Some Aspects of Generalized Dynamics of Quasiparticles in Crystals with Unit Cell of Arbitrary Complexity

Abstract

The conditions, under which the general description of the dynamical properties of quasiparticles is almost identical with those of real relativistic particles, are analyzed. Such analysis is, especially, actual today in connection with the growing interest in electronic properties of graphene and other nanostructures of carbon origin (fullerenes, nanotubes, etc.). The development of the traditional applications of quasiparticles (superfluidity, transfer of charge or energy) also requires a generalized analysis of dynamical properties of quasiparticles. The problem of the correlation of quantum and classical methods of description of the quasiparticles in the case of the excited states of crystals is considered. In order to focus attention on the discussed problem, the obtained results are demonstrated on the example of electronic excitations of crystals in the simplest case where other effects are neglected (phonons, defects, high density of excitations, which would require the account for interactions between them, the response of a lattice to excitations, and so forth). It is shown that such excitations can be described in three ways simultaneously. The first is the quantum description of the examined excitations in terms of wave functions and eigenvalues of energy. The second method is classical. It arises from the quantum method and is formulated in terms of the wave momentum. The third method, which follows from the second one, is also a description of the classical type, but is related to the other momentum -the mechanical one. The latter descriptions (the third or second one) make it possible to interpret the experimental data in terms of the usual relativistic dynamics. K e y w o r d s: quasiparticles, dispersive dependence, relativistic approximation, dynamical Dirac model, graphenes.c ○ A.D. SUPRUN, L.V. SHMELEVA, 2016ties of real particles, especially, in connection with the mentioned dynamical Dirac model in graphenes [12].The analysis of the general dynamical properties of free quasiparticles is fulfilled on the basis of one of the major characteristics of the excited states of condensed matter. This is the dispersive dependence of the energy or frequency on a wave vector [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15]. We will also examine the general conditions, under which the dynamical Dirac model becomes real. The problem of relationship between quantum and classical methods of description of quasiparticles is considered as well. Materials and Methods. Common Remarks About Dynamical Properties of Quasiparticles in Crystals Basic relationsWe will analyze the dependence (k) ≡ ( 1 , 2 , 3 ). Since the wave vector k is always associated with the

Features of the Generalized Dynamics of Quasiparticles in Graphene

“…At first glance, the dispersion relation (6) for the electron in graphene is not very different from the same relation for simple (one atom per the cell) rectangular lattice [24–27]. In fact, it complicates the situation so that it is not always possible to carry out a full analysis of the dynamic properties of electron in graphene.…”

“…These numerical and graphical results are interesting because in the area {| p x |, | p y |} ≤ π /3 , these relations are simplified almost to the form (8). Since in many applications such an area is enough, then in this case, the whole dynamics of a simple rectangular lattice, which was studied in details in papers [ 24 – 27 ], is reproduced. The region {| p x |, | p y |} ≤ π /3, usually meets research needs with a good margin at the low-energy electron injection into the conduction band of graphene and when using fields of not high intensity (much lower of electrostatic threshold of material).…”

“…It is known [ 28 ] that in the case of crystals with a complex unit cell (several atoms in one cell), the Hamilton functional for the one-electron excitations without account of lattice response on the excitation [ 27 ] is defined by the equation [ 24 ]:…”

“…It is shown that all the dynamic characteristics are significantly different from the standard ones for crystals with a simple lattice [ 24 – 27 ].)…”