Plasmonic performance of Au$_\mathbf{x}$Ag$_\mathbf{y}$Cu$_\mathbf{1-x-y}$ alloys from many-body perturbation theory

TL;DR

This paper investigates the optical and plasmonic properties of Au-Ag-Cu alloys using advanced first-principles many-body perturbation theory, highlighting improved modeling techniques for accurate predictions of their plasmonic behavior.

Contribution

It introduces a practical workflow combining G0W0 corrections with RPA for better spectroscopic predictions of alloy properties.

Findings

RPA with semi-local DFT poorly describes inter-band transitions.

Band-stretching operators effectively model G0W0 self-energy corrections.

Developed a method to calculate plasmon frequencies including self-energy effects.

Abstract

We present a detailed appraisal of the optical and plasmonic properties of ordered alloys of the form Au$_{x}$Ag$_{y}$Cu$_{1-x-y}$, as predicted by means of first-principles many-body perturbation theory augmented by a semi-empirical Drude-Lorentz model. In benchmark simulations on elemental Au, Ag, and Cu, we find that the random-phase approximation (RPA) fails to accurately describe inter-band transitions when it is built upon semi-local approximate Kohn-Sham density-functional theory (KS-DFT) band-structures. We show that non-local electronic exchange-correlation interactions sufficient to correct this, particularly for the fully-filled, relatively narrow $d$-bands that which contribute strongly throughout the low-energy spectral range ($0-6$ eV), may be modelled very expediently using band-stretching operators that imitate the effect of a perturbative G$_0$W$_0$ self-energy correction incorporating quasiparticle mass renormalization. We thereby establish a convenient work-flow for carrying out approximated G$_0$W$_0$+RPA spectroscopic calculations on alloys. We develop a pragmatic procedure for calculating the Drude plasmon frequency from first principles, including self-energy effects, as well as a semi-empirical scheme for interpolating the plasmon inverse lifetimes between stoichiometries. A range of optical and plasmonic figures of merit are discussed at three representative solid-state laser wavelengths.

Tables

Table 1. Table 1: Calculated values for stretching operators used to stretch KS band-structures in order to imitate perturbative G 0 W 0 QP band-structures for pure Au, Ag, and Cu.

	$s_v$	$s_c$
Au	1.419797	0.825253
Ag	1.376302	0.846172
Cu	1.735804	0.809883

Table 2. Table 2: The average Fermi velocities (in × 10 5 absent superscript 10 5 \displaystyle\times 10^{5} m/s units) for pure Au, Ag, and Cu. We have included results from the work of Gall  Gall ( 2016 ) (the third column), who evaluated the Fermi velocities using a 200 × 200 × 200 200 200 200 \displaystyle 200\times 200\times 200 Brillouin zone sampling.

	$v^{\mathrm{KS}}\left(E_F\right)$	$v^{G_0{W}_0}\left(E_F\right)$	$v^{\mathrm{Gall}}\left(E_F\right)$ Gall (2016)
Au	13.476	14.588	13.82
Ag	13.912	15.014	14.48
Cu	10.794	12.798	11.09

Equations

$$\mathcal{E}(\omega )={\mathcal{E}}^{\mathrm{inter}}(\omega )+{\mathcal{E}}^{\mathrm{intra}}(\omega ),$$

$$\mathcal{E}(\omega )=\frac{1}{{\lim }_{\mathbf{q}\to 0}{\left[{\epsilon }_{\mathbf{G},{\mathbf{G}}^{\prime }}^{-1}(\mathbf{q},\omega )\right]}_{\mathbf{G},{\mathbf{G}}^{\prime }=0}}.$$

$$\mathcal{E}(\omega )\approx \underset{\mathbf{q}\to 0}{\lim }{\left[{\epsilon }_{\mathbf{G},{\mathbf{G}}^{\prime }}(\mathbf{q},\omega )\right]}_{\mathbf{G},{\mathbf{G}}^{\prime }=0}.$$

$${\epsilon }_{\mathbf{G},{\mathbf{G}}^{\prime }}^{-1}(\mathbf{q},\omega )={\delta }_{\mathbf{G},{\mathbf{G}}^{\prime }}+v_{\mathbf{G},{\mathbf{G}}^{\prime }}(\mathbf{q}){\chi }_{\mathbf{G},{\mathbf{G}}^{\prime }}(\mathbf{q},\omega ),$$

$$v_{\mathbf{G},{\mathbf{G}}^{\prime }}(\mathbf{q})=\frac{4\pi }{|\mathbf{q}+\mathbf{G}||\mathbf{q}+{\mathbf{G}}^{\prime }|}.$$

$${\chi }_{\mathbf{G}{\mathbf{G}}^{\prime }}^0(\mathbf{q},\omega )=$$

$$\times \frac{|⟨{\psi }_{c,\mathbf{k}-\mathbf{q}}|e^{i(\mathbf{q}+\mathbf{G})\cdot \mathbf{r}}|{\psi }_{v,\mathbf{k}}⟩{|}^2}{\omega -{\omega }_{cv,\mathbf{kq}}+i\Gamma },$$

$${\chi }_{\mathbf{G}{\mathbf{G}}^{\prime }}(\mathbf{q},\omega )={\chi }_{\mathbf{G}{\mathbf{G}}^{\prime }}^0(\mathbf{q},\omega )\left(1+v_{\mathbf{G},{\mathbf{G}}^{\prime }}(\mathbf{q}){\chi }_{\mathbf{G}{\mathbf{G}}^{\prime }}(\mathbf{q},\omega )\right),$$

$${\mathbf{\chi }}^{-1}(\mathbf{q},\omega )={\mathbf{\chi }}_0^{-1}(\mathbf{q},\omega )-\mathbf{v}(\mathbf{q}).$$

$${\epsilon }_{\mathbf{G},{\mathbf{G}}^{\prime }}^0(\mathbf{q},\omega )={\delta }_{\mathbf{G},{\mathbf{G}}^{\prime }}-v_{\mathbf{G},{\mathbf{G}}^{\prime }}(\mathbf{q}){\chi }_{\mathbf{G}{\mathbf{G}}^{\prime }}^0(\mathbf{q},\omega ).$$

$${\mathcal{E}}^{\mathrm{intra}}(\omega )={\epsilon }_{\infty }-\frac{{\omega }_{\mathrm{p}}^2}{{\omega }^2-i{\eta }_{\mathrm{p}}\omega },$$

$${\omega }_{\mathrm{p}}^2=\frac{4\pi N(E_{\mathrm{F}})}{m_{\mathrm{eff}}},$$

$$m_{\mathrm{eff}}^{-1}$$

$$\displaystyle\displaystyle{}=\frac{1}{3}\Big{(}\sum_{i}\int_{S_{\mathrm{F_{i}}}}d\mathbf{k}\;v^{2}_{i}(\mathbf{k})\Big{)}\Big{(}\sum_{j}\int_{S_{\mathrm{F_{j}}}}d\mathbf{k^{\prime}}\;\Big{)}^{-1},$$

$$\text{mbox}if{v}_i^2(\mathbf{k})$$

$${\omega }_{\mathrm{p}}^2=\frac{4\pi }{3}N(E_{\mathrm{F}})⟨v^2(E_{\mathrm{F}})⟩.$$

$${\eta }_{\mathrm{p}}=\frac{N(E_{\mathrm{F}})}{{\sigma }_0{m}_{\mathrm{eff}}}\approx c_{\eta }N(E_{\mathrm{F}})⟨v^2(E_{\mathrm{F}}){⟩}^{1/2},$$

$$\mathfrak{Re}\left[{\mathcal{E}}^{\mathrm{intra}}\right]={\mathcal{E}}_1^{\mathrm{intra}}(\omega ,{\omega }_{\mathrm{p}},{\eta }_{\mathrm{p}},{\epsilon }_{\infty })$$

$$\mathfrak{Im}\left[{\mathcal{E}}^{\mathrm{intra}}\right]={\mathcal{E}}_2^{\mathrm{intra}}(\omega ,{\omega }_{\mathrm{p}},{\eta }_{\mathrm{p}})$$

$$[-\frac{1}{2m_{\mathrm{QP}}}{\nabla }_i^2+v_{\mathrm{ext}}(\mathbf{r})+v_H(\mathbf{r})$$

$$+\int d{\mathbf{r}}^{\prime }\Sigma (\mathbf{r},{\mathbf{r}}^{\prime },\omega )]{\psi }_i^{\mathrm{QP}}(\mathbf{r},\omega )={ϵ}_i^{\mathrm{QP}}{\psi }_i^{\mathrm{QP}}(\mathbf{r},\omega ),$$

$$\Sigma =iGW,$$

$${ϵ}_i^{\mathrm{QP}}$$

$$Z_i$$

$$G_0(\mathbf{r},{\mathbf{r}}^{\prime },\omega )=$$

$$\times \left(\frac{f_{i,\mathbf{k}}}{\omega -{ϵ}_{i,\mathbf{k}}+i\delta }+\frac{1-f_{i,\mathbf{k}}}{\omega -{ϵ}_{i,\mathbf{k}}-i\delta }\right)$$

$$W_0={\epsilon }^{-1}v,$$

$$W_{0\mathbf{G},\mathbf{G}}(\mathbf{q},\omega )={\epsilon }_{\mathbf{G},{\mathbf{G}}^{\prime }}^{-1}(\mathbf{q},\omega )\frac{4\pi }{|\mathbf{q}+\mathbf{G}||\mathbf{q}+{\mathbf{G}}^{\prime }|}.$$

$${\epsilon }_{\mathbf{G},{\mathbf{G}}^{\prime }}^{-1}(\mathbf{q},\omega )={\delta }_{\mathbf{G},{\mathbf{G}}^{\prime }}+\sum _{\lambda }\frac{{\Omega }_{\mathbf{G},{\mathbf{G}}^{\prime }}^{\lambda }}{{\omega }^2-{\left({\bar{\omega }}_{\mathbf{G},{\mathbf{G}}^{\prime }}^{\lambda }\right)}^2},$$

$${ϵ}_{\mathrm{v}}^{\mathrm{QP}}=s_{\mathrm{v}}{ϵ}_{\mathrm{v}},\mathrm{and}{ϵ}_{\mathrm{c}}^{\mathrm{QP}}=s_{\mathrm{c}}{ϵ}_{\mathrm{c}},$$

$$v^{{\mathrm{G}}_0{\mathrm{W}}_0}(E_{\mathrm{F}})=\sqrt{s_{\mathrm{v}}{s}_{\mathrm{c}}}{v}^{\mathrm{KS}}(E_{\mathrm{F}}).$$

$$\mathrm{EELS}(\omega )=-\mathfrak{Im}\left[{\mathcal{E}}^{-1}(\omega )\right].$$

$$\displaystyle\displaystyle p_{i}=\exp\left(\frac{E_{0}-E_{i}}{k_{B}T}\right)\Bigg{(}\sum_{j}\exp\left(\frac{E_{0}-E_{j}}{k_{B}T}\right)\Bigg{)}^{-1},$$

$$c_{\eta }^{(\mathrm{A}{\mathrm{u}}_{\mathrm{x}}\mathrm{A}{\mathrm{g}}_{\mathrm{y}}\mathrm{C}{\mathrm{u}}_{1-\mathrm{x}-\mathrm{y}})}=\left(x{c}_{\eta }^{\mathrm{Au}}+y{c}_{\eta }^{\mathrm{Ag}}+(1-x-y)c_{\eta }^{\mathrm{Cu}}\right)$$

Full text

Alloy Design for Plasmonics, Theoretical Spectroscopy, Many-Body Perturbation Theory

I Introduction

Noble metals and their alloys are compelling materials for opto-electronic applications Black et al. (2000); Lee and El-Sayed (2006); Jain et al. (2008); Zou et al. (2014); Medici et al. (2015); Sato, Oike, and Hanashima (2009) due to their strong plasmonic and optical response throughout the infrared-visible-ultraviolet spectral range. The tailoring of plasmonic and optical properties of metals via alloying is currently attracting interest due to a high demand for novel and more efficient nano-materials for opto-electronic applications Liu et al. (2008); Valodkar et al. (2011); Dastmalchi et al. (2016). Nano-structures of pure Au, Ag and Cu are used in diverse opto-electronic applications, due to their good chemical and mechanical stabilities, as well as their strong optical response in the low-energy spectral range. Hence, their alloys are naturally expected to be promising candidates for efficient opto-electronic applications. Spectroscopic measurements on alloys are mostly performed on their thin-film surfaces Wessel (1963); McAlister, Stern, and McGroddy (1965); Stahl, Spranger, and Aubauer (1969); Fukutani (1971); Beaglehole and Erlbach (1972); Rivory (1976); Nishijima and Akiyama (2012); Nishijima et al. (2014); Gong and Leite (2016); Hashimoto et al. (2016), which are highly dependent on the alloying technique used na Rodríguez et al. (2014). As a result, it is difficult to find consensus within the literature even on basic quantities such as the plasma frequency of an elemental metal.

Systematic first-principles studies of such alloys may, therefore, offer fundamental insights into the microscopic effects of alloying on optical properties, and potentially thereby even guide the tailoring of optical and plasmonic response for designated applications. This work is an exploratory investigation into the viability of such an approach using state-of-the-art theory. Specifically, in this article, we present a detailed investigation into the capabilities and limitations of contemporary theoretical spectroscopy for noble metals, and the development and testing of a set of computationally light techniques for studying the spectra of noble metals and their ordered alloys within the linear-response regime. Taking advantage of the resulting high-throughput-compatible approach, we provide various figures of merit for comparing the plasmonic performance of these alloys, from which the optimal stroichiometry for a given optical or plasmonic characteristic, at a given driving frequency, may be estimated.

II Theoretical Methodology

In the study and simulation of solid-state optical and energy-loss spectra, the macroscopic dielectric function $\displaystyle\mathcal{E}(\omega)$ is the central function due to its well-established connection to measured observables. In the low-energy spectral range, where the phenomena contributing to optical spectra are almost entirely electronic, the macroscopic dielectric function of a metallic system is constituted by two terms, explicitly by

[TABLE]

where $\displaystyle\mathcal{E}^{\mathrm{inter}}(\omega)$ and $\displaystyle\mathcal{E}^{\mathrm{intra}}(\omega)$ result from screening effects due to inter-band transitions and intra-band transitions (giving rise to the Drude plasmon), respectively.

When a solid is simulated as an infinite object, electronic transitions are envisaged as occurring simultaneously throughout the material, at a given energy of electro-magnetic (EM) radiation. Observable spectra are spatial averages of these transitions, whereas electronic transitions are microscopic events. Hence, an averaging process is required to connect microscopic quantities to macroscopic observables. Specifically, an averaging via the inverse dielectric function is the appropriate route to obtain the macroscopic dielectric function in the optical (vanishing momentum transfer) limit Adler (1962); Wiser (1963), and this is given, in terms of the microscopic dielectric matrix $\displaystyle\varepsilon$, by

[TABLE]

As the momentum $\displaystyle\mathbf{q}$ transferred from the incoming photon is assumed to be negligible, only vertical excitations at each point in reciprocal space are considered here. Notwithstanding, non-local screening ‘local field’ effects are explicitly incorporated by means of Eq. 2. Where such effects can be safely neglected, we may used the relatively simple, conveniently inversion-free formula

[TABLE]

II.1 Inter-band transitions

Inter-band transitions are electronic transitions between the valence (occupied) and conduction (unoccupied) bands. Within the linear-response regime that typically holds for photon energies in the IR-vis-UV range, the inverse microscopic dielectric function is given, in reciprocal space, by the expression

[TABLE]

where the bare Coulomb interaction takes the form

[TABLE]

In order to calculate $\displaystyle\varepsilon_{\mathbf{G},\mathbf{G}^{\prime}}(\mathbf{q},\omega)$, the non-interacting random-phase approximation (RPA) (Fermi’s Golden Rule) Dirac (1927) linear-response function for inter-band excitations is first calculated, in terms of independent-particle wave-functions $\displaystyle|\psi_{v,\mathbf{k}}\rangle$ with occupancies $\displaystyle f_{v,\mathbf{k}}$, as Adler (1962); Wiser (1963),

[TABLE]

where $\displaystyle v$ $\displaystyle(c)$ indicates valence (conduction) states and $\displaystyle\omega_{cv,\mathbf{k}\mathbf{q}}=(E_{c,\mathbf{k}-\mathbf{q}}-E_{v,\mathbf{k}})$ is a difference between single-particle energy eigenvalues (using atomic units). Here, $\displaystyle\Gamma$ is a small, positive-valued Lorentzian broadening factor, $\displaystyle\mathbf{q}$ is the transferred momentum vector, which lies in the first Brillouin zone, and the factor of $\displaystyle 2$ pre-supposes and accounts for spin degeneracy. Following this, the interacting RPA Bohm and Pines (1951); Pines and Bohm (1952); Bohm and Pines (1953); McLACHLAN and BALL (1964) response function $\displaystyle\boldsymbol{\chi}$ is given by Bohm and Pines (1951); Pines and Bohm (1952); Bohm and Pines (1953); McLACHLAN and BALL (1964)

[TABLE]

which is a Dyson equation. Eq. (7) may be rearranged into the compact form, involving matrix inversion, of

[TABLE]

for the interacting response function. Alternatively, the more approximate independent-particle RPA dielectric function may be calculated as

[TABLE]

When $\displaystyle\varepsilon\approx\varepsilon^{0}$ is invoked it is conventional to also neglect local-field effects, by means of Eq. 3.

II.2 Intra-band transitions

In addition to inter-band transitions, the promotion of electrons to higher energies within the same band, at finite temperatures, contributes very significantly to the macroscopic dielectric function of metallic solids within the low-energy regime. This is the intra-band transition effect, which gives rise to the prominent Drude plasmon divergence in the optical absorption spectrum in the static limit. This Drude plasmon can be thought of semi-classically as the collective oscillation of electrons at the Fermi level, in phase with the longitudinal part of the driving EM radiation. The Drude plasmon typically occurs at $\displaystyle\sim 10$ eV in elemental late transition metals, and it can be excited, e.g., by energy loss of incident electrons with kinetic energies in the $\displaystyle 1-20$ keV range, or by using lasers tuned to the plasmon wavelength.

The direct simulation of a well-converged Drude plasmon frequency starting a set of single-particle electronic states is computationally demanding, indeed extremely so due to the requirement for dense Brillouin-zone sampling at and around the Fermi surface, for example up to $\displaystyle\sim 16000$ grid points in Ref. Marini, 2001. This procedure is not commonly followed, as the resulting intra-band dielectric function remains excessively sensitive to the difficult-to-estimate excitation damping (or lifetime) factor that must be imposed. The Drude plasmon lifetime is limited by a multitude of physical processes, in reality, including scattering by phonons, defects, and grain boundaries; electron-electron (including electron-plasmon and plasmon-plasmon) scattering giving rise to decay, and quantum thermodynamic effects.

More commonly, the Drude plasmon is discussed in terms of a classical model for free electrons oscillating under the influence of external electric field, namely the Drude-Lorentz model Drude (1900a, b); Lorentz (1909); Fox (2010). The Drude plasmon contribution to the macroscopic dielectric function becomes

[TABLE]

where $\displaystyle\omega_{\mathrm{p}}$, $\displaystyle\eta_{\mathrm{p}}$, and $\displaystyle\varepsilon_{\infty}$ are the Drude plasmon energy (not to be confused with the actual net plasma frequency of the metal – where $\displaystyle\mathfrak{Re}\left[\mathcal{E}(\omega)\right]\;\approx 0$ eV), the phenomenological inverse life-time, and the electric permittivity in the infinite-frequency limit, respectively. The set $\displaystyle\{\omega_{\mathrm{p}},\eta_{\mathrm{p}},\varepsilon_{\infty}\}$ comprise the ‘Drude parameters’. Experimentally, it is standard practice is to perform optical (e.g., $\displaystyle n$ and $\displaystyle k$) measurements within the infrared and the far-infrared spectral range (corresponding to $\displaystyle\omega\approx 0-2$ eV), and to determine these parameters by fitting to the Drude-Lorentz model Ordal et al. (1985); Zeman and Schatz (1987); Hooper and Sambles (2002); Berciaud et al. (2005); Grady, Halas, and Nordlander (2004); Kreiter et al. (2002); Blaber, Arnold, and Ford (2009).

Inspired by this, here, we start from the Drude plasmon energy in Eq. (10), which can be expressed as Drude (1900a, b); Beach and Christy (1977)

[TABLE]

for a uniform non-interacting electron gas. In this, $\displaystyle N(E_{\mathrm{F}})$ is the density of states (DOS) at the Fermi level and $\displaystyle m_{\mathrm{eff}}$ is the electron effective mass. In practice, for real metals, this effective mass is also evaluated at the Fermi level and, if we further assume that the metallic bands have a parabolic dispersion normal to the Fermi surface Cohen (1958), we may write

[TABLE]

Here, $\displaystyle S_{\mathrm{F_{i}}}$ signifies the Fermi surface of the $\displaystyle i^{\textrm{th}}$ metallic band, and the factor of $\displaystyle 1/3$ results from the squared Fermi velocity being averaged (rather than summed) over Cartesian directions. Succinctly, the Drude plasmon energy can thus be approximated within a non-interacting, uniform-gas theory, simply and efficiently as Cohen (1958)

[TABLE]

As previously mentioed, the routine direct calculation of experimentally-relevant Drude plasmon lifetimes for is currently beyond the scope of state-of-the-art electronic structure simulation methodology. Electron-phonon and electron-impurity scattering typically dominantly contribute to the limiting DC conductivity $\displaystyle\sigma_{0}$, as compared to the more accessible electron-electron scattering processes Beach and Christy (1977). In order to circumvent this issue, we have developed a semi-empirical scheme based upon the Drude-Lorentz model, in which the scattering rate $\displaystyle\eta_{\mathrm{p}}$ is inversely proportional to the DC conductivity and to the effective mass of carriers, but proportional to their concentration. Noting a very plausibile linear dependence between $\displaystyle\sigma_{0}$ and $\displaystyle\sqrt{\langle v^{2}(E_{F})\rangle}$, we express the scattering rate (where the first equality is standard Drude-Lorentz) as

[TABLE]

where $\displaystyle c_{\eta}$ is our scaling coefficient to be determined.

Next, separating the real and imaginary part of $\displaystyle\mathcal{E}_{\mathrm{intra}}(\omega)$ from Eq. (10), we arrive at

[TABLE]

and note that, given the first-principles $\displaystyle\omega_{\mathrm{p}}$, the imaginary part is parametrized only by $\displaystyle\eta_{\mathrm{p}}$. Thus, in practice, we first determine the scaling factor $\displaystyle c_{\eta}$ for $\displaystyle\eta_{\mathrm{p}}$ in Eq. (14) by least-squares fitting against $\displaystyle\mathcal{E}^{\mathrm{intra}}_{2}$ from an appropriate experimental spectrum within the near infra-red spectral range (we find that $\displaystyle\omega\approx 0.8-1.2$ eV is very effective for noble metals – but we emphasize that the choice of regression domain does matter). Secondly, we determine $\displaystyle\varepsilon_{\infty}$ by fitting $\displaystyle\mathcal{E}^{\mathrm{intra}}_{1}$ to the same experimental spectrum, using the fixed values for $\displaystyle\omega_{\mathrm{p}}$ and $\displaystyle\eta_{\mathrm{p}}$ obtained in the previous step, within the same spectral range. In practice, we have found this two-step procedure to be quite reliable, whereas simultaneous least-squares regression of $\displaystyle c_{\eta}$ and $\displaystyle\varepsilon_{\infty}$ against the complex-valued $\displaystyle\mathcal{E}^{\mathrm{intra}}(\omega)$ can yield unphysical values for both parameters.

II.3 Treatment of non-local many-electron

effects within the quasi-particle formalism: Perturbative one-shot GW: G0W0

In order to calculate the aforementioned response functions, e.g. in Eq. 6, a sufficiently complete set of well-defined single-particle electronic states is required. Density functional theory (DFT) Hohenberg and Kohn (1964) is currently the almost-ubiquitously used approach for constructing ground-state electronic structures of solids within its Kohn-Sham formalism (KS-DFT) Kohn and Sham (1965). However, the DFT is limited by the accuracy of available, computationally feasible local and semi-local approximations for exchange and correlation Kohn and Sham (1965); Langreth and Perdew (1980); Perdew (1986); Perdew et al. (1992). Furthermore, the energy eigenvalues (band-structures) generated by the Kohn-Sham mapping have no formal meaning in terms of electron addition or removal energies (except in certain well-documented instances), in spite of their being widely interpreted as such. The RPA, although it is a true many-body approximation, is unable to build any electron or hole quasiparticle Landau (1957a, b, 1959) screening (e.g. electron-plasmon coupling) effects into an underlying Kohn-Sham eigensystem, as it treats only the screened interaction between pairs of such input particles. The fact that the absence of non-local quantum many-body effects in semi-local KS-DFT, and absent explicit long-ranged exchange in particular, often leads to inaccurate descriptions of the electronic bands in solids is well reported for various material classes, such as insulators and semi-conductors Rohlfing and Louie (1998); Marini, Del Sole, and Rubio (2003); Sottile (2003); Reining et al. (2002); Botti et al. (2004), transition-metal oxides Massidda et al. (1995, 1997); Continenza, Massidda, and Posternak (1999), and metallic solids Godby (1992); Aryasetiawan and Gunnarsson (1998).

In noble metals, the electronic bands that dominantly contribute to low-energy spectra are fully-filled $\displaystyle d^{10}$ bands tightly packed in a narrow energy window close to the Fermi level. It has previously been found that these electronic bands are poorly described within approximate KS-DFT for noble metals such as bulk Au Rangel et al. (2012), and then that such errors become more pronounced in spectral simulations using the RPA Eckardt, Fritsche, and Noffke (1984); Cohen, Mori-Sánchez, and Yang (2012); Marini, Onida, and Del Sole (2001); Marini, Del Sole, and Onida (2002); Marini et al. (2002). The formally correct approach to calculating band-structures from first principles is instead the quasiparticle (QP) formalism, fundamental to which is a mapping of the interacting many-body system to a weakly-interacting many-body system of virtual ones, namely the quasi-particles Pines (1966); Aulbur, Jönsson, and Wilkins (2000). QP wave-functions and corresponding energy levels can determined by self-consistently solving the QP equation

[TABLE]

where $\displaystyle\Sigma(\mathbf{r},\mathbf{r}^{\prime},\omega)$ is the energy-dependent, non-local, and non-Hermitian self-energy. The resulting $\displaystyle\Psi_{i}^{\textrm{QP}}(\mathbf{r})$ and $\displaystyle E_{i}^{QP}$ are the $\displaystyle i^{th}$ QP wave-function and the corresponding QP energy. In practice, QP energies are more commonly calculated using nonetheless demanding many-body perturbation theory (MBPT) Fetter and Walecka (2012) methods with Green’s functions instead of explicit solution of Eq. (II.3).

The GW approximation is the cornerstone of MBPT for electrons. In GW, the self-energy is calculated in one iteration formally as

[TABLE]

where the product here is in real space and time. The screened Coulomb interaction $\displaystyle W$ consistent with the GW approximation is that calculated within the RPA for a given Green’s function G. Self-consistent GW is a computationally expensive approach that requires the solution of a Dyson equation multiple times, and that involves the inversion of large, complex, and near-singular matrices. Furthermore, it has been shown that it fails for systems with over half-filled $\displaystyle 3d$ bands Shirley and Martin (1993). A more successful, further approximation is one-shot, non-self-consistent GW, or simply G0W0, which depends explicitly upon and input Green’s electronic function $\displaystyle G_{0}$, and stops at the first iteration of the self-energy, $\displaystyle\Sigma=\mathit{i}G_{0}W_{0}$.

In practice, this requires the choice of a suitable basis and, when all that is of interest are the QP energies, it is expedient to use the KS-DFT eigenbasis, on the grounds that the approximate KS-DFT density is usually reasonable, even if the KS-DFT eigenspectrum, represented in the form of G0, is unphysical. Assuming that $\displaystyle\langle\psi_{i}|\psi_{i}^{\mathrm{QP}}\rangle\approx 1$, the QP energies can furthermore be approximated as a first-order correction to the KS eigenvalues, as

[TABLE]

and $\displaystyle Z_{i}$ is called the QP re-normalization factor. This factor can be thought of as the absolute value of the charge of the QP (e.g., of the electron and its screening cloud). Here, $\displaystyle\hat{v}_{\mathrm{xc}}$ is the approximate exchange-correlation potential operator of KS-DFT. The method described by this final step is called perturbative G0W0, and it remains explicitly dependent on the choice of approximate functional in KS-DFT.

In practice, in the KS wave-function basis, the non-interacting single-particle Green function takes the form in the frequency domain, for $\displaystyle t^{\prime}>t$ Marini (2001),

[TABLE]

and, once $\displaystyle G_{0}$ is obtained, $\displaystyle W_{0}$ is calculated by using Dyson’s equation starting from the bare Coulomb interaction with

[TABLE]

where $\displaystyle W^{0}$ is a function of $\displaystyle\omega$ through the inverse dielectric function $\displaystyle\varepsilon^{-1}$. In reciprocal space, $\displaystyle W_{0}$ is expressed by Eq. (5) and Eq. (9) as the direct product

[TABLE]

Computationally, the inversion of the dielectric function, which is a large matrix with frequency-dependent complex entities, is troublesome. Hence, the frequency-dependent complex entities are approximated by Lorentzian peaks within the plasmon-pole approximation (PPA) Hybertsen and Louie (1986); Godby and Needs (1989). The idea behind the PPA is to replace the single-particle transitions making up $\displaystyle\varepsilon^{-1}_{\mathbf{G,G^{\prime}}}(\mathbf{q},\omega)$, which increase in number with the square of the system size, with a smaller number of effective plasmon modes. Thus $\displaystyle\varepsilon^{-1}$ is approximated in practice via

[TABLE]

where $\displaystyle\Omega_{\mathbf{G},\mathbf{G}^{\prime}}$ and $\displaystyle\bar{\omega}^{\lambda}_{\mathbf{G},\mathbf{G}^{\prime}}$ are the strength and the frequency of plasmons fitted to the RPA inverse dielectric function. A second advantage of the PPA is that analytical formulae for $\displaystyle\Sigma$ are available in the frequency domain.

III Computational details

Geometry optimization, self-consistent field (SCF) and non-self-consistent field (NSCF) simulations were performed using the Quantum ESPRESSO software (QE) Giannozzi et al. (2009, 2017). For this, norm-conserving PBE pseudo-potentials were produced using the pseudo-potential generator OPIUM opi . The initial crystallographic information for bulk Au, Ag, and Cu in their FCC structures were adopted from X-ray diffraction data at 1072 K from Ref. 75, for consistency with choice of smearing parameter for the Marzari-Vanderbilt cold smearing Marzari et al. (1999), namely $\displaystyle 0.1$ eV. Full geometry relaxations were then performed with variable cell parameters at over-converged plane-wave cutoff energies (Ecut $\displaystyle=75$ Ha) with automatically generated $\displaystyle 4\times 4\times 4$ uniform Monkhorst-Pack Brillouin zone sampling, without imposing any crystal symmetry.

For band-structure calculations, we moved down to a plane-wave cutoff energy of Ecut $\displaystyle=25$ Ha, which is sufficient to attain a total-energy tolerance of $\displaystyle\Delta E_{\mathrm{tot}}\leq 0.001$ Ha per atom, but up to a uniform Brillouin zone sampling of $\displaystyle 16\times 16\times 16$ at the NSCF level, which was necessary to converge the expectation value of the exchange self-energy. Experimental spectra have large inter-band smearing $\displaystyle\Gamma$ values due to finite temperature effects and impurities in practice Gurzhi (1958, 1959); Fox (2010). Hence, a Lorentzian smearing parameter of full-width $\displaystyle 0.2$ eV was adopted for spectral simulation with the Yambo code Marini et al. (2009). Final simulations with a common set of parameters were performed on Au, Ag, and Cu for benchmarking the various levels of theory studied, against experimental spectra. The work-flow for simulations of spectra with FGR, RPA and G0W0+RPA is illustrated in Fig. 1.

IV Work-flow optimization and

benchmarking on pure metals: Au, Ag, and Cu

For noble metals, we have found that even a converged inaccurate semi-local KS-DFT description of the relevant quasiparticle bands requires demanding run-time parameters and accurate pseudopotentials. Therefore, when larger crystallographic unit cells are of interest such as for the alloys central to this work, even perturbative G0W0 becomes computationally impractical, and we cannot routinely go very far beyond KS-DFT in terms of computational overhead. Hence, here, we have pursued an intermediate, compromise approach in which non-local quasiparticle screening effects are incorporated approximately, in a scaleable manner which incurs a minimum additional computational cost that is insignificant compared to the final RPA calculation for the spectrum.

In principle, G0W0+RPA spectra require a full QP band-structure as a starting point. To to obtain the QP band-structure, we would need to evaluate QP energies for each point in the Brillouin zone. for every band. For a given band and point in the Brillouin zone $\displaystyle\{i,\mathbf{k}\}$, such an operation consists of summing throughout the Brillouin zone and over bands to determine G0, as well as W0 through the inverse dynamic dielectric function for the self-energy operator in Eq. (II.3). Such a task is highly demanding both in terms of CPU hours as well as RAM. This operation needs to load all information about KS wave-functions in each processor unit, when using Yambo. Since our eventual goal here is to construct optical spectra rather than QP bands, an averaged stretching to underlying KS band-structure via stretching operators close to the Fermi level is sufficient, as well as more feasible for the spectral range of interest. The idea of stretching operators is to approximate QP energies as linear-functions of the KS energies, and for metals in particular we have simply

[TABLE]

where $\displaystyle s_{\mathrm{v}}$ and $\displaystyle s_{\mathrm{c}}$ are separate stretching factors for the valence and conduction bands, respectively. Such an approach introduces averaged corrections to the valence and conduction bands around Fermi level for the missing non-local electronic exchange-correlation effects, whilst keeping the Fermi level fixed.

The stretching operators were determined by linear regression on $\displaystyle\epsilon_{i}$ vs. $\displaystyle\epsilon_{i}^{\textrm{QP}}$. For pure metals, QP energies were calculated for 6 valence bands and 6 conduction bands at 10 points at and around $\displaystyle\Gamma$, and the stretching operators were determined by linear fitting as shown in Fig. 2 with the values listed in Table 1. In Fig. 2, two branches are observed in the valence manifolds. These distinctive branches are due to different non-local exchange contributions to $\displaystyle s$ and $\displaystyle d$-bands, but nonetheless a single stretching factor proved to be adequate.

The stretching operators modify the KS band-structure as shown in Fig. 3. The Fermi-Dirac distribution for the chosen electronic temperature was used to interpolate between the distinct stretching parameters for the valence and conduction bands. KS-DFT tends to excessively flatten the fully filled $\displaystyle d$-bands due to an absence of attractive non-local exchange. In a sense, the stretching operator approximately corrects the dispersions of the bands, particularly for the occupied $\displaystyle d$-bands but also for the half-occupied $\displaystyle s$-band, which is made less dispersive. In Fig. 3c for Cu, the bands close to the Fermi level are more narrowly packed and flattened compared to those of Ag and Au cases, reflecting the fact that the error in the KS-DFT treatment, and hence $\displaystyle s_{v}^{Cu}$, is the largest amongst the three. The inverse behaviour is seen, however, for the conduction stretching operators of pure metals, which is due to correlation effects only, as Ag has the largest stretching factor in the conduction manifold, albeit that the differences between the metals are less pronounced here.

IV.1 Drude parameters for pure metals

Before constructing optical spectra, the Drude parameters are needed for the intra-band part of the dielectric function in Eq. (II.2). Our first step is to calculate the Drude plasmon energies using Eq. (13). For this purpose, the energies of bands crossing the Fermi level at each k-point were extracted from the output of NSCF calculations and interpolated on a fine grid in the Brillouin zone (601 points in each reciprocal-space direction), and the Fermi surface was located on this grid with a $\displaystyle\pm 0.01$ eV tolerance for each system. Then, the square of the Fermi velocities, averaged over the Fermi surface, were calculated by means of Eq. (II.2). The calculated Fermi velocities for pure Au, Ag, and Cu are listed in Table 2 along with results of Ref. 80, which uses a similar procedure with an extremely dense Brillouin zone sampling as $\displaystyle 200\times 200\times 200$ rather than interpolating, as we have. Furthermore, the average Fermi velocity magnitudes for QP band-structures were approximated by applying the geometric averages (more appropriate to simulate intra-band response than the arithmetic mean) of the valence and conduction stretching operators, specifically using the formula

[TABLE]

Our averaged KS-DFT Fermi velocities slightly underestimate those of Ref. Gall, 2016, as shown in the third column of Table 2, and the origin of this discrepancy is not evident. Considering the much smaller computational cost of our interpolation scheme, our values are very reasonable estimates of the Drude plasmon energies. Lastly, the DOS based on the KS band-structure and QP band-structure were extracted by applying a $\displaystyle 0.1$ eV Lorentzian broadening using a post-processing tool of Yambo. Using Eq. (13), the Drude plasmon energies were estimated for the KS band-structures and QP band-structures with their respective DOS at the Fermi level.

The next step is to approximate the inverse life-time of the Drude plasmon, as well as the electric permittivity in the infinite-frequency limit, using our semi-empirical approach illustrated in Fig. 4. Scaling factors for the inverse life-times $\displaystyle\eta_{\mathrm{p}}$ in Eq. 14 were determined by fitting the imaginary part of the dielectric function to the experimental curves in Ref. 81, and then $\displaystyle\varepsilon_{\infty}$ values were determined by fitting the real part of the dielectric function with sets of $\displaystyle\{\omega_{\mathrm{p}},\eta_{\mathrm{p}}\}$ to the same experimental curves for Au, Ag, and Cu. The resulting values for FGR, RPA and G0W0+RPA are summarized in the Supporting Information (SI).

In experimental studies, the Drude parameters are commonly determined by least-squares fitting of the Drude-Lorentz model in Eq. (II.2) to measurements in the IR spectral range. Such measurements are highly sensitive to experimental details, and the resulting literature for the Drude parameters is not in good consensus as shown Fig. 5. This figure shows that, despite the relative simplicity of the approaches adopted in this work, our Drude parameters are comparable with experimental predictions (particularly those of Ordal Ordal et al. (1985)).

IV.2 Optical spectra of pure metals

The spectra of our elemental metals were obtained by applying FGR and RPA to the approximate KS-DFT band-structures, and RPA upon our approximate QP band-structures. The real and imaginary part of the total dielectric function are shown for pure metals in Figs. 6, 7, and 8, along with the experimental spectra from the detailed work by Babar and Weaver in Ref. 81. Also, the electron energy-loss spectrum (EELS) Boersch, Miessner, and Raith (1962); Raether (1965); Froitzheim (1977); Ferrell (1956); Bohm and Pines (1951); Pines and Bohm (1952); Bohm and Pines (1953) is shown, as calculated using the relation Sottile (2003)

[TABLE]

For all three systems, both FGR and RPA predict the lowest band-to-band absorptions to be at energies $\displaystyle\sim$ 0.5-1.5 eV lower than those of the experimental absorption spectra. Furthermore, for higher energies, both approaches produce strong absorption peaks that contradict experiments. On the other hand, G0W0+RPA locates the low-lying peaks at $\displaystyle\sim$3-4 eV in Au, at $\displaystyle\sim$4-5 eV in Ag, and at $\displaystyle\sim$2-3 eV in Cu more accurately, and does well for the overall curve trend, with respect to the experimental absorption spectra. All three approaches reproduce the behaviour of $\displaystyle\mathcal{E}_{1}^{\textrm{Cu}}$ and $\displaystyle\mathcal{E}_{1}^{\textrm{Au}}$ successfully, and particularly so for $\displaystyle\mathcal{E}_{1}^{\textrm{Ag}}$. Such improvements, along with the very substantial improvement in $\displaystyle\mathcal{E}_{2}$ given by G0W0+RPA, lead us to locate the first plasmonic peaks in all systems quite accurately in EELS, where FGR and RPA miss the the salient features completely (see Figs. 6, 7, and 8). As one can observe, the improvements provided by G0W0+RPA become less effective at higher energies, although some improvements are still achieved with respect to FGR and RPA. This depletion of performance at higher energies is to be expected, as the QP band-structures were approximated using an averaged stretching factor determined using only bands close to the Fermi level. Hence, by construction, our streamlines approach is more effective for transitions between bands close to the Fermi level, which constitute the lower part of the spectra, which are always those relevant to practical plasmonic applications.

V Spectra of Au${}_{\textbf{x}}$Ag${}_{\textbf{y}}$Cu${}_{\textbf{1-x-y}}$ alloys

Initial geometries for selected ordered alloys with compositions in multiples of $\displaystyle 12.5\%$ were constructed by using super cells of pure metals and substituting the desired number of atoms of other species to achieve primitive unit cells for each stoichiometric ratio. These geometries were optimised at the DFT level. Sample crystal structures for each stoichiometric ratio are shown in the SI. Alloys with the stoichiometric ratios of $\displaystyle\{2:1:1\}$ and $\displaystyle\{6:1:1\}$ have $\displaystyle 3$ and $\displaystyle 4$ possible phases, respectively, and in total $\displaystyle 39$ structures were studied through the standardized work-flow shown in Fig. 1. For systems with multiple primitive phases, we calculated our final spectra by means of a thermodynamic averaging process using the Boltzmann factor defined as

[TABLE]

where $\displaystyle j$ is the phase index, and $\displaystyle E_{0}$ and $\displaystyle E_{i}$, respectively, are the lowest ground-state energy among all phases and the ground-state energy of the $\displaystyle i^{th}$ phase. The final spectra of $\displaystyle\{2:1:1\}$ and $\displaystyle\{6:1:1\}$ are thus linear combinations of the spectra of their respective phases, with corresponding weighting constants.

V.1 Stretching operators for alloys

Band stretching operators were calculated for all valence bands and an equal number of conduction bands at 10 points at and around the $\displaystyle\Gamma$ point for each crystal structure of each stoichiometric ratio. The QP renormalization factors expressed in Eq. (II.3) show a steady decreasing trend with increasing Cu ratio, both in the valence and the conduction manifolds, as shown in Fig. 9a and Fig. 9b. The valence stretching operator in Fig. 9c grows significantly larger for increasing Cu ratios. Pure Cu has the largest reciprocal unit cell, where the $\displaystyle d$-bands are spuriously flattened and narrowly packed the most by KS-DFT, as seen in Fig. 3c. Hence, a larger QP stretching of the valence manifold is expected to result from an increasing Cu concentration. The conduction stretching operators show inverse trends, providing a slightly smaller stretching at the conduction manifold for increasing Cu concentrations. The stretching operators were applied to converged geometry of each ordered alloy cell, and perturbative G0W0 simulations were performed to construct the stretched (QP) band-structures individually. For simplicity, we will only discuss our approximated QP results from here, omitting FGR and RPA.

V.2 Drude parameters for alloys

Using a similar procedure to that of pure metals, the Fermi velocities and DOS at the Fermi level were computed for each alloy system. The stoichiometric dependence of the averaged QP Fermi velocities and DOS at the Fermi level are shown in Fig. 10. The Fermi velocities show some symmetric features, while they are lower in the case of predominantly Cu-based stoichiometric ratios. Conversely, the DOS becomes large for increasing Cu ratios as the volumes of the systems are also contracting with increasing Cu concentrations.

Competing trends in the Fermi velocities in Fig. 10a and DOS in Fig. 10b compensate each other, and lead to a symmetric trend in the interpolated contour-plot of the Drude plasmon energies in Fig. 11a. The inverse life-times in Fig. 11b were evaluated by using Eq. (14), where $\displaystyle\{c_{\eta}\}$ coefficients were produced via arithmetical averaging of the inverse-life times of pure metals with respect to the stoichiometric ratios as

[TABLE]

As the inverse life-times are proportional to the Fermi velocities and the DOS at the Fermi level, they show similar symmetries to the Drude plasmon energies. The electric permittivities at the infinite-frequency limit are simply approximated by the arithmetical averaging of values of pure metals with respect to the stoichiometric ratios; hence, the trend is a flat plane by construction.

VI Plasmonic performances of Au${}_{\textbf{x}}$Ag${}_{\textbf{y}}$Cu${}_{\textbf{1-x-y}}$ alloys

Even though, plasmon production is highly dependent on size and geometry of nano-materials, some fundamental criteria can be suggested as universal conditions for strong plasmonic response. The primary requirement of a strong plasmon is the presence of a high-density of free electrons such as in noble and alkali metals, which are prominent systems for plasmonic applications Blaber, Arnold, and Ford (2010). Moreover, plasmon quality is predominantly determined by loss Anantha Ramakrishna and Pendry (2003); Arnold and Blaber (2009), which can occur through various phenomena such as radiative dumping, surface scattering, thermal loss Kreibig and Vollmer (1995), and imperfections in materials such as surface roughness Raether (1988), and grain boundaries Kuttge et al. (2008). Some approximate methods are suggested to sum individual contributions of these conditions to determine overall plasmon quality Kreibig and Vollmer (1995). As we work on perfectly ordered bulk systems, our aim is to determine some universal preliminary optical merits, which are independent of size and structural properties, to measure plasmon qualities starting from bulk dielectric functions. Plasmons in bulk systems are predominantly bulk plasmons, which is a result of a combination of both intra-band and inter-band transitions Blaber, Arnold, and Ford (2010). The bulk plasmon energy $\displaystyle\Omega_{\mathrm{p}}$ is expected to be at a lower energy than the bare plasmon energy $\displaystyle\omega_{\mathrm{p}}$ due to screening of inter-band transitions. In addition to bulk plasmons, there are surface plasmons due to the finite size of realistic systems. EELS provides a signature for bulk plasmons, whereas it requires a slight modification due to the presence of $\displaystyle d$-bands to capture surface plasmons Rocca (1995)

[TABLE]

which create a condition as $\displaystyle\mathcal{E}_{1}\approx$ -1 and $\displaystyle\mathcal{E}_{2}\approx$ 0 for significant surface plasmons. Furthermore, some optical measures for more specific plasmons such as the localized surface-plasmon (LSP) Sönnichsen et al. (2002); Haes et al. (2004), and the surface-plasmon polaritons (SPP) Zayats, Smolyaninov, and Maradudin (2005), which are crucial to many plasmonic applications Zayats and Smolyaninov (2003) such as optical circuits Engheta (2007) and switching MacDonald et al. (2008); Krasavin and Zheludev (2004). In Ref. 88, various measures using the relation between $\displaystyle\mathcal{E}_{1}$ and $\displaystyle\mathcal{E}_{2}$ for LSP and SPP at low-loss and nearly an electrostatic limit have been investigated for their respective optimized geometries. $\displaystyle\mathcal{E}_{1}$ provides the necessary condition by checking the preliminary condition of a presence of free electrons, while $\displaystyle\mathcal{E}_{2}$ is related to loss due to the decaying of plasmons to particle-hole pairs via absorption. Hence, the number of electrons going through inter-band transitions is desired to be small around plasmon frequencies. Blaber, and et al. Blaber, Arnold, and Ford (2010) suggest some universal quality factors for LSP and SSP in metallic systems with optimized geometries as

[TABLE]

These quality factors provide some preliminary insights on the capacity of metals to produce surface plasmons and life-times of such plasmons determined by dumping due to inter-band transitions.

VI.1 Plasmonic response at common solid-state laser wavelengths

Three common solid-state laser wavelengths were chosen to demonstrate the plasmonic efficiencies of AuxAgyCu1-x-y . The first wavelength is a common red laser at $\displaystyle 650$ nm ($\displaystyle 1.9074$ eV) produced using InGaAIP  Koechner (2013) and used in a wide range of applications such as in imaging and sensing in biological systems in conjunction with a heterostructure of noble metals Jain et al. (2008); Küstner et al. (2009).

In Fig. 12, AuxAgyCu1-x-y alloys have weak bulk and surface plasmon resonances in orders of 10-2- 10-3. As this wavelength is commonly somewhere between Drude tails and inter-band transition edges of $\displaystyle\mathcal{E}_{2}$, these weak plasmons have long life-times due to small radiative dumping, which result in large LSP and SPP factors. Au6AgCu and AuAgCu6 exhibit relatively strong plasmon resonance, whereas LSP and SPP have poor quality factors at these stoichiometric ratios due to larger losses at this wavelength. Pure Ag exhibits large quality factors for LSP and SPP; however, plasmonic resonances are quite weak at 650 nm.

The second frequency is that of the common lasers used in blu-ray devices at $\displaystyle 405$ nm ($\displaystyle 3.0613$ eV), generally produced using InGaN Nakamura et al. (1996) for efficient optical recording in conjunction with noble metal nano-clusters Royon et al. (2010). In Fig. 13, {7:1:0} stoichiometric ratios show relatively significant plasmon resonance alongside {6:1:1} stoichiometric ratios in order of $\displaystyle\sim$ 10-1, which are comparable to these of maximums of plasmon peaks of binary alloys, with large quality factors of LSP and SPP. Ag predominantly has long-lived LSP, while AuAg3 shows some longevity.

Lastly, the deep-UV laser at $\displaystyle 290$ nm ($\displaystyle 4.2753$ eV) commonly produced in Ce:LiSAF / Ce:LiCAF media with Nd:YAG lasers Marshall et al. (1994) was presented for the sake of discussion. In Fig. 14, the bulk and surface EELS profiles here have structures but still a stronger response in Ag, Au, Au6AgCu, and AuAg6Cu, with a newly arising strong response in AuAg as shown. Correspondingly, AuAg has short-lived LSP and SPP as does Ag, which performs relatively better in the former case as at the blu-ray laser.

The plasmonic responses of AuxAgyCu1-x-y have complex behaviours, which cannot be simply interpolated based on the stoichiometric ratios alone, despite the similar electronic structures of constituting atoms. Consistent first-principles simulations are thus essential to engineer and predict comprehensive alloy systems with well-controlled approximations at reasonable computational costs.

VII Conclusion

It was shown that RPA starting from the KS band-structure fails to acceptably locate peaks in the absorption spectra of Au, Ag, and Cu, whereas RPA starting even from an approximate QP band-structure performs drastically better by locating low-energy peaks accurately in absorption spectra. Such an approximate QP band-structure can be achieved with little additional computational costs by applying some average stretching to bands via stretching operators, which can be obtained within G0W0 for a small set of grid points in the Brillouin zone and bands around the Fermi level. Despite their lack of finite size and surface effects, the bulk dielectric functions were used to determine some preliminary optic merits for LSP and SPP at plasmon resonance wavelengths. At the common solid-state red laser around 650 nm, AuAgCu6, Au6AgCu, AuCu7, and Au7Cu show relatively strong plasmon resonances, whereas higher Ag concentrations reduce plasmon resonance in general. AuAg6Cu and AuAu7 as well as AuAg, AuCu, and AgCu alloys start showing stronger plasmon resonances with significant quality factors for LSP and SPP at the common blu-ray laser at 405 nm. This trend at 405 nm becomes more distinctive at deep-UV laser wavelengths around 290 nm. Particularly, pure Ag, AuAg7, Ag7Cu produce significant plasmon resonances with high LSP and SPP quality factors at 290 nm. Combining RPA starting an approximate QP band-structure with the Drude-Lorentz model using the semi-classical Drude parameters provide a computationally feasible approach to investigate spectra and plasmonic responses of the noble metals and their alloys. Despite its simplicity, some preliminary optic merits can be obtained that are useful as a starting point for tailoring plasmonic responses in such systems.

We gratefully acknowledge the support of Trinity College Dublin’s Studentship Award and School of Physics. We acknowledge and thank Tonatiuh Rangel and Daniele Varsano for discussions. We also acknowledge the DJEI/DES/SFI/HEA Irish Centre for High-End Computing (ICHEC) for the provision of computational facilities and support. We finally acknowledge the Trinity Centre for High Performance Computing and Science Foundation Ireland for the maintenance and funding, respectively, of the Boyle cluster on which further calculations were performed.