Dielectric functions of Pd and Zr transition metals: an application of Drude–Lorentz models with simulated annealing optimization

William E. Vargas

doi:10.1364/AO.56.001266

Applied Optics
Vol. 56,
Issue 4,
pp. 1266-1275
(2017)
•https://doi.org/10.1364/AO.56.001266

Dielectric functions of Pd and Zr transition metals: an application of Drude–Lorentz models with simulated annealing optimization

William E. Vargas

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William E. Vargas, "Dielectric functions of Pd and Zr transition metals: an application of Drude–Lorentz models with simulated annealing optimization," Appl. Opt. 56, 1266-1275 (2017)

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Abstract

An accepted-probability-controlled simulated annealing (APCSA) method has shown to be a valuable tool to describe, in parametric form, by means of an extended Drude–Lorentz model, the dielectric function of several metals through infrared, visible, and ultraviolet photon energies [Appl. Opt. 37, 5271 (1998) [CrossRef] ]. In this work, an improved APCSA approach is used to estimate the parameters involved in an extended Drude–Lorentz type model which incorporates the dielectric constant due to a background electronic polarization in the Drude term and the normalization of the individual oscillation strengths involved in the Lorentz contributions to the dielectric function. This last approach allows us to introduce a new parameter $z$ to be optimized: the number density ratio, i.e., the ratio between number density of conduction electrons and number density of metal ions. From the optimization of the $z$ value within this novel approach, we evaluate other parameters: electrical resistivity, electron mean free path, effective mass of conduction electrons and relaxation time, Fermi energy, electronic density of states at the Fermi level, and electronic heat capacity coefficient. Application of the model is carried out to describe the dielectric functions of two transition metals, Pd and Zr, through ultraviolet, visible, and infrared photon energies.

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Rakic et al. Optimization (Model A)I				This Work (Model A)
$f_{o}$	0.330	$γ_{o}$	0.009	$f_{o}$	0.343	$γ_{o}$	0.008
$f_{1}$	0.769	$f_{3}$	0.309	$f_{1}$	1.197	$f_{3}$	0.244
$γ_{1}$	2.343	$γ_{3}$	2.022	$γ_{1}$	3.228	$γ_{3}$	2.258
$ω_{1}$	0.066	$ω_{3}$	2.432	$ω_{1}$	0.045	$ω_{3}$	4.864
$σ_{1}$	0.694	$σ_{3}$	1.167	$σ_{1}$	0.999	$σ_{3}$	0.584
$f_{2}$	0.093	$f_{4}$	0.409	$f_{2}$	0.114	$f_{4}$	0.765
$γ_{2}$	0.497	$γ_{4}$	0.119	$γ_{2}$	0.557	$γ_{4}$	0.165
$ω_{2}$	0.502	$ω_{4}$	5.987	$ω_{2}$	0.505	$ω_{4}$	9.901
$σ_{2}$	0.027	$σ_{4}$	1.331	$σ_{2}$	0.014	$σ_{4}$	1.784

$f_{o}$	0.343	$γ_{o}$	0.011
$f_{1}$	1.009	$ω_{3}$	14.81
$γ_{1}$	2.129	$σ_{3}$	3.837
$ω_{1}$	0.673	$f_{4}$	0.900
$σ_{1}$ ^b	0.949	$γ_{4}$	0.315
$f_{2}$	0.225	$ω_{4}$	22.17
$γ_{2}$	0.799	$σ_{4}$	2.010
$ω_{2}$	4.484	$f_{5}$	6.985
$σ_{2}$	1.249	$γ_{5}$	3.177
$f_{3}$	2.076	$ω_{5}$	59.49
$γ_{3}$	3.997	$σ_{5}$	3.516

$ω_{p}$	10.15	$γ_{o}$	0.091
$ϵ_{\infty}$	2.518	$f_{3}$	0.108
$f_{1}$	0.536	$γ_{3}$	2.617
$γ_{1}$	3.425	$ω_{3}$	4.949
$ω_{1}$	0.067	$σ_{3}$	0.315
$σ_{1}$	1.056	$f_{4}$	0.332
$f_{2}$	0.024	$γ_{4}$	0.204
$γ_{2}$	0.616	$ω_{4}$	9.757
$ω_{2}$	0.450	$σ_{4}$	1.586
$σ_{2}$	0.016	$z$	0.475
$F_{B}$	0.080	$ρ_{e}$ (μΩcm)	6.6
$α$	0.431	$E_{F}$ (eV)	8.57
$τ (f s)$	7.21	$N (E_{F})$ ^b	18.1
$L_{b}$ (nm)	19.1	$γ$ ^c	5.02

$ω_{p} \pm Δ ω_{p}$	$14.69 \pm 0.01$
$ϵ_{\infty} \pm Δ ϵ_{\infty}$	$5.88 \pm 0.03$
$γ_{o} \pm Δ γ_{o}$	$1.200 \pm 0.007$
$f_{1} \pm Δ f_{1}$	$0.166 \pm 0.002$
$γ_{1} \pm Δ γ_{1}$	$1.606 \pm 0.006$
$ω_{1} \pm Δ ω_{1}$	$1.579 \pm 0.001$
$σ_{1} \pm Δ σ_{1}$	$0.353 \pm 0.002$
$f_{2} \pm Δ f_{2}$	$0.092 \pm 0.001$
$γ_{2} \pm Δ γ_{2}$	$2.811 \pm 0.008$
$ω_{2} \pm Δ ω_{2}$	$3.685 \pm 0.004$
$σ_{2} \pm Δ σ_{2}$	$0.535 \pm 0.003$
$f_{3} \pm Δ f_{3}$	$0.742 \pm 0.002$
$γ_{3} \pm Δ γ_{3}$ ^b	$0.082 \pm 0.001$
$ω_{3} \pm Δ ω_{3}$	$10.77 \pm 0.04$
$σ_{3} \pm Δ σ_{3}$ ^b	$7.59 \pm 0.04$
$z \pm Δ z$	$0.96 \pm 0.02$
$α \pm Δ α$	$0.79 \pm 0.01$
$F_{B} \pm Δ F_{B}$	$0.11 \pm 0.03$
$τ \pm Δ τ$ (fs)	$0.550 \pm 0.003$
$ρ_{e} \pm Δ ρ_{e}$ (μΩcm)	$41.4 \pm 0.2$
$E_{F} \pm Δ E_{F}$ (eV)	$11.49 \pm 0.04$
$N (E_{F}) \pm Δ N (E_{F})$ ^c	$10.3 \pm 0.3$
$L_{b}$ (nm)	$1.25 \pm 0.01$
$γ$ ^d	$2.86 \pm 0.07$

Rakic et al. Optimization (Model A)I				This Work (Model A)
$f_{o}$	0.330	$γ_{o}$	0.009	$f_{o}$	0.343	$γ_{o}$	0.008
$f_{1}$	0.769	$f_{3}$	0.309	$f_{1}$	1.197	$f_{3}$	0.244
$γ_{1}$	2.343	$γ_{3}$	2.022	$γ_{1}$	3.228	$γ_{3}$	2.258
$ω_{1}$	0.066	$ω_{3}$	2.432	$ω_{1}$	0.045	$ω_{3}$	4.864
$σ_{1}$	0.694	$σ_{3}$	1.167	$σ_{1}$	0.999	$σ_{3}$	0.584
$f_{2}$	0.093	$f_{4}$	0.409	$f_{2}$	0.114	$f_{4}$	0.765
$γ_{2}$	0.497	$γ_{4}$	0.119	$γ_{2}$	0.557	$γ_{4}$	0.165
$ω_{2}$	0.502	$ω_{4}$	5.987	$ω_{2}$	0.505	$ω_{4}$	9.901
$σ_{2}$	0.027	$σ_{4}$	1.331	$σ_{2}$	0.014	$σ_{4}$	1.784

Abstract

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Tables (4)

Equations (8)

Applied Optics