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Journal ArticleDOI

A theory of electrodynamic response for bounded metals: surface capacitive effects

Hai-Yao Deng1
01 Jul 2020-Annals of Physics (Elsevier Masson)-Vol. 418, pp 168204
TL;DR: In this paper, a general macroscopic theory for the electrodynamic response of semi-infinite metals (SIMs) is presented, which includes the hitherto overlooked capacitive effects due to the finite spatial extension of a surface.
About: This article is published in Annals of Physics.The article was published on 2020-07-01 and is currently open access. It has received 14 citations till now. The article focuses on the topics: Surface roughness.

Summary (4 min read)

Introduction

  • The authors then employ the response function to evaluate the so-called dynamical structure factor, which plays an important role in particle scattering.
  • I. INTRODUCTION Electrodynamic responses, that is, the behaviors of charges in materials and the accompanied electromagnetic fields when subjected to external probes, underlie many physical processes involving the interaction of electron and photon with condensed matter.
  • On numerous occasions, e.g. in studying optical properties, a microscopic boundary is only of marginal importance and a macroscopic description could be more useful.
  • To remedy this deficiency, since 1950s additional boundary conditions (ABCs) have been invoked to supplement the MBCs3.

A. Overview of the literature

  • Electrodynamic response may be quantified by the charge density-density response function, which measures the amount of charges induced in a material due to certain probing potential.
  • Their work established the concept of collective electronic oscillations – known as plasma waves or more precisely volume plasma waves (VPWs, sometimes called bulk plasma waves) – in the bulk of metals.
  • While it accounts for the surface effects in a self-consistent manner and might even provide a microscopic knowledge of the surface itself, the computational approach does not always make transparent the underlying physics and often presumes an ideal surface, such as those modeled by a hard-or-soft-wall-type infinite barrier potential24.
  • Where non-local effects are intended, i.e. in dispersive medium, the HDM and the SRM are usually invoked68,69.
  • Nevertheless, it was pointed out long ago that ABCs are physically superficial having no general physical basis76,77.

B. Outline of main results

  • The main purpose of the present work is to derive a macroscopic electrodynamic response theory for semi-infinite metals (SIMs) that is free from the usual boundary conditions, and then employ it to calculate the density-density response function (Sec. II).
  • It is shown that the response function naturally contains two components, one being essentially the same as for an infinite system whereas the other solely due to the presence of surfaces (Secs. II A and II C).
  • The authors find that under ABCs the surface contribution would be totally lost and hence no SPWs would exist, in agreement with their recent work92 showing that the apparent SPW solution admitted in ABC-based HDM is incompatible with that of the DM.
  • The SCM unveils two interesting yet natural features unseen in other models.
  • As an illustration, the authors have evaluated the distribution of these charges induced by an exterior charged particle grazing over the surface at constant speed [Fig. 1 (a)].

II. THEORY OF DYNAMICAL RESPONSES

  • The authors derive the macroscopic electrodynamic response theory and calculate the charges induced by external stimuli, and from this the density-density response function is extracted including contributions from both the SPWs and VPWs.
  • In studying dynamical responses for bounded medium, it is customary to work directly with the electrostatic potential – or more generally the electromagnetic field in the case of nonnegligible retardation effects – and write down its expressions on the vacuum side and the material side separately.
  • Considering that a real microscopic surface can hardly be specified even for the simplest material, one might deem it hopeless.
  • The authors may characterize this layer by a surface potential φs, which should quickly decay to zero in the bulk regions outside the interfacial layer.
  • To recapitulate, Eq. (1) elegantly captures two important physical consequences of an interface: the rapid variation of the current density through the step function Θ(z) and the surface scattering effects on electron dynamics through the parameters contained in the bulk values JA/B.

A. Generic formulation

  • With the macroscopic limit of physical interfaces, Eq. (1), the authors now formulate a general theory of electrodynamic response for the SIM.
  • As to be seen, boundary conditions, i.e. both MBCs and ABCs, are no longer needed.
  • Here the authors obtain the induced charge densities from the theory derived above.
  • Analogously, the authors may split G, the kernel in Eq. (15), into two parts, Gb and Gs, which originate from σb,µν and σs,µν, respectively.

C. The density-density response function

  • The authors discuss two cases of special importance in many applications such as particle and light scattering.
  • This relation shows that χ is more fundamental than P, namely the latter can be completely determined if the former is known while the converse is not true.
  • In experiments such as electron transmission through metal foils and where penetration is not negligible as well as optical experiments, the full structure of χ should be taken into account.

III. RESPONSES WITHIN COMMON MODELS

  • The theory presented in Sec. II is generic and applicable to any electron dynamics models, dispersive or non-dispersive.
  • In the literature, there are a few models that have been proposed and widely used for describing electron dynamics in metals.
  • Here the authors discuss the most common ones, i.e. the DM, the HDM and the SRM, leaving the SCM to be systematically treated in Sec. IV.
  • The authors consider the responses due to conduction electrons only.
  • In Table I, the authors summarize the defining quantities for each of the models to facilitate a quick comparison.

A. The local dielectric model (DM)

  • The authors begin the survey with the non-dispersive DM.
  • Symmetry breaking effects due to the surface are obviously excluded from this model.
  • Instead, only the resonance near the zero of ǫs exists with P. Equation (49) is one of the most used results for analyzing surface excitations and other surface phenomena such as the energy absorption of grazing particles and photon drag effect.
  • The responses to an electrostatic potential – case (ii) – can be similarly dealt with.

B. The hydrodynamic model (HDM)

  • The DM assumes a local dependence of the current density on the electric field.
  • There are several paths, which are not always equivalent, to the HDM106.
  • The responses to exterior charges can easily be obtained using Eq. (57).
  • Again there is nearly perfect cancellation between P1 and P2 near the VPW resonances, as seen in the lower panel of Fig.
  • For ω < ωp, the charges are localized within a layer of thickness around v0/ωp.

C. The specular reflection model (SRM)

  • The VPW dispersion relation is obtained by solving the equation that ω̄ = Ω(K, ω) while the SPW dispersion relation by the following equation ǫs,SRM(k, ω) = 0. (71) which is nothing but the SRM equation for SPWs first proposed by Ritchie and Marusak48 in 1966.
  • The present derivation makes it clear that the SRM can be regarded as an extension of the HDM.
  • In contrast to its original contrivance, the SRM does not simply assume a specularly reflecting surface in actuality; otherwise, one would have no surface contribution and Eq. (71) would not have been reached.
  • The responses within the SRM will be briefly discussed in the next section, in parallel with the SCM.

IV. RESPONSES BY THE SEMI-CLASSICAL MODEL

  • The authors should also remark that Eq. (79) assumes a global relaxation term.
  • If the authors keep only gb, the SRM equation (71) will be revisited, making it evident that the SRM does not correspond to the limit of p = 1 (specularly reflecting surface).
  • A crucial improvement of the SCM over the SRM comes through the quantity Gs(K, ω).

V. DISCUSSIONS

  • The authors have developed a general dynamical response theory for SIMs.
  • This theory is straightforwardly extendable to other bounded systems such as films and spheres.
  • Introduced over six decades ago and having been adopted in innumerable work, ABCs are widely regarded as superficial without a generic physical basis and should not play any role in a complete theory21,76.
  • In the simplest case of local dielectric models, their approach is actually identical to the present one103.

A. Dynamical structure factor and SPW peak narrowing

  • A systematic analysis of its properties being reserved for a separate publication, here the authors briefly discuss its use in the study of charged particles (e.g. electrons) reflected off a metal surface.
  • As shown in Ref.93, upon decreasing 1/τ, ǫs(k, ω) can be made to vanish and hence S2 can be made singular around the SPW pole whereas S1 is dominated by Landau damping via the VPW pole and much less affected.
  • The dipole approximation, despite its widespread use, is incapable of satisfactorily reproducing the experimental observations.
  • S(k, ω) displays only the SPW peak, though an additional broad peak due to VPWs has been seen in numerous scattering experiments23,101.

B. Charges induced by a grazing particle

  • For moderate z0 (the panels with z0ωp/vF = 15), however, ρ‖(r, t) strongly depends on the model: in the DM it is symmetric about the grazing particle along its motion but in the SCM the charges are more concentrated in front of the particle.
  • In this figure, the panels are organized in eight pairs, each pair consisting of two panels in the same model and with the same z0.
  • The induced charge density profiles are of experimental interest for two reasons.
  • Firstly, they may be directly measured116 to discriminate existing models against one another.

VI. SUMMARY

  • In summary, the authors have presented a general macroscopic theory of electrodynamic response for bounded systems without the use of ABCs and MBCs.
  • 92 H.-Y. Deng, “A unified macroscopic theory of surface plasma waves and their losses.”.
  • The inclusion of Jp in the calculations directly modifies the expressions of only two quantities: Ω(K, ω) and G(K, ω).

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Citations
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Journal ArticleDOI
TL;DR: In this paper, the surface response function of a metal is retrieved via the metal's non-equilibrium response to an external perturbation, and a complementary approach where one of the most appealing surface response functions, namely the Feibelman $d$-parameters, yield a finite contribution even in the case where they are calculated directly from the equilibrium properties described under the local-response approximation (LRA), but with a spatially varying equilibrium electron density.
Abstract: Surface-response functions are one of the most promising routes for bridging the gap between fully quantum-mechanical calculations and phenomenological models in quantum nanoplasmonics. Within all the currently available recipes for obtaining such response functions, \emph{ab initio} calculations remain one of the most predominant, wherein the surface-response function are retrieved via the metal's non-equilibrium response to an external perturbation. Here, we present a complementary approach where one of the most appealing surface-response functions, namely the Feibelman $d$-parameters, yield a finite contribution even in the case where they are calculated directly from the equilibrium properties described under the local-response approximation (LRA), but with a spatially varying equilibrium electron density. Using model calculations that mimic both spill-in and spill-out of the equilibrium electron density, we show that the obtained $d$-parameters are in qualitative agreement with more elaborate, but also more computationally demanding, \emph{ab initio} methods. The analytical work presented here illustrates how microscopic surface-response functions can emerge out of entirely local electrodynamic considerations.

10 citations

Journal ArticleDOI
TL;DR: In this paper, a Boltzmann equation is used to describe the scattering between the two charge types. But the model is not suitable for the case of metal surfaces with disorder or with nanostructure modifications, allowing for a localized charge layer (CL) in addition to continuous charges (CC) in the bulk.
Abstract: Metal surfaces with disorder or with nanostructure modifications are studied, allowing for a localized charge layer (CL) in addition to continuous charges (CC) in the bulk, both charges having a compressional or diffusive non-local response. The notorious problem of "additional boundary conditions" is resolved with the help of a Boltzmann equation that involves the scattering between the two charge types. Depending on the strength of this scattering, the oscillating charges can be dominantly CC or CL; the surface plasmon (SP) resonance acquires then a relatively small linewidth, in agreement with a large set of data. With a few parameters our model describes a large variety of SP dispersions corresponding to observed data.

5 citations

Journal ArticleDOI
Hai-Yao Deng1
TL;DR: In this paper, surface roughness effects on the conduction of electrons in metals using both the quantal Kubo-Greenwood formalism and the semi-classical Fuchs-Sondheimer method are discussed.
Abstract: We discuss surface roughness effects on the conduction of electrons in metals using both the quantal Kubo-Greenwood formalism and the semi-classical Fuchs-Sondheimer method. The main purpose here is to compare these methods and clarify a few subtle conceptual issues. One of such issues is concerned with the conditions under which the broken translation symmetry along a rough surface may be restored. This symmetry has often been presumed in existing work but not always with proper justifications. Another one relates to the physical meaning of a phenomenological parameter (denoted by $p$) intuitively introduced in the semi-classical theory. This parameter, called the specularity parameter or sometimes the \textit{Fuchs} parameter, plays an important role in the experimental studies of surface roughness but has so far lacked a rigorous microscopic definition. The third issue arises as to the domain of validity for the electrical conductivity obtained in those methods. A misplacement of the domain may have resulted in erroneous analysis of surface effects in a variety of electrodynamic phenomena including surface plasma waves.

3 citations

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Frequently Asked Questions (2)
Q1. What have the authors contributed in "A theory of electrodynamic response for bounded metals: surface capacitive effects" ?

In this paper, the authors discuss the problem of finding a good copy-write version of a text in a copy-edit setting. 

Applying the theory to these physical processes should be a fascinating subject of future study. As mentioned in the beginning section of this paper, the earliest effort perhaps dated back to 1970s based on Ewald-Oseen extinction theorem within the dielectric approximation, which has recently been further developed by Schmidt et al.