Plasmon resonances of Ag(001) and Ag(111) studied by power density absorption and photoyield
Highlights
► Ampère–Maxwell equation modeling of plasmon resonances in Ag(001) and Ag(111) ► Good agreement between calculated observables and experimental photoelectron spectra. ► PES experimental–theoretical disagreement in Ag(001), poor quality final state wf ► Vacuum maximum and zero surface induced by the laser electron density as in TDDFT
Introduction
The plasmon resonances appear on the Ag(001) and Ag(111) surfaces in the 3–4 eV region. These resonances are used in a wide class of applications ranging from physics to medicine (see e.g. Sarid and Challener [6]).
Experimentally the photoelectron spectra (PES) of low index silver surfaces have been obtained by several authors [4], [5], [7], [8], [9], [10], [11], [12]. To lower the work function of the clean silver surfaces from 4.5 to 2.8 eV and observe the surface plasmon resonances, Barman et al. [4], [5], [12] adsorbed 1–180 ML of Na. Then these authors measured the 2.8–10 eV PES and found a multipole surface plasmon resonance m at about 3.70 eV. One can avoid the use of sodium by performing two photon photoemission experiments (see e.g. Ueba and Gumhalter [13] and Winkelmann et al. [11]). However to our knowledge no measurements of the two photon PES have been performed in the region of interest.
The electron energy loss spectra (EELS) [14], [15], [16], [17], [18], [19] also revealed a resonance in the 3–4 eV region first identified as a monopole s then as a multipole m surface plasmon resonance. Using the inverse photoemission technique Himpsel and Ortega [20] found the multipole m surface plasmon resonance at about 4 eV. Using the material constants taken from Palik's tables [21], [22] Marini et al. [23] and other authors calculated at 3.90 eV a dip in the reflectance corresponding to the bulk plasmon resonance.
To study surface plasmon resonances of metallic surfaces Liebsch et al. [24], [25], [26], [27], [28], [29] have developed a “s–d” model. This model is based on the dynamical response function, Poisson equation and the time dependent density functional theory (TDDFT) in its local density approximation form (TDLDA). In the valence region of silver two bands “5s” and “4d” participate in the interaction. The “5s” electron band, modeled as a semi-infinite jellium system, is characterized by a nonlocal response function χ(z,z′,q∥,). The “4d” electron band is modeled as a polarizable medium using the dielectric function of the bulk. When calculating the EELS, the “s–d” model predicts: bulk plasmon eV, monopole surface plasmon resonance and multipole surface plasmon resonance m∗ ≈ 0.8p ≈ 6. 7 eV(p = 8. 375 eV)p. The EELS experiments [14], [15], [16], [17], [18], [19] do not cover the region above 6 eV whereas the PES of Barman et al. [4], [5], [12] does but no resonance appears at 6.7 eV. In the region 2.8–10 eV Marini et al. [23] calculated the EELS and the reflectance using a GW method. Their spectrum presents a sharp feature at about 4 eV and another large feature above 8 eV. Contrary to a DFT-LDA calculation and in agreement with the experiment, their GW reflectance presents a single bulk plasmon resonance dip at 3.92 eV.
A “s–d” model, calculating the ‘5s’ band using DFT-GW approximation and the ‘4d’ one as a polarizabile background, has been successfully used by Garcia-Lekue et al. [30] to calculate the widths of the surface and image states of Ag(001) and Ag(111). This demonstrates that a model studying silver surfaces should take into account these two bands.
In this paper we use the recently developed vector potential from electron density-coupled integro-differential equations (VPED-CIDE, Raseev [1], [2]) model to study the laser excitation of the low index silver surfaces in the 2.8–10 eV region. Section 2 summarizes our model and presents the observables. Section 3 discusses the Chulkov et al. [3] DFT-LDA potential modified to include the adsorbed sodium, the resulting projected band structure and the choice of the material functions. Section 4 presents our results and Section 5 discusses and concludes the paper.
Section snippets
Summary of the VPED-CIDE model
Consider the Ampère–Maxwell equation in the classical approximation written in SI or in the atomic system of units, i.e. ε0μ0 = 1/c2 with ε0 and μ0 the vacuum permittivity and permeability and c the speed of lightwhere , and are the magnetic field, the displacement and the current density. Assuming linear interactions, the matter appears in the Ampère–Maxwell equations through the constitutive tensor relations for the displacement , the polarization and the current
Preliminary calculations
The technical details of the VPED-CIDE model are given in our preceding papers, refs. [1], [2]. Briefly, in the numerical calculations of the vector potential function of the coordinate z, one uses a large slab with the laser incident on the left surface. The wave functions entering in the calculation of the unperturbed electron density ρ(z), Eq. (4), and of the transition moment, Eqs. (16), (17), (18), are obtained by solving the Schrödinger equation with the model potential of Chulkov et al.
Plasmon resonances in Ag(001) and Ag(111)
Before discussing the spectra of the observables of Ag(001) and Ag(111) surfaces calculated using VPED-CIDE model let us recall the retained parameters of our model: i) for the V5(z) potential (Eq. (20)) we use ζ = 0.2 and Vδϕ = 1.63 eV; ii) for the separation of the bound and conduction electron polarizabilities the parameters rs = 3.0135 a.u., m∗ = 0.96 and τ = 3.1 · 10− 14 s− 1 taken from the work of Johnson and Christy [44]; iii) for the bound electron polarizability the Palik's experimental material
Discussion and conclusions
This paper presents a theoretical study of the plasmon resonances in the PES spectrum of Ag(001) + Na and Ag(111) + Na low indexed surfaces using the recently developed VPED-CIDE model [1], [2]. We have calculated the EEPY-PDA (Y, Eq. (11)), Feibelman's parameter (d⊥, Eq. (12)), the non local reflectance (R, Eq. (14)) and the Fermi photoelectron cross section (σF, Eq. (15)). To analyze the surface plasmon resonances we have also calculated the induced electron density (δρ Eq. (13) from the
Acknowledgments
We are indebted to Andrew Mayne for the critical reading of the manuscript. We acknowledge the use of the computing facility cluster GMPCS of the LUMAT federation (FR LUMAT 2764).
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