Multipole surface plasmon resonance of an aluminium surface

Abstract

The surface photoelectric effect and the surface plasmon resonances appear when a $\mathfrak{p}$/transverse magnetic polarized laser hits a gas–solid interface. We model this effect in the long wave length (LWL) domain (λ_vac > 10 nm, ħω < 124 eV) by combining the Ampère–Maxwell equation, written in classical approximation, with the material equation for the susceptibility. The resulting model, called the vector potential from the electron density (VPED), calculates the susceptibility as a product of the bulk susceptibility and the electron density of the actual system. The bulk susceptibility is a sum of the bound electron scalar susceptibility taken from the experiment and of the conduction electron non-local isotropic susceptibility tensor in a jellium metal (Lindhard, 1954 [1]). The electron density is the square of the wave function solution of the Schrödinger equation. The analysis of observables, the reflectance R and the photoelectron yield Y as well as the induced charge density permits to identify and characterize the multipole surface plasmon resonance of Al(111) appearing at ω_m ∼ 0.8ω_p or 11–12 eV.

Introduction

Consider a finite solid illuminated by a linearly polarized laser of a wave length λ_vac > 10 nm (ℏω < 124 eV) corresponding to the long wave length (LWL) spectral region. For simplicity, here we restrict the temporal dependence of the laser to harmonic, i.e. the laser is continuous with an angular frequency ω. We concentrate our discussion on the $\mathfrak{p}$ or transverse magnetic (TM) linear polarization, i.e. the electric field E⃗ or the vector potential A⃗ (E⃗||A⃗) of the laser are in the plane of polarization POI or xOz (see Fig. 1). In this $\mathfrak{p}$ polarization, the surface photoelectric effect is a supplementary contribution to the photoelectron yield due to the spatial variation of the laser field. Equivalently, in the surface coordinate system this effect is due to the breakdown of the Coulomb gauge ∇⃗ · E⃗ ≠ 0.

The surface photoelectric effect is not a minor contribution neither in the photoelectron (PES: for Al(001) see Levinson et al. [3], [4] and for Al(111) see Barman et al. [5]) nor in the electron energy loss (EELS; for Al(001) see Refs. [6], [7]) spectra. In the angle resolved PES (ARPES, arbitrary incident angle) or angle and energy resolved photoelectron yield (AERPY, arbitrary incident angle with normal to the surface ejection of the electron) of Al(001) and Al(111) one measures a multipole (several nodes) surface plasmon resonance at ω_m ∼ 0.8ω_p or 11–12 eV. In the EELS one finds not only this multipole plasmon resonance but also the standard surface monopole (one node) resonance at ${\omega }_s={\omega }_p/\sqrt{2}$ or 10.6 eV. This last resonance is forbidden in PES. Both resonances have a significant contribution from the surface photoelectric excitation term and motivated us to develop a new model calculating the laser field and the observables, as for example the photoelectron yield and the reflectance.

When considering a model for the laser–matter interaction at an interface, the first idea in mind is to use the Fresnel continuity relations. For an incident $\mathfrak{s}$ or transverse electric (TE) polarized light, the classical Fresnel continuity relations give a correct description of the optical properties. The susceptibility function reduces to the local transverse component and the laser electric field to a continuous component E_x parallel to the surface. The longitudinal component E_z of the electric field vanishes everywhere, the interaction is local and there are no induced charge densities at the interface.

For $\mathfrak{p}$ or TM polarized light, the situation is different. In the surface coordinate system, the Fresnel continuity relations give a transverse electric field component E_x and a displacement vector D⃗ that change continuously at the surface whereas the longitudinal field E_z is discontinuous. As a result, there is a singular induced charge at the interface described by a δ-function. In a valid microscopic theory of the surface photoelectric effect the longitudinal field E_z is continuous and the induced charge density varies smoothly. Due to the dynamical anisotropy of the electric field in the matter, the susceptibility is a non-local tensor.

In this paper, the laser–matter interaction is modeled in the framework of the classical theory of electromagnetism. Matter appears in the Ampère–Maxwell equations through the constitutive relations (Si units)

$$\overrightarrow{D}={\epsilon }_0\left(\tilde{1}+{\tilde{\chi }}^b\right)\cdot \overrightarrow{E}={\epsilon }_0\overrightarrow{E}+\overrightarrow{P}$$

$$\overrightarrow{J}=\tilde{\sigma }\cdot \overrightarrow{E}=-i{\epsilon }_0\omega {\tilde{\chi }}^c\cdot \overrightarrow{E}\text{.}$$

Here D⃗ is the displacement, ε₀ the vacuum permittivity, χ˜ the susceptibility tensor, E⃗ the electric field of the laser, P⃗ the polarization, J⃗ the current density and σ̃ the conductivity tensor. Recall also that “·” is the dot product between a tensor and a vector. In a microscopic theory of matter, one has to take into account the interaction of the light with electrons and holes, a system of particles presenting non-local interactions. Assuming linear interactions, this complicated physics corresponds to tensor and non-local material functions. In particular the current density J⃗ is related to the conduction electron susceptibility tensor ${\tilde{\chi }}^c$ through the relation

$$\overrightarrow{J}(\overrightarrow{r},\omega )=-i{\epsilon }_0\omega \int d{\overrightarrow{r}}^{\prime }{\tilde{\chi }}^c(\overrightarrow{r},{\overrightarrow{r}}^{\prime },\omega )\cdot \overrightarrow{E}({\overrightarrow{r}}^{\prime },\omega )\text{,}$$

where r⃗ and ${\overrightarrow{r}}^{\prime }$ are independent variables. For an isotropic system (e.g. cubic metal) this general non-local current density simplifies to

$$\overrightarrow{J}(\overrightarrow{r},\omega )\simeq -i{\epsilon }_0\omega \int d{\overrightarrow{r}}^{\prime }{\tilde{\chi }}^c(\overrightarrow{r}-{\overrightarrow{r}}^{\prime },\omega )\cdot \overrightarrow{E}({\overrightarrow{r}}^{\prime },\omega )\text{.}$$

In the momentum space the corresponding equation reads

$$\overrightarrow{J}(\overrightarrow{q},\omega )=-i{\epsilon }_0\omega {\tilde{\chi }}^c(\overrightarrow{q},\omega )\cdot \overrightarrow{E}(\overrightarrow{q},\omega )\text{,}$$

where q⃗ and ${\tilde{\chi }}^c(\overrightarrow{q},\omega )$ are the wave vector and the susceptibility tensor in momentum space. Therefore this last susceptibility is non-local in real space and local in momentum space. The pairs of Eqs. (4), (5) express the same physics.

We will consider the specific case of a flat structureless surface. Then, parallel to the surface, the spatial dependence of the laser fields can be modeled as plane waves. The transverse electric field of the laser E_x is isotropic and from the laws of optics its wave number q_x is conserved in the two phases: gas and solid. The longitudinal electric field E_z is anisotropic and the associated wave vector q_z is not conserved in the bulk.

In the literature the question of how one can model the laser–matter interaction at flat surfaces has been raised in many papers [1], [2], [8], [9], [10], [11], [12], [13], [14], [15], [16], [17]. Usually, one first combines the material equations ((1) and (2)) with Ampère-Maxwell equation, then one integrates the result over x and y. The resulting differential system of equations is solved in momentum space. The local behavior of the material (Eq. (5)) is thus exploited.

More precisely the models in the literature can be divided into two categories: (i) microscopic explicitly calculating the permittivity quantum mechanically; (ii) macroscopic assuming that the microscopic and macroscopic fields are locally similar and therefore one can use the macroscopic permittivity in the microscopic Ampère–Maxwell equation.

A microscopic approach, based on the jellium model for conduction electrons, has been developed by Feibelman [8], [9]. In Feibelman's model the microscopic dielectric tensor is obtained from the Random Phase Approximation (RPA). Other microscopic models, developed by Maniv and Metiu [12], [13], [14] and Gerhardts and Kempa [15], combine a RPA with an infinite barrier (IB) for the electrons at the surface. The IB model leads to a much steeper electron density profile than the more realistic Lang and Kohn [18] density functional theory (DFT) type potential used by Feibelman. As a consequence, some surface response properties show strong disagreement with Feibelman's results. The microscopic model of Gies, Gerhardts and Maniv [16] extends the IB model to an arbitrary potential well, introducing two auxiliary infinite potential barriers far from the jellium edge. Unfortunately, this treatment suppresses the emission of photoelectrons [17]. Gies, Gerhardts and Maniv [16] calculate the surface response functions d_∥ and d_⊥ whereas Maniv and Metiu [14] calculate photoelectron spectrum using a laser field power density absorption expression and the reflectance.

The second category includes the classical hydrodynamic [19], [20], [21], [22], [23] and the semi-classical infinite barrier (SCIB) [2], [24], [25] models. These models combine the bulk susceptibility with the electron density of the actual system. The models of Forstmann et al. [20], [22], [23] use a modified Drude theory of the bulk susceptibility to include a simple dispersion relation in the longitudinal susceptibility. Kliewer and Fuchs [2], [24] and Mermin [26] re-derived and corrected the Lindhard [1] quantum susceptibility functions for jellium. Then they used this susceptibility to obtain the laser fields in various numerical models for example SCIB and the three step model (see the review papers of Kliewer [11], [27]). Raseev [28], [29] used the Drude or the Lindhard model respectively for the scalar and tensor susceptibility of the conduction electrons and the experimental susceptibility for the bound electrons in the Ampère–Maxwell equation in real space. Apell et al. [30], [31] and Gies et al. [16] used a dielectric function where the non-locality appears as a consequence of a δ function: ε(z,z′, ω) = δ(z − z′) ε(z,ω).

In real space there are several ways to model the electron density of the actual system. First, one can use a smooth analytic step function but such a function discards the Friedel oscillations of a jellium solid. Smooth analytic step functions are used in the hydrodynamic models of Eguiluz and Quinn [19] and Sipe [21], the extended SCIB model of Georges [25], the model of Apell et al. [31] and the model of Bagchi, Kar and Barrera [32]. Second, one can explicitly calculate the electron density from the wave functions solution of the Schrödinger equation. In this equation, the input potential is either a jellium solid with interface (Lang and Kohn [18] and Raseev [28]) or a potential with the atomic structure included calculated from a DFT with local density approximation for the exchange-correlation functional (LDA; Raseev [29]).

Because in momentum space the material equations are local (see Eq. (5)), the Ampère–Maxwell equation in momentum space is also local. The majority of the models, except the one advocated by Raseev [28], [29] solve the Ampère–Maxwell equation in momentum space. But working in momentum space has the disadvantage that it is difficult to enforce the asymptotic boundary conditions at the boundaries. To overcome this difficulty each of the models cited above has its own recipe.

As mentioned in the Abstract, in the vector potential from electron density (VPED) model we have developed [28], [29] combines the material and Ampère–Maxwell equations. All the material equations are written in term of susceptibility itself a function of the electron density and the susceptibility of the bulk. The electron density of the actual material system with interface is calculated from the Schrödinger equation. For the conduction electrons this density is Fourier transformed in momentum space, multiplied by the bulk susceptibility of these electrons that is a non-local tensor and the resulting susceptibility is reverse Fourier transformed in real space. For the bound electrons the susceptibility extracted from the experiment is local and it is multiplied by the electron density of the actual system. These bound and conduction electrons susceptibilities are inserted in Ampère–Maxwell equation which is solved in real space. The complete derivation of the VPED model has been already presented in a preceding publication [29] and we will refer to that paper's equations when needed. The main advantage of the VPED model is this calculation in real space that permits a straightforward interpretation of the results. In the next section we detail the VPED model for the calculation of the susceptibility. We analyze the nature of the Al(111) multipole surface plasmon resonance ω_m ∼ 0.8ω_p appearing at 11–12 eV. This resonance is characterized by studying the reflectance calculated from the vector potential, the electron escaping probability obtained from the power density absorption and the charge density induced by the laser calculated from the derivative of the Coulomb-Maxwell equation. These observables permit to unravel the position, the multipole character and the surface nature of the multipole surface plasmon resonance. A discussion and the conclusions are presented in the last section.