Optical scatterings in layered systems

Optical properties, reﬂectance, absorbance and transmittance spectra, are fully investigated for the layered structures through the development of theoretical framework. The transverse dielectric function, which characterizes the dynamic charge screenings, can cover all the intralayer and interlayer atomic interactions under the electro-magnetic wave perturbation. By the continuous reﬂection and transmission scatterings at two surfaces, their analytic formulas are established from the vertical valence-state transitions and the boundary conditions. They are also suitable for ﬁnite-width bulk materials. Most important, this study is fully combined with the generalized tight-binding model with all the intrinsic interactions and external ﬁeld.

a multi-/few-layer graphene system will be evaluated under the generalized tight-binding model [14], being suitable under perpendicular gate voltages and uniform magnetic fields.
In this work, a full optical theory is developed for the 2D layered graphene systems.
Their band structures and wave functions are calculated within the generalized tightbinding model in the presence/absence of external fields [4,14]. They are responsible for the dynamic charge screenings of an electromagnetic field through the evaluated dielectric function at long wavelength limit. The perturbation of an electric dipole moment is calculated by the gradient approximation [4,5]. By the direct combination of (ω ) and two boundary conditions at the upper and lower surfaces, the whole propagating paths can be characterized by the optical excitation process inside the multi-layer honeycomb lattices and the continuous boundary scatterings of incidence, reflection and refraction. As a result, both reflectance and transmittance are expressed as an analytic formula, covering the useful information about the crystal symmetries [27] and the main feature of electronic properties [4,27,28].

Theories
The featured optical excitations are directly reflected from the main characteristics of electronic structures. In general, the low-lying electronic energy spectra of layered graphene systems, with E c,v ≤ 3 eV [4][5][6][7], are well characterized by the extended C-2p z π bondings on each honeycomb lattice. The first-principles calculations would become too cumbersome to deal with the other essential properties for thin film beyond the layer number of one hundred. On the other side, the generalized tight-binding model is suitable for a lot layers and even in perpendicular electric and magnetic fields [4,5]. For a N L -layer ABstacked graphene, the C-2p z -based Hamiltonian, the 2N L × 2N L Hermitian matrix,, could be expressed as where h = exp(ik x b) + cos(k y √ 3b/2)exp(−ik y b/2). When any condensed-matter system is present in an electromagnetic field, all the charges very efficiently screen the external perturbations by creating the induced current density.
Their electronic states are vertically excited from the occupied to unoccupied ones under the quasiparticle absorption process [32]. That is, a valence electron absorbs one photon becomes an excited conduction one. The electric dipole is responsible for driving all the optical transition channel. By the delicate calculations on its inner products, the imaginarypart dielectric function could be evaluated from the Fermi-golden rule, as expressed in where h (h ) represents the initial (final) state, n F (E(k)) is the Fermi-Dirac distribution function, and δ (=0.01 eV), the energy width due to various deexcitation mechanisms, is treated as a free parameter in the calculations. The excitation energy Equation (2) covers the joint density of states and the velocity matrix elements, respectively, dominating the number of excitation channels and scattering amplitudes {optical selection rules; [4][5][6][7]}. The gradient approximation is very convenient and reliable in resolving the second term, e.g., the successful evaluations on graphite intercalation compounds [33], carbon nanotubes [34], and few-layer graphenes [4,5,[35][36][37]. This linear Kubo formula, which is directly combined with the generalized model, is successful in exploring the diverse magneto-optical excitations of 2D layered structures, e.g., the systematic investigations on magnetic quantization phenomena of layered group-IV and group-V materials [38].
Very interesting, the significant couplings of carrier charges and electromagnetic waves are rather sensitive to position and time. In general, their dynamic screening and propagating behaviors are dominated by the Helmholtz equation [32]. The linear energy dispersion relation in free space would become a non-linear one of (ω ) ω 2 = c 2 k 2 in any material.
The real-and imaginary-part dielectric function could lead to the reduce of group velocity and the decline of spectrum intensity, mainly owing to the imaginary wave vector through the specific relations k = [ω /c]N (ω ) and N (ω ) = (ω ) = n(ω ) + iκ(ω ) [32]. n and κ, respectively, represent the refractive index and extinction coefficient. The absorption coefficient, α = [κ ω /c] −1 , is very useful in understanding the characteristic decay length under the efficient photon absorptions of the valence quasiparticles. Apparently, layered graphene systems provide the effective inter-π-band transitions in the particle-field couplings [ Fig.   1(b)], even dominating the continuous scatterings at the upper and lower boundaries. It should be noticed that such π valence electrons absorb the electromagnetic-field energy and then release them to heat reservoir by gradually enhancing temperature. The energy transfer between them is an interesting open issue [39,40].
An electromagnetic wave with in a free space [a dielectric constant of 0 ] is assumed to be normally incident on the upper surface a finite-width sample, as clearly illustrated in Fig. 2. Electric and magnetic fields behave as Its propagation experiences the first incidence, reflection and transmission, indicating the significant coupling of fields and charges. Obviously, this scattering event is mainly determined by two independent boundary conditions, namely, the continuities of tangential electric and magnetic fields The transmitted wave will propagate along the z-direction, but with the declines intensity by the Landau damping of the inter-π-band excitations [32]. After reaching the opposite surface, the similar scattering behavior come to exist there. Part of wave can penetrate into the outside environment and then make certain contribution to the transmitted field.
The others are directly reflected and continue their interactions with the whole valence charges. With the next scattering at the initial surface, the second contribution to the reflected field is achieved through the same boundary conditions. A series of reflections and transmissions, which, respectively, occur at the upper and lower surfaces, could be analytically calculated and expressed as m+1 j The total electric fields of the reflected [Eq.

Results
The  1) and (2). They could be calculated in the presence of perpendicular electric and magnetic fields, since the sublattice-based superposition is utilized in the generalized tight-binding model.
Also, the rich intrinsic interactions, which arise from the stacking symmetries [4], the multi-orbital hybridizations [38] and spin-orbital couplings [38], are included in the various Hamiltonians. Reflectance and transmittance spectra of few-layer graphene systems are thus expected to exhibit the diverse magnetic quantization phenomena under the various stacking configurations [38]. But when a bulk system gradually reduces its thickness, electronic energy spectra and wave functions will display a dramatic transform and thus dynamic charge responses. The width-created drastic changes of essential properties become an extra challenge. Such issue is worthy of the systematic investigations. Apparently, the current optical scatterings of layered materials has been covered in the theoretical development of quasiparticle framework [14]. That is, electronic, magnetic, optical, Coulombexcitation and transport properties could be explored simultaneously from the modified and unified phenomenological models.

Conclusions
Optical scatterings are fully explored for layered materials under the theoretical framework of quasiparticles [14], being suitable for the rich and unique intrinsic interactions and the external fields. The fundamental properties are unified together and could be understood simultaneously. Specifically, layered graphene systems, with AB stackings, clearly show the unusual reflectance and transmittance spectra through the quantum-size effects. The strong frequency and thickness dependences illustrate the prominent π-electronic excitations and their significant couplings with electromagnetic waves.