Analytic representations of the dielectric functions of materials for device and structural modeling

The frequency dependent dielectric permittivity of dispersive materials is commonly modeled as a rational polynomial based on multiple Debye, Drude, or Lorentz terms in the finite-difference time-domain (FDTD) method. We identify a simple effective model in which dielectric polarization depends both on the electric field and its first time derivative. This enables nearly exact FDTD simulation of light propagation and absorption in silicon in the spectral range of 300-1000 nm. where ε ∞ is permitivitty at infinite frequency, and a p;j , b p;j are real fitting coefficients that do not necessarily have a physical meaning. There are recent reports on successful application of the critical point model (with a p;1 ≠ 0) for the description of the dielectric function of gold [6,7], silver [7,8], aluminum, and chromium [7] in the wide wavelength range. This model was implemented in FDTD with the help of the recursive convolution (RC) technique [9]. Dispersion profiles of complex materials cannot always be fitted using a small number, P, of terms in (1). Alternately, one can divide the required wavelength range into subranges for separate fittings in each subrange. In this case, multiple FDTD simulations should be performed, followed by merging of the separate results. This makes simulation cumbersome and inefficient.In traditional fitting, each dielectric susceptibility term (2) consists of either a single imaginary pole (Debye model), two complex poles (Lorentz model), or an imaginary pole plus pole at zero (Drude model) in the complex ω-plane. More flexible fitting (some of which is captured in the critical point model [5]

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“…Instead, it can be approximated using multiple Debye, Drude, Lorentz [1], or critical point [5] terms of the form …”

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confidence: 99%

“…More flexible fitting (some of which is captured in the critical point model [5]), is enabled by additional terms proportional to −iω n in both numerators and denominators of (2). In the numerator, they describe direct polarization of the medium by time derivatives of the applied electric field.…”

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confidence: 99%

Effective optical response of silicon to sunlight in the finite-difference time-domain method

Deinega

John

2011

Opt. Lett.

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“…Throughout the calculation, we use the data [9] available in the existing literature for the dispersion of the permittivity of silicon [10], silicon oxide [11] and silicon carbide [12]. With reference to Fig.…”

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confidence: 99%

Visibility of graphene flakes on a dielectric substrate

Abergel

Russell

Falko

2007

Applied Physics Letters

284

204

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We model the optical visibility of monolayer and bilayer graphene deposited on a SiO2/Si substrate or thermally annealed on the surface of SiC. Visibility is much stonger in reection than in transmission, reaching the optimum conditions when the bare substrate transmits light resonantly. In the optical range of frequencies a bilayer is approximately twice as visible as a monolayer thereby making the two types of graphene distinguishable from each other.Monolayer graphene is a single two-dimensional honeycomb lattice of carbon atoms. Although the rst graphene-based structures were only recently fabricated [1] they have quickly become the subject of an extensive research eort [2,3,4]. Monolayer graphene is a zero-gap semiconductor with a Dirac-like dispersion of chiral quasiparticles near the K points of the hexagonal rst Brillouin zone [5]. Bilayer graphene is a pair of graphene sheets with the Bernal (AB) stacking arrangement. In the low-energy spectrum of this material [6] the conduction and valence bands both consist of two quadratic branches split by the inter-layer coupling γ 1 . Measurements of the quantum Hall eect [1,2,7] and ARPES experiments [8] have conrmed that these are the low-energy band structures of these materials.The widespread microcleavage technique used to fabricate graphene-based devices requires a visual inspection of the substrate [1] to nd akes of one or two layers thickness. In this Letter, we aim to determine the optimum conditions for making these akes optically visible when they are deposited on various substrates. The parameters at one's disposal (see Fig. 1) are the frequency ω, angleᾱ and aperture δα of the focused incident radiation, as well as the thicknesses of the various layers of the underlying dielectric materials.Below we calculate the reection of non-polarized incident light taking the geometry of the substrate intoBare substrate, σ = 0Bilayer flake, σ2 substrate geometry: Fig. 1, we analyze the reection R of light from a substrate with a ake on it and compare this to the reection R 0 of a bare (graphenefree) substrate. The optical visibility of a ake is then determined as the contrast between two such parts of the sample studied using a monochromatic light source:The scattering of light is analyzed using the electromagnetic wave equations in vacuum and dielectric media and the standard boundary conditions at interfaces between dierent materials,The superscripts and ⊥ stand for the components of the eld parallel and perpendicular to the interface respectively, n is the unit vector normal to the interface, the subscript 1 (2) denotes the eld below (above) the interface, and σ(ω) is the frequency-dependent conductivity of a graphene ake and D = (ω) E. One more boundary condition (on the perpendicular components of H) duplicates Snell's law.Having in mind an optical setup used to locate a small ake, we consider a beam of light focused by a lens, so that the light in the beam arrives at the substrate surface with some aperture δα (see Fig. 1). Therefore the measurable reecta...

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“…Effectiveness in suppressing endpoint-discontinuity effects is indicated by the values of n at which the dependences diverge from the information [12], obtained as a piecewise-continuous, Kramers-Kronig-consistent set of analytic functions determined self-consistently from data for oxidized (001)Si samples covering a range of oxide thicknesses with assumptions made about the Si±SiO 2 interface. We also investigate the fully analytic 4-oscillator, 21-parameter representation of Leng et al [13], whose parameters were determined by fitting to the results of [10]. The data of [8, 10, and 12] were obtained with rotating-analyzer ellipsometers, while those of [9] were obtained with a photoelastic-modulator system.…”

Section: Application To Crystalline Simentioning

confidence: 99%

“…We consider only he 1 i. The spectra analyzed either consist of 250 data points equally spaced in energy from 1.5 to 6.0 eV ( [8,10]), were projected onto these energies by linear interpolation of published tables ( [9,12]), or were evaluated directly at these energies from the published analytic function ( [13]). We analyzed the segment extending from j 1 80 through j 2 130 inclusive, which corresponds to energies of 2.9277 and 3.8313 eV, respectively.…”

Section: Application To Crystalline Simentioning

confidence: 99%