Research paperDelocalized nonlinear vibrational modes in fcc metals
Introduction
Many important physical phenomena observed in solid state physics are due to nonlinearity of interatomic interactions, which comes into play with increasing amplitudes of atomic vibrations. Among them are thermal expansion, temperature dependence of heat capacity, thermal conductivity, elastic constants, and other macroscopic properties of crystals [1]. Spatial localization of vibrational energy in the form of discrete breathers, also called intrinsic localized modes, is also possible only in nonlinear lattices [2], [3], [4], [5], [6], [7].
There are no general methods for finding exact solutions describing dynamics of nonlinear lattices. On the other hand, some families of exact solutions can be found by considering the symmetry of the lattice. With the aid of the approach developed in [8], [9], [10], it is possible to find exact solutions to nonlinear dynamical equations either for crystals with space symmetry [11], [12], [13], [14] or for molecules with point symmetry [15]. In crystals such solutions have the form of delocalized nonlinear vibrational modes (DNVMs) with relatively small translational cell. In the small amplitude limit DNVMs transform into short-wavelength phonon modes. That is why the problem of finding DNVMs can be formulated as the problem of finding such linear phonon modes that preserve their pattern of vibrations even for large amplitudes. A trivial example of such a phonon in a chain of anharmonically coupled particles is the zone-boundary mode with the maximal displacements , which, due to the mode symmetry, can be described by single equation of motion and thus exists as an exact solution for any amplitude and for any type of anharmonicity. Other DNVMs in nonlinear chains have been reported in [11].
DNVMs can have degrees of freedom described by coupled equations of motion; they are called -component DNVMs or -dimensional bushes of nonlinear normal modes (BNNMs) [8], [9], [10]. The most interesting are DNVMs with the number of degrees of freedom much smaller than the total number of degrees of freedom of the system, . One-component DNVMs () represent symmetry-determined nonlinear normal modes by Rosenberg [16]. According to the definition, each Rosenberg mode is a periodic vibrational regime for which all degrees of freedom vary in time proportionally to the same time-dependent function, Substituting the ansatz Eq. (1) into the dynamical system and taking into account symmetry of the system one obtains single governing differential equation for the function . DNVMs can exist in a system regardless the type of interparticle interactions and for any amplitude because their existence is determined solely by the lattice symmetry. DNVMs with components generalize the one-component Rosenberg modes [8], [9], [10].
The group-theoretical approach [8], [9], [10] was applied to the analysis of one- and multi-component DNVMs in S molecule [15]. DNVMs have also been considered in nonlinear chains [11], [17], [18], [19], [20], [21], carbyne [12], triangular Morse lattice [22], hexagonal lattice of graphene [14], [23], [24], [25], [26], [27], diamond [13], fcc metals [18], [28], [29], [30] and other lattices.
DNVMs are interesting not only from the pure theoretical point of view. New types of discrete breathers can be obtained by superimposing a localizing function on a DNVM having frequency outside the phonon spectrum [23], [29], [31], [32], [33], [34], [35], [36]. Excitation of DNVMs affects elastic constants of nonlinear lattices [37]; it can result in a second harmonic generation or induce negative pressure [24]. Modulational instability of DNVMs with frequency outside the phonon band leads to energy localization in the form of long-living chaotic discrete breathers [38], [39], [40], [41], [42], [43], [44], [45], [46], [47].
For the first time, the ab initio method of studying the dynamics of DNVMs was applied to the hexagonal lattice of graphene [14]. Here the ab initio method is used to analyze one-component DNVMs in fcc lattice of three metals, Cu, Ni and Al. These metals are chosen because they have different stacking fault energy , which is very important in the discussion of dislocation dynamics and mechanisms of plastic deformation [48]. The value of depends on the purity of the metal, temperature, and other parameters, but it is generally recognized that Cu, Ni, and Al have relatively low, intermediate, and high values, respectively. For example, in [49] for Cu, Ni and Al one can find , 125 and 166 mJ/m, respectively. Besides, for these metals a number of phenomenological interatomic potentials have been developed for molecular dynamics simulations, and they will be used here for the analysis of DNVMs together with the first-principle modeling.
In this study, we demonstrate that DNVMs can be used to test interatomic potentials, since they have a short spatial period and can be modeled ab initio using a small computational cell. Since the ab initio method takes into account electronic structure of metals, it is believed to be more accurate than the molecular dynamics method. Comparison of the results obtained by the two different methods will provide information on the accuracy of the interatomic potentials at large atomic displacements. Fitting the potentials for large displacements of atoms is important for modeling radiation damage to metals [50] and many other problems.
In Section 2, twelve one-component DNVMs that exist in fcc lattice are described. Simulation methods are described in Section 3. Numerical results are presented in Section 4, in particular, frequency as the function of amplitude for all studied DNVMs is reported in Section 4.1, DNVM energy in Section 4.2 and stress components induced in the lattice by DNVMs in Section 4.3. Section 5 concludes the work and provides directions for future research.
Section snippets
Symmetry-determined DNVMs for fcc lattice
Copper, nickel, and aluminum analyzed in this work have an fcc lattice shown in Fig. 1. DNVMs for the fcc lattice were obtained using the theory of bushes of nonlinear normal modes, a brief introduction to which is given in Appendix. In this work, DNVMs in three fcc metals are analyzed at large vibration amplitudes, when the effects of anharmonicity appear, for example, the frequency of vibrational modes turns out to be amplitude dependent.
Twelve one-component DNVMs (or Rosenberg modes [16])
Numerical simulation methods
Computer modeling of DNVMs is carried out by the method of molecular dynamics based on empirical interatomic potentials and by the more accurate (and more resource consuming) ab initio method based on the density functional theory (DFT) [51].
DFT simulations are performed using Quantum Espresso program package [52], [53]. The Perdew–Burke–Ernzerhof (PBE) exchange–correlation functional under the generalized gradient approximation (GGA) are used. The cutoff energy for plane waves are set to
Numerical results
Simulation results include the examples of atomic displacements as the functions of time, the frequency–amplitude dependencies for all one-component DNVMs in three metals, Cu, Ni and Al, and also the information about DNVM energy and mechanical stresses induced by DNVMs in the lattice.
We begin with the examples of the time evolution of the distance from moving atoms to their equilibrium lattice positions, , see Fig. 3. Here we choose Å and give the results for Cu modeled with the
Conclusions and future research directions
Twelve one-component DNVMs of fcc lattice are presented as exact solutions of the equations of motion of atoms, which exist regardless of the type of interatomic interactions and for any vibration amplitude. It has been proven that the fcc lattice does not support other one-component DNVMs.
Frequency–amplitude relations for the twelve one-component DNVMs in fcc Cu, Ni and Al were calculated using ab initio simulations based on the density functional theory. The same results were obtained with
CRediT authorship contribution statement
S.A. Shcherbinin: Software, Data curation, Investigation, Funding acquisition. K.A. Krylova: Software, Data curation, Investigation, Visualization. G.M. Chechin: Conceptualization, Methodology, Writing – original draft, Writing – review & editing, Funding acquisition, Supervision. E.G. Soboleva: Conceptualization, Investigation, Funding acquisition. S.V. Dmitriev: Conceptualization, Methodology, Writing – original draft, Writing – review & editing, Funding acquisition, Supervision.
Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
Acknowledgments
The work of S.A.Sh. (derivation of DNVMs, ab initio simulations) was supported by the Government of the Russian Federation (state assignment 0784-2020-0027). G.M.Ch. (derivation of DNVMs, writing the Appendix) acknowledges the financial support by the Ministry of Education and Science of the Russian Federation (state task in the field of scientific activity, scientific project No. 0852-2020-0032 (BAS0110/20-3-08IF)). Research of E.G.S. was carried out at Tomsk Polytechnic University within the
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