, Phys. Rev. Lett. 57, 2725 (1986) ADSCrossRefGoogle Scholar
  20. 20.
    R.J. White, R. Luque, V.L. Budarin, J.H. Clark, D.J. Macquarrie, Chem. Soc. Rev. 38, 481 (2009) CrossRefGoogle Scholar
  21. 21.
    M. Kang, K.J. Baeg, D. Khim, Y.Y. Noh, D.Y. Kim, Adv. Funct. Mater. 23, 3503 (2013) CrossRefGoogle Scholar
  22. 22.
    D. Mongin, V. Juvé, P. Maioli, A. Crut, N. Del Fatti, F. Vallée, A. Sànchez-Iglesias, I. Pastoriza-Santos, L.M. Liz-Marzán, Nano Lett. 11, 3016 (2011) ADSCrossRefGoogle Scholar
  23. 23.
    P. Hui, D. Stroud, Phys. Rev. B 33, 2163 (1986) ADSCrossRefGoogle Scholar
  24. 24.
    J.P. Perdew, P. Ziesche, C. Fiolhais, Phys. Rev. B 47, 16460 (1993) ADSCrossRefGoogle Scholar
  25. 25.
    J. Villain, J. Phys. (Paris) 36, 581 (1975) CrossRefGoogle Scholar
  26. 26.
    H. Knops, L. Den Ouden, Ann. Phys. 138, 155 (1982) ADSCrossRefGoogle Scholar
  27. 27.
    B. Nienhuis, J. Phys. A: Math. Gen. 15, 199 (1982) ADSMathSciNetCrossRefGoogle Scholar
  28. 28.
    M. Den Nijs, M. Nightingale, M. Schick, Phys. Rev. B 26, 2490 (1982) ADSCrossRefGoogle Scholar
  29. 29.
    B. Nienhuis, Phys. Rev. Lett. 49, 1062 (1982) ADSMathSciNetCrossRefGoogle Scholar
  30. 30.
    J. Kosterlitz, J. Phys. C 7, 1046 (1974) ADSCrossRefGoogle Scholar
  31. 31.
    S.M. Girvin, in Aspects topologiques de la physique en basse dimension (Topological aspects of low dimensional systems) (Springer, Berlin, Heidelberg, 1999), pp. 53–175 Google Scholar
  32. 32.
    H.E. Stanley, N. Ostrowsky, in Random fluctuations and pattern growth: experiments and models (Springer Science & Business Media, Netherlands, 2012), Vol. 157 Google Scholar
  33. 33.
    M. Najafi, A. Tavana, Phys. Rev. E 94, 022110 (2016) ADSCrossRefGoogle Scholar
  34. 34.
    J. Cheraghalizadeh, M. Najafi, H. Dashti-Naserabadi, H. Mohammadzadeh, Phys. Rev. E 96, 052127 (2017) ADSCrossRefGoogle Scholar
  35. 35.
    M. Najafi, arXiv:1801.08978 (2018)
  36. 36.
    G. Delfino, Nucl. Phys. B 818, 196 (2009) ADSCrossRefGoogle Scholar
  37. 37.
    S. Fortunato, Phys. Rev. B 66, 054107 (2002) ADSCrossRefGoogle Scholar
  38. 38.
    S. Fortunato, Phys. Rev. B 67, 014102 (2003) ADSCrossRefGoogle Scholar
  39. 39.
    M. Najafi, Phys. Lett. A 380, 370 (2016) ADSCrossRefGoogle Scholar
  40. 40.
    A.R. Kose, B. Fischer, L. Mao, H. Koser, Proc. Natl. Acad. Sci. 106, 21478 (2009) ADSCrossRefGoogle Scholar
  41. 41.
    P. Francesco, P. Mathieu, D. Sénéchal, Conformal field theory (Springer, New York, 1996) Google Scholar
  42. 42.
    J. Cardy, Ann. Phys. 318, 81 (2005) ADSCrossRefGoogle Scholar
  43. 43.
    A.L. Barabási, H.E. Stanley, Fractal concepts in surface growth (Cambridge University Press, New York, 1995) Google Scholar
  44. 44.
    H. Kirchner, Pure Appl. Geophys. 160, 1370 (2003) Google Scholar
  45. 45.
    K. Falconer, Fractal geometry: mathematical foundations and applications (John Wiley & Sons, New York, 2004) Google Scholar
  46. 46.
    J. Kondev, C.L. Henley, Phys. Rev. Lett. 74, 4580 (1995) ADSCrossRefGoogle Scholar
  47. 47.
    R.J. Adler, in The geometry of random fields (SIAM, Philadelphia, New York, 1981), Vol. 62 Google Scholar
  48. 48.
    J. Kondev, C.L. Henley, D.G. Salinas, Phys. Rev. E 61, 104 (2000) ADSCrossRefGoogle Scholar
  49. 49.
    Y. Gefen, B.B. Mandelbrot, A. Aharony, Phys. Rev. Lett. 45, 855 (1980) ADSMathSciNetCrossRefGoogle Scholar
  50. 50.
    M. Najafi, J. Phys. A: Math. Theor. 49, 335003 (2016) CrossRefGoogle Scholar
  51. 51.
    M. Najafi, M. Ghaedi, S. Moghimi-Araghi, Phys. A: Stat. Mech. Appl. 445, 102 (2016) CrossRefGoogle Scholar
  52. 52.
    H. Kikura, J. Matsushita, M. Matsuzaki, Y. Kobayashi, M. Aritomi, Sci. Technol. Adv. Mater. 5, 703 (2004) CrossRefGoogle Scholar
  53. 53.
    M. Matsuzaki, H. Kikura, J. Matsushita, M. Aritomi, H. Akatsuka, Sci. Technol. Adv. Mater. 5, 667 (2004) CrossRefGoogle Scholar
  54. 54.
    J. Philip, P. Shima, B. Raj, Appl. Phys. Lett. 91, 203108 (2007) ADSCrossRefGoogle Scholar
  55. 55.
    J.H. Kim, F.F. Fang, H.J. Choi, Y. Seo, Mater. Lett. 62, 2897 (2008) CrossRefGoogle Scholar
  56. 56.
    P.Y. Keng et al., ACS Nano 3, 3143 (2009) CrossRefGoogle Scholar
  57. 57.
    H. Kikura, J. Matsushita, N. Kakuta, M. Aritomi, Y. Kobayashi, J. Mater. Process. Technol. 181, 93 (2007) CrossRefGoogle Scholar
  58. 58.
    N. Goldenfeld, Lectures on phase transitions and the renormalization group (CRC Press, Florida, 2018) Google Scholar
  59. 59.
    S. Lübeck, K.D. Usadel, Phys. Rev. E 56, 5138 (1997) ADSCrossRefGoogle Scholar
  60. 60.
    J. Hoshen, R. Kopelman, Phys. Rev. B 14, 3438 (1976) ADSCrossRefGoogle Scholar
  61. 61.
    J. Cheraghalizadeh, M. Najafi, H. Mohammadzadeh, A. Saber, arXiv:1801.08962 (2018)
  62. 62.
    M. Najafi, J. Stat. Mech. Theory Exp. 2015, P05009 (2015) CrossRefGoogle Scholar
  63. 63.
    M. Najafi, S. Moghimi-Araghi, S. Rouhani, Phys. Rev. E 85, 051104 (2012) ADSCrossRefGoogle Scholar

Copyright information

© EDP Sciences, SIF, Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  • Jafar Cheraghalizadeh
    • 1
  • Morteza N. Najafi
    • 1
  • Hossein Mohammadzadeh
    • 1
  1. 1.Department of PhysicsUniversity of Mohaghegh ArdabiliArdabilIran

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