Overview of Drude-Lorentz Type Models and Their Applications

DOI 10.2478/nsmmt-2014-0001 Ë Nanoscale Syst.: Math. Model. Theory Appl. 2014; 3:1–13

Review Article Open Access

Paolo Di Sia* Overview of Drude-Lorentz type models and their applications

Abstract: This paper presents an overview of mathematical models for a better understanding of mechanical processes, as well as dynamics, at the nanoscale. After a short introduction related to semi-empirical and ab initio formulations, molecular dynamics simulations, atomic-scale nite element method, multiscale computational methods, the paper focuses on the Drude-Lorentz type models for the study of dynamics, considering the results of a recently appeared generalization of them for the nanoscale domain. The theoretical framework is illustrated and some examples are considered.

Keywords: Theoretical Modeling; Drude-Lorentz type Models; Computational Methods; Finite Element Meth- ods; Nano-Bio-Technology

MSC: 97Mxx, 74Axx, 82Dxx, 83Axx, 92Fxx PACS: 73.63.-b, 05.60.-k, 62.20.-x, 02.70.-c, 03.30.+p

ËË *Corresponding Author: Paolo Di Sia: Faculty of Education, Free University of Bolzano-Bozen, Viale Ratisbona 16, 39042 Bressanone-Brixen, Italy, E-mail: [email protected]

1 Introduction

The study of the mechanical behavior of nanomaterials utilizes solid (or materials) mechanics, which searches the determination of the materials responses to the action of external forces, such as elongations, deformations and fracture behavior [1]. Among the methods of reality simulation (virtual modeling), the Finite Element Method (FEM) is often used, because the fundamental aspects of the studied phenomenon are preserved and it permits the decomposition of a complex problem into a set of many small problems through the process of discretization. One solves them and after reassembles the individual solutions of simple problems in order to reconstruct the initial one [2]; the initial problem is reduced to the solution of a system of linear equations to be numerically solved. Its action goes in the direction of the reduction of the system freedom degrees number, passing from an innite number of parameters (continuous) to a nite number, corresponding to the state of some points (nodes of the elements). The equilibrium conguration of a solid corresponds to the minimum energy state of the system; the location of all nodes is determined by minimizing its energy [3]. For each element a rigidity matrix is written; assembling all of them, it is possible to obtain the matrix of the entire system and the unknown displacements are obtainable by solving a system of the form K v = P of N algebraic equations, where v is the vector of the nodal displacements and P is the vector of the unbalanced forces. Experimental results at mesoscopic scale, not explainable with the classical mechanics of materials, nd a valid explanation with nanomechanics and the study at atomic level [4–10]. A single model is often not sucient to fully describe the behavior of the medium at nanoscale; it is so born the concept of multiscale modeling [11–14]. At dynamical level, charge transport is one of the most important aspects at nanoscale, inuenced by particles dimensions and with dierent characteristics with respect to those of bulk [15–18]. In mesoscopic systems the mean free path of charges, related to scattering phenomena, can become larger than the particle dimensions; the transport depends therefore by dimensions and, in principle, corrections of the transport

© 2014 Paolo Di Sia, licensee De Gruyter Open. This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivs 3.0 License. Unauthenticated Download Date | 1/23/18 11:06 AM 2 Ë Paolo Di Sia bulk theories, for considering this peculiarity, are possible. The smallest nanostructure dimension can be less than the mean free path, requiring therefore modications to existing theoretical transport models. Various techniques have been used for the knowledge of transport phenomena, in particular analytical descriptions based on transport equations and numerical approaches, like classical and quantum Monte Carlo simulations [19, 20]. An important set of models regards the Drude-Lorentz type models, particular variations of the Drude- Lorentz and Smith models. Recently it has appeared a new theoretical model, based on the complete Fourier transform of the frequency-dependent complex conductivity of the studied system [21, 22], which makes possible the exact calculation of the analytical expressions for the three most important dynamical functions:

1. the velocities correlation function h~v(t) · ~v(0)iT at the temperature T; h i2 2. the mean square deviation of position R2(t) = ~R(t) − ~R(0) ;

3. the diusion coecient D(t) = 1/2 (dR2(t)dt). The model was tested in the last years with the most commonly and promising utilized nanomaterials at today, i.e. Si, ZnO, TiO2, GaAs, CNTs, CdTe, CdS, CIGS [23, 30]; it is in accuracy agreement with experimental data [31, 40] and oer also new interesting peculiarities, as high initial diusivity and oscillating initial behavior, to be veried with experimental techniques, such as TRTS spectroscopy [37, 43].

2 Semi-empirical and ab initio formulations

Among the most used semi-empirical formulations, we remember: a) the “Tight-Binding” Method (T-BM), a semi-empirical method normally preferred when the computational calculation is substantial [44]; it must be applied with attention, because its accuracy is related to the choice of used parameters. It is based on the method of linear combination of atomic orbitals (LCAO), originally proposed by Bloch (1928) and later revised by Slater and Koster (1954) [45]. The Schrödinger equation utilized for T-B method is:

H |Ψi = E |Ψi , (1)

where |Ψi is the wave function of the electron in the Dirac notation and H is a Hamiltonian. The resolution of the Schrödinger equation is equivalent to the minimization of the energy functional:

hΨ| H |Ψi E = , (2) hΨ | Ψi

with hΨ | Ψi = 1. The minimization carries to the expression:

δ (hΨ| H − E |Ψi) = 0. (3) b) The “ab-initio” formulations, which produce accurate results, although dependent by the made choices, for example the choice of the basis functions.

b1) Density Functional Theory (DFT): it was formulated by Hohenberg, Kohn (1964), Kohn and Sham (1965), expressing the total energy of the system by means of the functional of the electrons total density [46, 47]. The energy of a system constituted by a determined number of electrons is writable as: E = T ρ (~r ) + V ρ(~r) + Eexch ρ(~r) , (4)

where T and V are the kinetic energy of the electrons and the potential energy of the electrons-nuclei and nucleus-nucleus respectively, and~r denotes the coordinates of the electrons. Only the outer electrons (valence electrons) are considered, while the inner electrons and the nuclei are treated together as ions.

Eexch is the exchange functional; it depends by the adopted approach.

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b2) Local Density Functional Theory (LDFT): in this theory the exchange functional is written as: Z Eexch = ρ ~r εexch ρ ~r d~r, (5)

where εexch is the exchange energy for the electron inside an electron cloud with constant density [47]. For a system of N electrons, the electrons density function is expressed by the Kohn-Sham form for the

orbital of a single electron Ψi:

N N X 2 X * ρ ~r = Ψi ~r = Ψi (~r ) Ψi (~r ). (6) i = 1 i = 1 LDFT is a rather rough approximation of the molecular system, because it assumes a uniform density of electrons in the molecular system. Other used approaches are: c1) the “Non-Local Functional Approach” (N-LFA), in which the energy of the system depends also on a density gradient of the electrons; c2) the “Car-Parrinello Molecular Dynamic Method” (C-PMDM), a type of ab initio molecular dynamics usually employing periodic boundary conditions, plane wave basis sets, and density functional theory [48, 49]; c3) the “Conjugate-Gradient Method” (C-GM), an iterative method for the numerical solution of particular systems of linear equations, in particular those whose matrix is symmetric and positive-denite. It can be applied to sparse systems, that are too large to be handled by direct methods [50]; c4) the “Augmented Plane Wave Method” (APWM), for approximating the energy states of electrons in a crystal lattice; the potential is assumed to be spherically symmetric within spheres centered at each atomic nucleus and constant in the interstitial region. The wave functions are constructed by matching solutions of the Schrödinger equation within each sphere with plane wave solutions in the interstitial region; linear combinations of these wave functions are so determined by the variation method [51]; c5) the “Korringa-Kohn-Rostoker Method“ (K-K-RM), used for calculating the electronic band structure of solids; it is a Green’s function method matching the dierent types of one-electron wave functions and their derivatives, which are used in a mun-tin approximation [52, 53]; c6) the “Linearized-Mun-Thin-Orbital Method” (L-M-T-OM), a specic implementation of density functional theory within the local density approximation (LDA). For mathematical convenience the crystal is divided up into regions inside mun-tin spheres, where Schrödinger’s equation is numerically solved. In all LMTO methods the wave functions in the interstitial region are Hankel functions [54]; c7) the “Full Potential Linearized Augmented Plane Wave Method” (FPLAPWM), used as ab initio electronic structure technique with reasonable computational eciency, for simulating the electronic properties of materials on the basis of density functional theory (DFT)[55].

3 Molecular Dynamics (MD) simulations

3.1 Introduction

The simulations of the behavior at atomic level are used for problems in which the dimensions of the model are too large for a simulation at quantum level, but an atomic scale of the problem must be kept. These simulations were initially used in thermodynamics and in chemical physics, for calculating the average thermo- chemical properties of various physical systems, such as gas, liquids, solids [56, 57]. The two basic assumptions of the MD simulation are: i) the molecules or atoms are described as a system of interacting material points, whose motion is dynam- ically described by a positions vector and instantaneous velocities. The interactions among atoms are strictly dependent by the spatial distribution and the atoms distances;

Unauthenticated Download Date | 1/23/18 11:06 AM 4 Ë Paolo Di Sia ii) there is no mass changes, then the number of atoms in the system remains unchanged. The system is treated as an isolated domain in which the energy is conserved.

3.2 Lagrange equations of motion

The equations of motion for a system of interacting material points (particles or atoms), with f degrees of freedom, can generally be written in terms of a Lagrangian:

d ∂L ∂L − = 0, (7) dt ∂q˙ f ∂qf with f = 1, 2, 3,...,k. The MD simulation refers to a Cartesian coordinate system, where the above equation is simplied:

d ∂L ∂L − = 0, (8) dt ∂r˙i ∂ri with i = 1, 2, 3,...,N. Considering the homogeneity of time and space and the isotropy of inertial systems, the Lagrange equations of motion must not depend on the initial choice of the instant of observation, from the origin of the reference system and the direction of the axes, but by the absolute value of the velocity vectors. A possible function for non-interacting particles in R3 is the following:

N N X m r˙2 X m L = i i = i x˙ 2 + y˙ 2 + z˙ 2 . (9) 2 2 i i i i = 1 i = 1

Considering the interaction among particles, the Lagrangian becomes:

N X m r˙2 L = i i − U (r , r , ... , r ), (10) 2 1 2 N i = 1 with U the potential. The motion equation in Newtonian form is:

2 d ri ∂U (ri) Fi = mi 2 = − , (11) dt ∂ri with Fi inner force on the i-th atom.

3.3 Hamilton equations of motion

An alternative to the Lagrangian function in terms of generalized coordinates and momentum is the formulation of Hamilton. The independent variables are then the generalized coordinate q and the canonical momentum p. Considering the dierential dL of the full Lagrangian, one can write:

∂L ∂L d pf dq˙ f − L = − dqf + q˙ f dpf − dt. (12) ∂qf ∂t

The Hamiltonian function is writable as: ∂L ∂L d H = − dqf + q˙ f dpf − dt, (13) ∂qf ∂t with: X H (p, q, t) = q˙ f pf − L (p, q, t). (14) i

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Using the Lagrange equation, dH can be rewritten and, by comparison, one obtains the equations of motion of Hamilton: ∂H ∂H ∂H ∂L p˙ f = − ; q˙ f = ; = − . (15) ∂qf ∂pf ∂t ∂t For a conservative system of M interacting atoms in a Cartesian system, the Hamiltonian function can be written as:

H (r1, r2, ... , rM , p1, p2, ... , pM) , (16) with: ∂H ∂H r˙i = ; p˙ i = − . (17) ∂pi ∂ri If the Hamiltonian function and the initial state of the system atoms are known, one can instantly compute the positions and the momentum in successive instants.

3.4 Interatomic potentials

For representing the interatomic interaction of a system, the quantum mechanical eects should be considered, evaluating the energy due to an outer gravitational or electrostatic eld, in which the system is im- mersed. For the choice of potentials, one can consider the following features: i) accuracy in reproducing the characteristics of interest; ii) transferability, i.e. the potentials might be able to be used for dierent characteristics; iii) computational velocity, i.e. possibly fast calculations with simple potentials. The chosen potential must be appropriate to describe the studied phenomenon. Two great potentials classes can be considered: a) “Pair Potentials”, between pairs of atoms; b) “Multi-Body Potentials”. a) Pair-Potentials a1) “Hard/Soft Spheres” is the simplest potential without any cohesive interaction; it is useful for theoretical investigations of some ideal problems. The form is as follows:

∞ for rij 6 r0 U (rij) = (Hard), (18) 0 for rij > r0

− n rij U (rij) = (So), (19) r0

with: rij = ri − rj [58]. a2) Lennard and Jones proposed this potential to describe the interaction between pairs of atoms:

 !12 !6 D D U (rij) = 4 Eb  −  , (20) ri − rj ri − rj

with Eb binding energy and D diameter of collision. The minimum of this function represents the equilibrium state for a pair of atoms. The rst term represents the atomic repulsion, dominant over short distances, while the second term represents the at- traction. The Lennard-Jones potential describes well the interaction of van der Waals forces in molecular

systems and inert gases (Ar, Kr, CH4,O2,H2,C2H4, etc.) [59]. Also for half metallic materials considerable progresses were made [60]. a3) Morse Potential has the form: 2a(Leq −r) a(Leq −r) U (rij) = Ed e − 2e , (21)

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where Leq is the equilibrium length, Ed is the dislocation energy and a is a scale factor as inverse of length. It is normally used when the attractive interaction is determined by a chemical bond [61]. The potential functions, such as Lennard-Jones, have an innite range of interaction, so it is established a

cut-o radius Rc for which the interactions among atoms, separated by a distance greater than Rc , are ignored. b) Multi-Body Potentials are used to explain phenomena of “stiction“ and “snap-back”, so as for the analysis of the molecular mechanisms under large deformation and for the resistance to fracture [63, 64].

4 Atomic-scale Finite Element Method (AFEM)

The equilibrium conguration of a system of M atoms corresponds to the state of minimum total energy.

Indicating with Etot (x) = Etot (x1, x2, ... , xM) the function which denes the total system energy, the state of equilibrium is identied by: ∂E (x) tot = 0. (22) ∂x In the context of molecular dynamics simulation (MD), an interatomic potential must be considered as a tool for determining the forces acting among the atoms. For nding the equilibrium conguration, one uses potentials whose values depend on the material parameters and the mutual position of the atoms. The equation governing the problem is the same equation that governs the continuous FEM modeling: K v = P. A subset of atoms is called “element”; the elements composition depends by the atomic structure and the nature of atomic interactions. AFEM is specic for dierent materials, while the FEM modeling is generic for all materials, because an AFEM element depends by the interatomic potentials, dierent for each material [65, 66].

5 Multiscale computational methods and applications

The conventional continuous FEM approach works for problems at micro and macroscale, but does not describe the behavior of atoms interacting with each other (multi-body interactions). AFEM simulation is instead used to analyze the behavior of the material at atomic level, but it is dicult to be implemented for problems with great dimension and with high degrees of freedom. The multi-scale computational methods have been designed for the study of materials and systems at dierent scales of investigation [67]. One of the mechanics elds in which the molecular dynamics modeling has given interesting results is the fracture mechanics [68], including the crack propagation [69] and the dislocation emission [70]. In the representation of crystalline materials, for obtaining mono crystalline materials consisting of elements with dierent porosity, not random and well dened, without having huge computational costs, scientists used a Lennard-Jones potential to describe the interaction of the grains at mesoscopic scale [71]. AFEM proceeds in each iteration: – to the calculation of the rigidity matrix and the vector of unbalanced forces of order N; – to the resolution of the equation K v = P, of order N too. The method is used in the case of carbon nanotubes with simulated patterns consisting of a dierent number of atoms. Nanotubes are blocked at the ends and subjected to a lateral applied force; the deformation is studied as a function of the atoms number of the simulated pattern. It can simultaneously take into consid- eration the multiple interactions among atoms, so as the covalent bonds and van der Waals forces; it permits to determine the frequency and vibration modes of the nanotubes.

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6 Drude-Lorentz type models

The Drude model is based on kinetic theory model of an “electronic gas” in a solid. It is assumed that all 2 electrons have the same average kinetic energy Em = (3/2)kB T = mvm/2 [72]. ~ ~ The current density is dened as J = σcond E, with σcond electric conductivity. This result, obtained by Drude, is an important goal of the classic theory for the conduction of metals (said “Drude theory”). The Drude-Lorentz model is an improvement of Drude model, with explanations about statistical aspects. The electrons are considered as free charges, with charge “-e”; they are described by a Maxwellian velocity − i ω t distribution. In presence of an electric eld of the form E (t) = E0 e , the complex conductivity assumes the form:

σω = σ0 / (1 − i ω τ). (23) The model has received great success and is still used, but has also underlined series diculties [73].

The Smith model is a variation of the Drude-Lorentz model, in which appears a parameter cn accounting for the anisotropy of scattering upon the rst scattering event. The analytical form of the complex conductivity is [74]: * 2 " ∞ # n e τ X cn σ (ω) = 1 + , (24) m (1 − i ω τ) (1 − i ω τ)n n = 1 with n* eective electron density and τ relaxation time. For cn = 0 it reduces to Drude-Lorentz model. The “Eective Medium Theories” (EMTs) are variations in which the electromagnetic interactions between pure materials and host matrixes are approximately taken into account. Among EMTs, the “Maxwell- Garnett model” (MG) usually describes an isotropic matrix containing spherical inclusions, that are isolated from each other, such as the metal particles dispersed in a surrounding host matrix; the “Bruggeman model” (BR) is used mainly for the study of eective conductivity in systems constituting of spherical multicompo- nent inclusions with dierent conductivity [75–77].

7 Recent generalizations of Drude-Lorentz type models

A recent extension of the Drude-Lorentz model showed to t very well with experimental scientic data and oers interesting new predictions of various peculiarities at nanolevel [18, 21–40]. The model contains also a gauge factor, which permits its use to study the dynamics of reality processes from sub-nano to macrolevel, which present oscillations in time, so as characteristics of diusivity in time [18]. It is based on the complete Fourier transform of the frequency-dependent complex conductivity σ (ω) of a system, which can be deduced from linear response theory (Green-Kubo formula) [78, 79]:

∞ β Z Z 2. iωt α β σβα(ω) = (e } V) dte dλh~v (t − iλ)~v (0)i. (25) 0 0

The inversion of Eq. 25 permits to nd the velocities correlation function h~v (t) · ~v (0)iT inside the integral. The presence of an integration from 0 to ∞ constitutes a problem for the analytical inversion; it can be overcame evaluating the integral on the entire time axis (−∞, +∞). The new idea was to consider the real part of the complex conductivity in Eq. 25; doing it, the extension to the entire time axis is possible and the complete Fourier transform can be performed, obtaining directly real velocities [21, 22]. The integral can be resolved in the complex plane considering a Cauchy integration; the velocities correlation function can be exactly eval- uated by the residue theorem [80]. The analytical form of h~v (t) · ~v (0)iT permits to obtain also the analytical 2 form of R (t) and D(t) , both writable as a function of h~v (t) · ~v (0)iT :

t Z D E R2(t) = 2 dt′ t − t′ ~v (t′) · ~v (0) , (26)

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t dR2(t) Z D E D(t) = = dt′ ~v (t′) · ~v (0) . (27) 2 dt 0 The core of this formulation is:

+ ∞ KTV Z h~vα (0) ~vβ (t)i = dω Re σ (ω) ei ω t . (28) T π e2 βα − ∞

The entire time axis (−∞, +∞) is involved, not the half time axis (0, +∞), as usually considered in litera- ture [81]. Eq. 28 is the nal step of an analytical calculation, which starts by linear response theory, considering a system with an Hamiltonian written as:

H = H0 + H1, (29)

− i ω t η t with H1 = λ A e e having small eects with respect to H0. The parameter λ is a real quantity and η a positive number, so that in remote past the perturbation is negligible (adiabatic representation: limt→−∞H1 = 0). In the case of an electric eld of frequency ω, it is therefore: ~ H1 = e E · ~r. (30) If the electric eld is constant in space and its evolution depends on time, it is writable as follows:

~ ~ i ω t E = E0 e , (31) and the time dependent corresponding current is:

~J (t) = σ ( ω ) ~E ( t ). (32)

Following the standard time-dependent approach, it is possible to nd a general formula for the linear response of a dipole moment density ~B = e~rV in the β direction with the electric eld ~E in the α direction, where V is the volume of the system. This permits to deduce the susceptibility χ(ω), which is correlated to σ (ω) through the relation: σ(ω) 1 + 4 π χ(ω) = 1 + 4 π i . (33) ω Analytical calculations permit to write the real part of the complex conductivity σ (ω) as:

2 e ω π − } ω/KT Re σβα(ω) = Sβα(ω) 1 − e , (34) V } with: + ∞ Z D α β E − i ω t Sβα(ω) = dt ~r (0) ~r (t) e . (35) T − ∞

The part h···iT in the integral 35 is the thermal average, and the exponential factor arises from equilibrium thermal weights for Fermi particles. By considering the identity v = d r = i [H, r], Eq. 34 can be written in dt } a form containing the velocities correlation function instead of the position correlation function. Assuming the high temperature limit } ω KT, valid in such contests), we obtain:

+ ∞ 2 Z e D α β E − i ω t Re σβα(ω) = dt ~v (0)~v (t) e . (36) 2 VKT T − ∞ The integral in Eq. 36 spans the entire t axis, so the complete inverse Fourier transform is possible giving Eq. 28.

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8 Relativistic motion in nanostructures

The starting point for the analysis of a possible relativistic motion of carriers travelling in a nanostructure (charged particles in general, not only electrons), in a rst step in which gravitational eects are not considered (special relativity), is to consider the (special) relativistic Newton’s law: d X (m ~v) = ~F . (37) dt part i i About the forces operating in Eq. 37, it has been considered an outer passive elastic-type force, a passive friction-type force and an outer oscillating electric eld (Eq. 31) [30, 82, 83], canonical “ansatz” in such situ- ations. With the procedure used in the classical case, it has been searched for values of ω which vanish the denominator of the real part of the complex conductivity σ (ω) [21, 63]. The calculation of the velocities correlation function ~v (0) · ~v (t) is the rst step of this procedure. For useful comparison, these functions, for the classical, quantum and relativistic cases respectively, are reported below: KT t α t 1 α t ~v (0) · ~v (t) = exp − cos R − sin R , (38) m 2 τ 2 τ αR 2 τ √ p 2 2 with αR = ∆ = 4 τ ω0 − 1 (τ = relaxation time; ω0 = center frequency) [21]; KT X t αi R t 1 αi R t ~v (0) · ~v (t) = fi exp − cos − sin , (39) m 2 τi 2 τi αi R 2 τi i √ q 2 2 with αi R = ∆ = 4 τi ωi − 1 (τi = relaxation time of the i-th state; ωi = (Ei − E0) } = frequency of the i-th state) [22]; KT 1 t α t 1 α t ~v (0) · ~v (t) = exp − cos Rrel − sin Rrel , (40) m0 γ ρ 2 τ ρ 2 ρ τ αRrel 2 ρ τ

√ q 2 2 4 ρ ω0 τ p 2 2 2 2 with αRrel = ∆ = γ − 1 .(β = v/c, γ = 1/ 1 − β , ρ = 1 + β γ = γ ) [30]. Eqs 38 - 40 are related to the case ∆ > 0; for ∆ = 0 we obtain the Drude model, for ∆ < 0 we have a Smith behavior of ~v (0) · ~v (t) [21, 22, 30].

9 Examples of application

About the main goals of the new considered model, we underline: a) its ability to conrm previous results, without further particular assumptions, such as the presence of

the cn parameters in the expression of the conductivity of the model (Eq. 24) [74]; b) the possibility to nd new features at the nanoscale [24, 28], and to make predictions about the behavior of nanodevices [23]; c) the elegance of the analytical approach, with respect to the most utilized models at the nanoscale (numerical approaches, Monte Carlo simulations). As example of application and for conrming the above said, we consider the motion of electrons, at dierent velocities, in a nanostructure of ZnO [37, 38, 84, 85]. Currently is not easy, from an experimental point of view, to consider and evaluate ultrafast velocities of carriers in a nanostructure. Fig. 1 represents the evolution of

~v (0) · ~v (t) in time for a xed value of αR, in relation to three dierent velocities of electrons. We xed τ = 10− 13 s and T = 300 K. The classical “Drude” velocity v = 107 cm/s implies a negligible variation in mass for the electrons (in Fig. 1 it is superposed to blue solid line). The increase in velocity tends to raise the wavelength of the damped oscillation, reducing its amplitude. The same situation is presented in Fig. 2,

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7 Fig. 1. h~v (0) · ~v (t)i vs t for xed value αR = 10; the considered velocity of the carrier is v = 10 cm/s (blue solid line), v = 1010 cm/s (red dashed line) and v = 2 · 1010 cm/s (green dots line) (color online).

7 Fig. 2. h~v (0) · ~v (t)i vs t for xed value αR = 50; the considered velocity of the carrier is v = 10 cm/s (blue solid line), v = 1010 cm/s (red dashed line) and v = 2 · 1010 cm/s (green dots line) (color online).

with a dierent value of the parameter αR [21]. The initial more marked compression of the curve (blue solid line) obeys to the variations indicated in the previous case.d Analogous results can be found considering the parameter αI [22]. The typical Smith behavior of ~v (0) · ~v (t) tends to become negative in longer times and the curves approach the x-axis when the velocity of the carrier increases.

10 Conclusions

In this work an overview of the current state-of-the-art about the mechanical and dynamical study of systems with application in the nanodomain has been considered. The main models, which govern the transport in nanostructures, were considered; in particular, starting by the Drude model, the most important Drude- Lorentz type models were listed, arriving to a recent extension, formulated both in classical and in quantum level, and currently under construction at relativistic level, for the explanation of possible ultrafast motion of charged particles inside nanostructures. This last extension is mathematically very elegant, because of the analytical approach, and is giving interesting conrmations with respect to the previous models, so as new informations about the dynamics of systems at nanoscale, which could be appropriately tested through experimental time-resolved techniques, like TRTS.

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7 Fig. 3. h~v (0) · ~v (t)i vs t for xed value αI = 0.5; the considered velocity of the carrier is v = 10 cm/s (blue dashed line) (superposed to the red dots line, representing the classical case), v = 1010 cm/s (violet dot-dashed line) and v = 2.5 · 1010 cm/s (clear blue solid line) (color online).

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Received July 9, 2012; accepted October 30, 2013.

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