Skip to main content
SpringerLink
Log in

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

  • Regular Article
  • Published:
The European Physical Journal B Aims and scope Submit manuscript

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 T − T 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 T − T 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.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. A. Moisala, A.G. Nasibulin, E.I. Kauppinen, J. Phys. Condens. Matter 15, S3011 (2003)

    Article  ADS  Google Scholar 

  2. K. Subrahmanyam, A.K. Manna, S.K. Pati, C. Rao, Chem. Phys. Lett. 497, 70 (2010)

    Article  ADS  Google Scholar 

  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)

    Article  Google Scholar 

  4. P.V. Kamat, J. Phys. Chem. Lett. 1, 520 (2009)

    Article  Google Scholar 

  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

  6. R. Haglund, R. Magruder, K. Becker, R. Zuhr, J. Wittig, L. Yang, Opt. Lett. 18, 373 (1993)

    Article  ADS  Google Scholar 

  7. V.M. Shalaev, A.K. Sarychev, Phys. Rev. B 57, 13265 (1998)

    Article  ADS  Google Scholar 

  8. W. Cai, D.A. Genov, V.M. Shalaev, Phys. Rev. B 72, 193101 (2005)

    Article  ADS  Google Scholar 

  9. V. Kravets, S. Neubeck, A. Grigorenko, A. Kravets, Phys. Rev. B 81, 165401 (2010)

    Article  ADS  Google Scholar 

  10. K.H. Kim, S.H. Choe, Plasmonics 12, 855 (2017)

    Article  Google Scholar 

  11. K.H. Kim, Plasmonics (2018), DOI:https://doi.org/10.1007/s11468-017-0687-x

  12. K.H. Kim, Laser Phys. 23, 115401 (2013)

    Article  ADS  Google Scholar 

  13. X. Meng, K. Fujita, Y. Moriguchi, Y. Zong, K. Tanaka, Adv. Opt. Mater. 1, 573 (2013)

    Article  Google Scholar 

  14. V. Subramanian, E. Wolf, P.V. Kamat, J. Phys. Chem. B 105, 11439 (2001)

    Article  Google Scholar 

  15. M.E. Franke, T.J. Koplin, U. Simon, Small 2, 36 (2006)

    Article  Google Scholar 

  16. Y.F. Huang et al., RSC Adv. 3, 16080 (2013)

    Article  Google Scholar 

  17. P. Yu, Y. Yao, J. Wu, X. Niu, A.L. Rogach, Z. Wang, Sci. Rep. 7, 7696 (2017)

    Article  ADS  Google Scholar 

  18. X. Li, O. Niitsoo, A. Couzis, J. Colloid Interface Sci. 465, 333 (2016)

    Article  ADS  Google Scholar 

  19. K. Arya, Z. Su, J.L. Birman, Phys. Rev. Lett. 57, 2725 (1986)

    Article  ADS  Google Scholar 

  20. R.J. White, R. Luque, V.L. Budarin, J.H. Clark, D.J. Macquarrie, Chem. Soc. Rev. 38, 481 (2009)

    Article  Google Scholar 

  21. M. Kang, K.J. Baeg, D. Khim, Y.Y. Noh, D.Y. Kim, Adv. Funct. Mater. 23, 3503 (2013)

    Article  Google Scholar 

  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)

    Article  ADS  Google Scholar 

  23. P. Hui, D. Stroud, Phys. Rev. B 33, 2163 (1986)

    Article  ADS  Google Scholar 

  24. J.P. Perdew, P. Ziesche, C. Fiolhais, Phys. Rev. B 47, 16460 (1993)

    Article  ADS  Google Scholar 

  25. J. Villain, J. Phys. (Paris) 36, 581 (1975)

    Article  Google Scholar 

  26. H. Knops, L. Den Ouden, Ann. Phys. 138, 155 (1982)

    Article  ADS  Google Scholar 

  27. B. Nienhuis, J. Phys. A: Math. Gen. 15, 199 (1982)

    Article  ADS  MathSciNet  Google Scholar 

  28. M. Den Nijs, M. Nightingale, M. Schick, Phys. Rev. B 26, 2490 (1982)

    Article  ADS  Google Scholar 

  29. B. Nienhuis, Phys. Rev. Lett. 49, 1062 (1982)

    Article  ADS  MathSciNet  Google Scholar 

  30. J. Kosterlitz, J. Phys. C 7, 1046 (1974)

    Article  ADS  Google Scholar 

  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

  32. H.E. Stanley, N. Ostrowsky, in Random fluctuations and pattern growth: experiments and models (Springer Science & Business Media, Netherlands, 2012), Vol. 157

  33. M. Najafi, A. Tavana, Phys. Rev. E 94, 022110 (2016)

    Article  ADS  Google Scholar 

  34. J. Cheraghalizadeh, M. Najafi, H. Dashti-Naserabadi, H. Mohammadzadeh, Phys. Rev. E 96, 052127 (2017)

    Article  ADS  Google Scholar 

  35. M. Najafi, arXiv:1801.08978 (2018)

  36. G. Delfino, Nucl. Phys. B 818, 196 (2009)

    Article  ADS  Google Scholar 

  37. S. Fortunato, Phys. Rev. B 66, 054107 (2002)

    Article  ADS  Google Scholar 

  38. S. Fortunato, Phys. Rev. B 67, 014102 (2003)

    Article  ADS  Google Scholar 

  39. M. Najafi, Phys. Lett. A 380, 370 (2016)

    Article  ADS  Google Scholar 

  40. A.R. Kose, B. Fischer, L. Mao, H. Koser, Proc. Natl. Acad. Sci. 106, 21478 (2009)

    Article  ADS  Google Scholar 

  41. P. Francesco, P. Mathieu, D. Sénéchal, Conformal field theory (Springer, New York, 1996)

  42. J. Cardy, Ann. Phys. 318, 81 (2005)

    Article  ADS  Google Scholar 

  43. A.L. Barabási, H.E. Stanley, Fractal concepts in surface growth (Cambridge University Press, New York, 1995)

  44. H. Kirchner, Pure Appl. Geophys. 160, 1370 (2003)

    Google Scholar 

  45. K. Falconer, Fractal geometry: mathematical foundations and applications (John Wiley & Sons, New York, 2004)

  46. J. Kondev, C.L. Henley, Phys. Rev. Lett. 74, 4580 (1995)

    Article  ADS  Google Scholar 

  47. R.J. Adler, in The geometry of random fields (SIAM, Philadelphia, New York, 1981), Vol. 62

  48. J. Kondev, C.L. Henley, D.G. Salinas, Phys. Rev. E 61, 104 (2000)

    Article  ADS  Google Scholar 

  49. Y. Gefen, B.B. Mandelbrot, A. Aharony, Phys. Rev. Lett. 45, 855 (1980)

    Article  ADS  MathSciNet  Google Scholar 

  50. M. Najafi, J. Phys. A: Math. Theor. 49, 335003 (2016)

    Article  Google Scholar 

  51. M. Najafi, M. Ghaedi, S. Moghimi-Araghi, Phys. A: Stat. Mech. Appl. 445, 102 (2016)

    Article  Google Scholar 

  52. H. Kikura, J. Matsushita, M. Matsuzaki, Y. Kobayashi, M. Aritomi, Sci. Technol. Adv. Mater. 5, 703 (2004)

    Article  Google Scholar 

  53. M. Matsuzaki, H. Kikura, J. Matsushita, M. Aritomi, H. Akatsuka, Sci. Technol. Adv. Mater. 5, 667 (2004)

    Article  Google Scholar 

  54. J. Philip, P. Shima, B. Raj, Appl. Phys. Lett. 91, 203108 (2007)

    Article  ADS  Google Scholar 

  55. J.H. Kim, F.F. Fang, H.J. Choi, Y. Seo, Mater. Lett. 62, 2897 (2008)

    Article  Google Scholar 

  56. P.Y. Keng et al., ACS Nano 3, 3143 (2009)

    Article  Google Scholar 

  57. H. Kikura, J. Matsushita, N. Kakuta, M. Aritomi, Y. Kobayashi, J. Mater. Process. Technol. 181, 93 (2007)

    Article  Google Scholar 

  58. N. Goldenfeld, Lectures on phase transitions and the renormalization group (CRC Press, Florida, 2018)

  59. S. Lübeck, K.D. Usadel, Phys. Rev. E 56, 5138 (1997)

    Article  ADS  Google Scholar 

  60. J. Hoshen, R. Kopelman, Phys. Rev. B 14, 3438 (1976)

    Article  ADS  Google Scholar 

  61. J. Cheraghalizadeh, M. Najafi, H. Mohammadzadeh, A. Saber, arXiv:1801.08962 (2018)

  62. M. Najafi, J. Stat. Mech. Theory Exp. 2015, P05009 (2015)

    Article  Google Scholar 

  63. M. Najafi, S. Moghimi-Araghi, S. Rouhani, Phys. Rev. E 85, 051104 (2012)

    Article  ADS  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Physics, University of Mohaghegh Ardabili, P.O. Box 179, Ardabil, Iran

    Jafar Cheraghalizadeh, Morteza N. Najafi & Hossein Mohammadzadeh

Authors
  1. Jafar Cheraghalizadeh
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Morteza N. Najafi
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Hossein Mohammadzadeh
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Morteza N. Najafi.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Cheraghalizadeh, J., Najafi, M.N. & Mohammadzadeh, H. Gaussian free field in the background of correlated random clusters, formed by metallic nanoparticles. Eur. Phys. J. B 91, 81 (2018). https://doi.org/10.1140/epjb/e2018-90086-8

Download citation

  • Received:

  • Revised:

  • Published:

  • DOI: https://doi.org/10.1140/epjb/e2018-90086-8

Keywords

Search

Navigation