Kinetic Monte Carlo method: mathematical foundations and applications to physics of low-dimensional nanostructures

S. V. Kolesnikov ; A. M. Saletsky ; S. A. Dokukin ; A. L. Klavsyuk

Matematičeskoe modelirovanie, Tome 30 (2018) no. 2, pp. 48-80.

Résumé

The kinetic Monte Carlo method is essential tool for investigation of atomic and molecular systems. It is applicable for the wide range of problems such as the atomic diffusion, the formation of crystal defects and chemical compounds, the growth and the self-organization of nanostructures. In the present review we consider the basic principles of the kinetic Monte Carlo method and its modern modifications both lattice and non-lattice. The special attention is focused on the self-learning algorithms constructed from different saddle point finding methods and the algorithms for kinetic Monte Carlo acceleration. All methods are illustrated by the actual examples, the most of them are connected with the physics of metal surfaces.

Détail

Keywords: kinetic Monte Carlo, self-organization
Mots-clés : nanostructures.

Bibliographie

[1] N. Metropolis et al., “Equation of state calculations by fast computing machines”, J. Chem. Phys., 21 (1953), 1087 | DOI

[2] I.M. Sobol, The Monte Carlo method, Nauka, M., 1968

[3] K. Binder, D.W. Heerman, Monte Carlo simulations in statistical physics, Springer-Verlag, Berlin–Heidelberg–New York–London–Paris–Tokyo, 1988 | MR

[4] M.P. Allen, D.J. Tildesley, Computer simulations of liquids, Oxford Univ. Press, Oxford, 1989

[5] M.E.I. Newman, G.T. Barkema, Monte Carlo methods in statistical physics, Oxford Univ. Press, Oxford, 2001 | MR

[6] D. Frenkel, B. Smith, Understanding molecular simulations: from algorithm to applications, Academic Press, N.-Y., 2002

[7] I.Z. Fisher, “Applications of the Monte Carlo method in statistical physics”, Phys. Usp., 2 (1960), 783 | DOI | MR

[8] C. Jacoboni, L. Reggiani, “The Monte Carlo method for the solution of charge transport in semiconductors with applications to covalent materials”, Rev. Mod. Phys., 55 (1983), 645 | DOI

[9] M. Creutz, L. Jacobs, C. Rebbi, “Monte Carlo computations in lattice gauge theories”, Phys. Rep., 95 (1983), 201 | DOI | MR

[10] Yu.M. Makeenko, “The Monte Carlo method in lattice gauge theories”, Phys. Usp., 27 (1984), 401 | DOI | DOI

[11] H. De Raedt, A. Lagendijk, “Monte Carlo simulation of quantum statistical lattice models”, Phys. Rep., 127 (1985), 233 | DOI | MR

[12] M.A. Suhm, R.O. Watts, “Quantum Monte Carlo studies of vibrational states in molecules and clusters”, Phys. Rep., 204 (1991), 293 | DOI

[13] W. von der Linden, “A quantum Monte Carlo approach to many-body physics”, Phys. Rep., 220 (1992), 53 | DOI

[14] V.P. Kandidov, “Monte Carlo method in nonlinear statistical optics”, Phys. Usp., 39 (1996), 1243 | DOI | DOI

[15] I.K. Kamilov, A.K. Murtazaev, Kh.K. Aliev, “Monte Carlo studies of phase transitions and critical phenomena”, Phys. Usp., 42 (1999), 689 | DOI | DOI

[16] W.M.C. Foulkes, “Quantum Monte Carlo simulations of solids”, Rev. Mod. Phys., 73 (2001), 33 | DOI

[17] A.S. Mishchenko, “Diagrammatic Monte Carlo method as applied to the polaron problem”, Phys. Usp., 48 (2005), 887 | DOI | DOI

[18] E. Gull et al., “Continuous-time Monte Carlo methods for quantum impurity models”, Rev. Mod. Phys., 83 (2011), 349 | DOI

[19] J. Carlson et al., “Quantum Monte Carlo methods for nuclear physics”, Rev. Mod. Phys., 87 (2015), 1067 | DOI | MR

[20] J.M. Sellier, M. Nedjalkov, I. Dimon, “An introduction to applied quantum mechanics in the Wigner Monte Carlo formalism”, Phys. Rep., 577 (2015), 1 | DOI | MR

[21] D.K. Belashchenko, “Diffusion mechanisms in disordered systems: computer simulation”, Phys. Usp., 42 (1999), 297 | DOI | DOI

[22] A.F. Voter, “Classically exact overlayer dynamics: Diffusion of rhodium cluster on Rh\{100\}”, Phys. Rev. B, 34 (1986), 6819 | DOI

[23] K.A. Fichthorn, W.H. Weinberg, “Theoretical foundations of dynamical Monte Carlo simulations”, J. Chem. Phys., 95 (1991), 1090 | DOI

[24] A.F. Voter, “Parallel replica method for dynamics of infrequent events”, Phys. Rev. B, 57 (1998), R13985 | DOI

[25] A.F. Voter, “A method for accelerating the molecular dynamics simulation of infrequent events”, J. Chem. Phys., 106 (1997), 4665 | DOI

[26] A.F. Voter, “Hyperdynamics: Accelerated molecular dynamics of infrequent events”, Phys. Rev. Lett., 78 (1997), 3908 | DOI

[27] M.D. Sorensen, A.F. Voter, “Temperature-accelerated dynamics for simulation of infrequent events”, J. Chem. Phys., 112 (2000), 9599 | DOI

[28] V. Bochenkov, N. Suetin, S. Shankar, “Extended temperature-accelerated dynamics: Enabling longtime full-scale modeling of large rare-event systems”, J. Chem. Phys., 141 (2014), 094105 | DOI

[29] G.H. Vineyard, “Frequency factors and isotope effects in solid state rare processes”, J. Chem. Phys., 3 (1957), 121

[30] D.E. Sanders, A.E. DePristo, “Predicted diffusion rates on fcc (001) metal surfaces for adsorbate/substrate combinations of Ni, Cu, Rh, Pd, Ag, Pt, Au”, Surf. Sci., 260 (1992), 116 | DOI

[31] U. Kürpick, T.S. Rahman, “Diffusion processes relevant to homoepitaxial growth on Ag\{100\}”, Phys. Rev. B, 57 (1998), 2482 | DOI

[32] U. Kürpick, T.S. Rahman, “Monovacancy diffusion on Ag\{100\}, Cu\{100\}, and Ni\{100\}: Prefactors and activation barriers”, Phys. Rev. B, 59 (1999), 11014 | DOI

[33] U. Kürpick, “Self-diffusion on \{100\}, \{110\}, and \{111\} surfaces of Ni and Cu: a detailed study of prefactors and activation energies”, Phys. Rev. B, 64 (2001), 075418 | DOI

[34] U. Kürpick, “Effect of adsorbate interactions on adatom self-diffusion on Cu\{111\} and Ni\{111\} surfaces”, Phys. Rev. B, 66 (2002), 165431 | DOI

[35] G. Boisvert, L.J. Lewis, “Self-diffusion of adatoms, dimmers, and vacancies on Cu\{100\}”, Phys. Rev. B, 56 (1997), 7643 | DOI

[36] N.N. Negulyaev et al., “Bilayer growth of nanoscale Co islands on Cu\{111\}”, Phys. Rev. B, 77 (2008), 125437 | DOI

[37] G. Boisvert, L.J. Lewis, A. Yelon, “Many-body nature of the Meyer-Neldel compensation law for diffusion”, Phys. Rev. Lett., 75 (1995), 469 | DOI

[38] J. Heinonen et al., “Island diffusion on metal fcc \{100\} surfaces”, Phys. Rev. Lett., 82 (1999), 2733 | DOI

[39] O. Biham et al., “Models for diffusion and island growth in metal monolayers”, Surf. Sci., 400 (1998), 29 | DOI

[40] H. Mehl et al., “Models for adatom diffusion on fcc (001) metal surfaces”, Phys. Rev. B, 60 (1999), 2106 | DOI

[41] H. Mehl et al., “Electromigration-induced flow of islands and voids on the Cu(001) surface”, Phys. Rev. B, 61 (2000), 4975 | DOI

[42] O. Pierre-Louis, T.L. Einstein, “Electromigration of single-layer clusters”, Phys. Rev. B, 62 (2000), 13697 | DOI

[43] J. Kauttonen, J. Merikoski, “Single-layer metal-on-metal islands driven by strong time-dependent forces”, Phys. Rev. E, 85 (2012), 011107 | DOI

[44] Latz et al., “Simulation of electromigration effects on voids in monocrystalline Ag films”, Phys. Rev. B, 85 (2012), 035449 | DOI

[45] R.S. Sorbello, “Theory of electromigration”, Solid State Phys., 51 (1997), 159 | DOI

[46] O. Mironets et al., “Direct evidence for mesoscopic relaxations in cobalt on Cu(001)”, Phys. Rev. Lett., 100 (2008), 096103 | DOI

[47] S. Pick et al., “Magnetism and structure on the atomic scale: Small cobalt clusters in Cu(001)”, Phys. Rev. B, 70 (2004), 224419 | DOI

[48] V.S. Stepanyuk et al., “Ab Initio approach for atomic relaxations in supported magnetic clusters”, Phase transitions, 78 (2005), 61 | DOI

[49] S.V. Kolesnikov, A.L. Klavsyuk, A.M. Saletsky, “Anisotropy of energy barriers for diffusion of the Co adatom in the vicinity of the Co islands on the Cu \{100\} surface”, Phys. Solid State, 53 (2011), 2504 | DOI

[50] J. Jacobsen et al., “Island shape-induced transition from 2D to 3D growth for Pt/Pt \{111\}”, Phys. Rev. Lett., 74 (1995), 2295 | DOI

[51] A. Bogicevic, J. Strömquist, B. I. Lundquist, “Low-symmetry diffusion barriers in homoepitaxial growth of Al \{111\}”, Phys Rev. Lett., 81 (1998), 2608

[52] S. Ovesson, A. Bogicevic, B.I. Lundquist, “Origin of compact triangular islands in metal-on-metal growth”, Phys. Rev. Lett., 83 (1999), 2608 | DOI

[53] J.M. Pomeroy et al., “Kinetic Monte Carlo-molecular dynamics investigations of hyperthermal copper deposition on Cu \{111\}”, Phys. Rev. B, 66 (2002), 235412 | DOI

[54] M. Müller et al., “Island shapes, island densities, and stacking-fault formation on Ir \{111\}: Kinetic Monte Carlo simulations and experiments”, Phys. Rev. B, 71 (2005), 075407 | DOI

[55] P. Hyldgaard, M. Persson, “Long-ranged adsorbate-adsorbate interactions mediated by a surface-state band”, J. Phys.: Condens. Matter, 12 (2000), L13 | DOI

[56] A.G. Syromyatnikov, A.L. Klavsyuk, A.M. Saletsky, “Analysis of interactions between Co adatoms on the vicinal Cu \{111\} surface”, JETP Lett., 100 (2014), 24 | DOI

[57] A.G. Syromyatnikov et al., “Ab initio study of interaction between 3d adatoms on the vicinal Cu \{111\} surface”, Mod. Phys. Lett. B, 30 (2016), 1650218 | DOI

[58] J.M. Rogowska, M. Maciejewski, “Dilute nanostructures built of dimmers: Kinetic Monte Carlo study of Co on Cu \{111\}”, Phys. Rev. B, 82 (2010), 035444 | DOI

[59] Bogicevic et al., “Nature, strength, and consequences of indirect adsorbate interactions on metals”, Phys. Rev. Lett., 85 (2000), 1910 | DOI

[60] J.M. Rogowska, M. Maciejewski, “Dilute Cu nanostructure stabilized by substrate-mediated interactions on Cu \{111\}: Kinetic Monte Carlo simulations”, Phys. Rev. B, 74 (2006), 235402 | DOI

[61] A.S. Smirnov et al., “Effect of quantum confinement of surface electrons on an atomic motion on nanoislands: Ab initio calculations and kinetic Monte Carlo simulations”, Phys. Rev. B, 78 (2008), 041405(R) | DOI

[62] N.N. Negulyaev et al., “Effect of strain relaxations on heteroepitaxial metal-on-metal island nucleation and superlattice formation: Fe on Cu \{111\}”, Phys. Rev. B, 79 (2009), 195411 | DOI

[63] N.N. Negulyaev et al., “Self-organization of Ce adatoms on Ag \{111\}: A kinetic Monte Carlo study”, Phys. Rev. B, 74 (2006), 035421 | DOI

[64] S.V. Kolesnikov, A.L. Klavsyuk, A.M. Saletsky, “Formation of cobalt bilayer islands on Cu \{100\} surface”, Phys. Solid state, 51 (2009), 1254 | DOI

[65] U. Ramsperger et al., “Growth of Co on a stepped and on a flat Cu (001) surface”, Phys. Rev. B, 53 (1996), 8001 | DOI

[66] S.A. Dokukin et al., “Influence of burrowing the atoms on the density of the Fe and Co nanoclusters on the Cu \{100\} surface”, Phys. Solid state, 55 (2009), 1505 | DOI

[67] R. Pentcheva et al., “Non-Arrhenius behavior of the island density in metal heteroepitaxy: Co on Cu (001)”, Phys. Rev. Lett., 90 (2003), 076101 | DOI

[68] O. Kurnosikov, J.T. Kolhepp, W.J.M. de Jonge, Can surface atoms be moved with an STM tip?, Europhys. Lett., 64 (2003), 77 | DOI

[69] R. van Gastel et al., “Vacancy-mediated diffusion of Co atoms embedded in Cu (001)”, Surf. Sci., 605 (2011), 1956 | DOI

[70] S.V. Kolesnikov, A.L. Klavsyuk, A.M. Saletsky, “Atomic-scale self-organization of Co nanostructures embedded into Cu \{100\}”, Phys. Rev. B, 79 (2009), 115433 | DOI

[71] S.V. Kolesnikov, A.L. Klavsyuk, A.M. Saletsky, “Vacancy formation on stepped Cu \{100\} accelerated with STM: Molecular dynamics and kinetic Monte Carlo simulations”, Phys. Rev. B, 80 (2009), 245412 | DOI

[72] S.V. Kolesnikov, A.L. Klavsyuk, A.M. Saletsky, “Simulation of the formation of vacancies upon scanning of Cu \{100\} surface”, JETP Lett., 89 (2009), 471 | DOI

[73] M. Giesen, “Step and island dynamics at solid/vacuum interfaces”, Progr. Surf. Sci., 68 (2001), 1 | DOI

[74] A.L. Klavsyuk, A.M. Saletsky, “Formation and properties of metallic atomic contacts”, Phys. Usp., 58 (2015), 933 | DOI | DOI

[75] F. Sato et al., “Computer simulations of gold nanowire formation: the role of outlayer atoms”, Appl. Phys. A, 81 (2005), 1527 | DOI

[76] A.L. Klavsyuk et al., “Molecular dynamics simulation of the formation of metal nanocontacts”, Phys. Solid State, 53 (2011), 2356 | DOI

[77] S.V. Kolesnikov et al., “Formation of gold nanocontact in an ultrahigh vacuum transmission electron microscope: A kinetic Monte Carlo simulation”, Europhys. Lett., 103 (2013), 48002 | DOI

[78] S.V. Kolesnikov, A.L. Klavsyuk, A.M. Saletsky, “Simulation of self-organization of nanocontacts in thin gold film”, Phys. Solid State, 55 (2013), 1950 | DOI

[79] V. Rodrigues, T. Fuhrer, D. Ugarte, “Signature of atomic structure in the quantum conductance of gold nanowires”, Phys. Rev. Lett., 85 (2000), 4124 | DOI

[80] S.V. Kolesnikov, A.L. Klavsyuk, A.M. Saletsky, “Self-organisation and magnetic properties of Co nanostructures embedded in a Cu \{100\} surface”, Surf. Sci., 612 (2013), 48 | DOI

[81] O. Trushin et al., “Self-learning kinetic Monte Carlo method: application to Cu \{111\}”, Phys. Rev. B, 72 (2005), 115401 | DOI

[82] O.S. Trushin, M. Kotrla, F. Maca, “Energy barriers on stepped IrIr \{111\} surfaces: a molecular statics calculation”, Surf. Sci., 389 (1997), 55 | DOI

[83] H. Jónson, G. Mills, K.W. Jacobsen, “Nudged elastic band method for finding energy paths of transitions”, Classical and quantum dynamics in condensed phase simulations, World Scientific, Singapore, 1998, 385 | DOI

[84] A. Karim et al., “Diffusion of small two-dimensional Cu island on Cu \{111\} studied with a kinetic Monte Carlo method”, Phys. Rev. B, 73 (2006), 165411 | DOI

[85] O.S. Trushin i dr., “Issledovanie diffuzionnykh protsessov na poverkhnosti metallov metodom samoobuchaemogo kineticheskogo Monte-Karlo”, Matem. modelirovanie, 19 (2007), 116–126

[86] A. Karim et al., “The crossover from collective motion to periphery diffusion for two-dimensional adatom-island on Cu \{111\}”, J. Phys.: Condens. Matter, 23 (2011), 462201 | DOI

[87] G. Nandipati et al., “Parallel kinetic Monte Carlo simulations of Ag \{111\} island coarsening using a large database”, J. Phys.: Condens. Matter, 21 (2009), 084214 | DOI

[88] G. Nandipati et al., “Kinetically driven shape changes in early stages of two-dimensional island coarsening: Ag/Ag \{111\}”, Phys Rev. B, 88 (2013), 115402 | DOI

[89] S.I. Shah et al., “Self-diffusion of small Ni clusters on the Ni \{111\} surface: A self-learning kinetic Monte Carlo study”, Phys Rev. B, 88 (2013), 035414 | DOI

[90] A. Latz, L. Brendel, D.E. Wolf, “A three-dimensional self-learning kinetic Monte Carlo model: application to Ag \{111\}”, J. Phys.: Condens. Matter, 24 (2012), 485005 | DOI

[91] M. Jongmanns, A. Latz, D.E. Wolf, “Impurity-induced island pinning during electromigration”, EPL, 110 (2015), 16001 | DOI

[92] A. Latz et al., “Anisotropy of electromigration-induced void and island drift”, J. Phys.: Condens. Matter, 26 (2014), 055005 | DOI

[93] G. Henkelman, H.J. Jonson, “Improved tangent estimate in the nudged elastic band method for finding minimum energy path and saddle point”, J. Chem. Phys., 113 (2000), 9978 | DOI

[94] E.A. Bleda, X. Gao, M.S. Daw, “Calculations of diffusion in FCC binary alloys using on-the-fly kinetic Monte Carlo”, Computational Materials Science, 43 (2008), 608 | DOI

[95] S.V. Kolesnikov, A.L. Klavsyuk, A.M. Saletsky, “The role of the diffusion of dimmers in the formation of Co nanostructures embedded into Cu \{100\} surface”, Eur. Phys. J. B, 86 (2013), 399 | DOI

[96] S.V. Kolesnikov, “Self-organization of iron-atom nanostructures in the first layer of the \{100\} copper surface”, JETP Lett., 99 (2014), 286 | DOI

[97] S.V. Kolesnikov, A.L. Klavsyuk, A.M. Saletsky, “Fe and Co nanostructures embedded into the Cu \{100\} surface: Self-Organization and magnetic properties”, JETP, 121 (2015), 616 | DOI

[98] G. Henkelman, H.J. Jonson, “A dimer method for finding saddle points on high dimensional potential surfaces using only first derivatives”, J. Chem. Phys., 111 (1999), 7010 | DOI

[99] G. Henkelman, H.J. Jonson, “Long time scale kinetic Monte Carlo simulations without lattice approximation and predefined event table”, J. Chem. Phys., 115 (2001), 9657 | DOI

[100] M. Basham, F. Montalenti, P.A. Mulheran, “Multiscale modeling of island nucleation and growth during Cu \{100\} homoepitaxy”, Phys. Rev. B, 73 (2006), 045422 | DOI

[101] C. Harris et al., “Calculations of diffusion-limited processes in Ni3Al using accelerated molecular dynamics”, Comp. Material Sci., 37 (2005), 462 | DOI

[102] K. Salo, S. Takizawa, T. Mohri, “On-the-fly kinetic Monte Carlo simulation of atomic diffusion in L10 structure”, Materials Transactions, 52 (2011), 391 | DOI

[103] L.J. Karssemeijer et al., “Long-timescale simulations of diffusion in molecular solids”, PCCP, 14 (2012), 10844 | DOI

[104] L.J. Karssemeijer et al., “Dynamics of CO in amorphous water-ice environments”, The Astrophysical journal, 781 (2014), 16 | DOI

[105] F. Hontinfinde, A. Rapallo, R. Ferrando, “Numerical study of growth and relaxation of small C60 nanoclusters”, Surf. Sci., 600 (2006), 995 | DOI

[106] H. Xu, Y.N. Osetsky, R.E. Stoller, “Simulation complex atomistic processes: On-the-fly kinetic Monte Carlo scheme with selective active volumes”, Phys. Rev. B, 84 (2011), 132103 | DOI

[107] H. Xu et al., “Self-evolving atomistic kinetic Monte Carlo simulations of defects in materials”, Comp. Material Sci., 100 (2015), 135 | DOI

[108] S.A. Trygubenko, D.J. Wales, “A doubly nudged elastic band method for finding transition states”, J. Chem. Phys., 120 (2004), 2082 | DOI

[109] A. Kushima et al., “Computing the viscosity of supercooled liquids”, J. Chem. Phys., 130 (2009), 224504 | DOI

[110] L.K. Beland et al., “Kinetic activation-relaxation technique”, Phys. Rev. E, 84 (2011), 046704 | DOI

[111] O.S. Trushin et al., “Searching for transition paths in multidimensional space with a fixed repulsive bias potential”, Phys. Rev. B, 69 (2004), 033405 | DOI

[112] B. Puchala, M.L. Falk, K. Garikipati, “An energy basin finding algorithm for kinetic Monte Carlo acceleration”, J. Chem. Phys., 132 (2010), 134104 | DOI

[113] M.A. Novotny, “Monte Carlo algorithms with absorbing Markov Chains: Fast local algorithms for slow dynamics”, Phys. Rev. Lett., 74 (1995), 1 | DOI

[114] C.S. Deo, D.J. Srolovitz, “First passage time Markov Chain analysis of rare events for kinetic Monte Carlo: double kink nucleation during dislocation glide”, Modeling Simul. Mater. Sci. Eng., 10 (2002), 581 | DOI

[115] D.R. Mason, R.E. Rudd, A.P. Sutton, “Stochastic kinetic Monte Carlo algorithms for long-range Hamiltonians”, Computer Physics Communication, 160 (2004), 140 | DOI

[116] S.A. Trygubenko, D.J. Wales, “Graph transformation method for calculating waiting times in Markov chains”, J. Chem. Phys., 124 (2006), 234110 | DOI

[117] S.V. Kolesnikov, “Low-temperature study of the magnetic properties of finite atomic chains”, Jetp. Lett., 103 (2016), 588 | DOI

[118] K.A. Fichthorn, Y. Lin, “A local superbasin kinetic Monte Carlo method”, J. Chem. Phys., 138 (2013), 164104 | DOI

[119] La Magna, S. Coffa, “Accelerated Monte Carlo algorithms for defect diffusion and clustering”, Comp. Material Sci., 17 (2000), 21 | DOI

[120] C.D. van Siclen, “Stochastic method for accommodation of equilibrating basins in kinetic Monte Carlo simulations”, J. Phys.: Condens. Matter, 19 (2007), 072201 | DOI

[121] W. Kohn., “Nobel Lecture: Electronic structure of matter - wave functions and density functionals”, Rev. Mod. Phys., 71 (1999), 1253 | DOI | MR

[122] R. Car, M. Parrinello, “Unified approach for molecular dynamics and density-functional theory”, Phys. Rev. Lett., 55 (1985), 2471 | DOI

[123] Y. Li, B.-G. Liu, “Long-range ferromagnetism in one-dimensional monatomic spin chains”, Phys. Rev. B, 73 (2006), 174418 | DOI

[124] Y. Li, B.-G. Liu, “Current controlled spin reversal of nanomagnets with giant unaxial anisotropy”, Phys. Rev. Lett., 96 (2006), 217201 | DOI

[125] A.S. Smirnov et al., “Magnetic behavior of one-and two-dimensional nanostructures stabilized by surface-state electrons: a kinetic Monte Carlo study”, New J. Phys., 11 (2009), 063004 | DOI

[126] P.A. Ignatiev et al., “Magnetic ordering of nanocluster ensembles promoted by electronic substratemediated interactions: Ab initio and kinetic Monte Carlo studies”, Phys. Rev. B, 80 (2009), 165408 | DOI

[127] K.-C. Zhang, B.-G. Liu, “Dynamical ferromagnetism of interacting tiny magnets with strong anisotropy”, Phys. Lett. A, 374 (2010), 2058 | DOI

[128] K.M. Tsysar, S.V. Kolesnikov, A.M. Saletsky, “Magnetization dynamics of mixed Co-Au chains on Cu \{110\} substrate: Combined ab initio and kinetic Monte Carlo study”, Chin. Phys. B, 24 (2015), 097302 | DOI

[129] Y. Li, B.-G. Liu, “Monte Carlo simulated dynamical magnetization of single-chain magnets”, J. Magn. Magn. Mat., 378 (2015), 186 | DOI

[130] S.V. Kolesnikov, K.M. Tsysar, A.M. Saletsky, “Magnetic properties of one-dimensional Au-Co on the copper \{110\} surface”, Phys. Solid State, 57 (2015), 1513 | DOI