Anisotropic optical response of gold-silver alloys

Gold and silver alloys enable novel opportunities for engineering materials with distinct optical responses. Here we investigate the optical properties of gold and silver (Ag x Au 1 − x ) structures using First-Principle Density Functional Theory (DFT) for gold concentrations varying from 0% up to 100% with steps of 25%. Results of the optical permittivity are analyzed with the independent particle approximation and compared with previously reported theoretical and experimental works. The pure systems and the ones with unbalanced concentrations exhibit isotropic optical responses. The Ag 0 . 50 Au 0 . 50 shows an anisotropic response among the y-direction and the xz-direction, mainly in the intraband transition energy range. The anisotropy is elucidated in terms of the d-orbitals density of states and the charge distribution with the structure. The anisotropic optical response can be the origin of the discrepancies among reported experimental results for structures with the same stoichiometry.


Introduction
Alloying in photonics is a promising route for obtaining materials properties and functionalities otherwise unachievable with single material-based structures [1][2][3][4] . Combining different chemical elements opens up avenues of possibilities to fine-tune the optical and electronic response of devices for a broad range of applications such as energy harvesting 5 , continuous-wave and pulsed lasing 3,6 , and non-linear optics 7 . In the last decade, the rapid progress of the plasmonic field occurred due to the advance of new techniques of fabrication and characterization of pure metallic structures 4,8 . More recently, a change of paradigm driven by the maturing of nanofabrication methods and the systematic analysis of alloyed nanostructured systems [9][10][11] resulted in the rise of applications of this novel system as plasmonic tweezers 12 , hydrogen sensors 13 , photocatalystis 14 , and bio-sensors 15 .
The complex permittivity of materials is directly related to their optical properties. The control of the permittivity through alloying allowed the engineering of surface plasmon polariton (SPP) resonances, diversifying and increasing the efficiency of devices such as sensors, nanoscale lasers, and solar cells 10,16,17 .
For pure metals, the permittivity is well described in the literature. However, for alloys, there is a lack of fundamental knowledge and agreement about the physical processes underlying their optical response 4 .
To predict the permittivity of a certain compositions, theoretical approaches employed linear interpolation between the alloy's counterparts. Nevertheless, especially for noble metals, there was no good agreement with experiments, leading to spurious effects as negative absorptions 18 . This is mainly due to complex mechanisms such as band hybridization 19 . Also, effective multi-parametric Drude-Lorentz-like models were used to fit experimental results, accounting for frequency-dependent corrections of interband transitions 9,18,20,21 . Here, there was no consensus among fitted parameters, leading to differences in the description of the visible and near ultra-violet optical responses 9 . Finally, in a parameter-less fashion, ab-initio methods such as the density functional theory (DFT) were used to elucidate several aspects of the optical properties of alloys 4,21,22 . The advantage of using computational approaches lies in the fact that it is a much faster route to predict materials properties than the traditional fabrication and measurement methods. This has lead to unprecedented development in materials discovery 4,23 . The possibilities of combining materials and properties are numerous, and there are always unexplored aspects of it.
From the optical point of view, materials with unique properties can lead to applications that explore new degrees of freedom for optical processes. That is the case of birefringent materials, whose response depends on the polarization of the incident light, with applications as optical waveguide-polarizers with exceptional extinction ratios 24,25 . At the nanoscale, new degrees of freedom in the optical response can be achieved by manipulating the localized surface plasmon polariton (LSP) via the shape and/or size of nanoparticles 26 , nanorods 27,28 , nanowires 29 , and ultrathin metal films 30 . Particles with different shapes lead to anisotropic linear optical responses 26,29,30 . Anisotropic non-linear properties also play a meaningful contribution due to their dependence on the shape of the nanostructure 26,31 . However, we 2/17 believe the analysis of the anisotropic optical behavior of bulk alloys is still underestimated and should receive prior attention to the nanostructured systems.
In this work, we report on the application of DFT to obtain the bulk permittivity of gold and silver alloys with concentrations ranging from the pure metals to 25% Au (75% Ag), 50% Au (50% Ag), and 75% Au (25% Ag). For the Ag 0.50 Au 0.50 alloy, we show that the optical response is anisotropic due to an unbalanced charge distribution among the Au and Ag atoms. Using DFT with the Generalized-Gradient-Approximation (GGA) exchange-correlation functional, we calculate the influence of stoichiometry on the band structures, the density of states, and the energy-dependent permittivity with the independent particle approximation 23 . We observe a drastic difference between the y-direction and the xz-direction of the permittivity in the intraband energy regime. The intraband transitions are heavily dependent on the shape of the d-orbitals in the valence band. Here, we confirm that the d xz orbital differs from the d xy and d yz due to an unbalanced charge distribution in the structure. For all other chemical compositions, the optical response is isotropic, as expected for a cubic periodic unit cell.

Band Structure and Density of States
To obtain the permittivity of the Ag-Au alloyed structures, we first perform their structural relaxation and calculate their band structures using the Quantum ESPRESSO package 32   properties. 33 The optical response of the bimetallic systems are direct dependent on their chemical composition 10,21,36 . For instance, in energy-dependent reflectance measurements one can clear see an

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abrupt reduction of the spectra transitioning from the ultra-violet around 3.5 eV (Ag) to the visible around 2 eV (Au) 23 . For pure Ag and Au, reflectance evaluated with DFT is found to have a slight blue shifted drop ( Supplementary Fig. S1) due to the underestimation of the d − sp interband transition obtained with the GGA-PBE functional, but still in good agreement with experimental responses 36 . Overall, our results nicely agree with the ones found in the literature 4, 17, 19-22, 36, 37 . Figure 2 shows the imaginary part of the permittivity (ε 2 ) for the Ag 0.50 Au 0.50 composition where an anisotropic behavior is observed in the calculated results (Figure 2 (a)). Conversely, the remaining structures are isotropic (see Supplementary Fig. S2). We evaluate ε 2 via the independent particle approximation (IPA) (see Methods for details). At the intraband transition region (energies < 3.5 eV) we see a split between the out-of-plane (y-direction) from the in-plane (xz-direction) component of ε 2 .

Complex Dielectric Function for Ag-Au Alloys
To the best of our knowledge, for the aforesaid concentration, this is the first time that the anisotropic response is reported for the bulk configuration. The real part of the permittivity (ε 1 ), obtained from the

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Cao Long and co-workers showed that the structure of the Ag 0.50 Au 0.50 system is not cubic, but triclinic 22 .
In both works, no evidence of an anisotropic optical response was reported. It is known that distinct methods of fabrication provide materials with different morphology and permittivity 38 . When comparing reported values of ε 2 found in the literature we notice a larger discrepancy that reflects the same trend observed in our calculations ( Fig. 2(b)). 4, 19-21 Therefore, we attribute such difference as a result of the anisotropic optical response for the Ag 0.50 Au 0.50 system. As shown in Fig. 2(a), the separation between the in-and out-of-plane components of the permittivity is in agreement with the literature results ( Fig. 2(b)).  and (d)), the charge distribution slightly changes, with an nearly evenly distribution of the main charge over the unit cell (Supplementary table S1 and Supplementary table S2). However, for the Ag 0.50 Au 0.50 alloy (Fig. 4(c)), we find the charge to change in the xz-direction due to the distribution of the atoms in the unit cell. is clearly different from that of the out-of-plane (Fig. 4(m) 8/17 Figure 5 shows the birefringence of the Ag 0.50 Au 0.50 structure, where one can observe significant anisotropy for wavelengths above 600 nm. Here, we report the difference between the y-and the xzdirection of the refractive index (n) and the extinction coefficient (k).For comparison, n and k of all systems can be found on Supplementary Fig. S7. As expected, the pure structures and the remaining alloys shows isotropic response. The structure with 50% Au and Ag could be employed as a birefringent material, broadening its range of applications as those in linear and non-linear anisotropic optics 26,28,29 .

Conclusions
In summary, we used the density functional theory with the GGA exchange-correlation functional to determine the electronic structure, the density of states, and the optical response of gold-silver bimetallic structures in bulk form. For the pure gold and silver structures, as well as the alloys with 25% (75%) silver and 75% (25%) we obtained isotropic optical responses as expected for the cubic periodic unit cells. For the alloy with equal concentrations of gold and silver, namely Ag 0.50 Au 0.50 , the same was not valid. For such structure, we observed a difference between the y-direction and the xz-direction responses of the optical permittivity, mainly in the intraband energy regime. Since the dominant contribution to the intraband transitions comes from the d-orbitals in the valence band, we analyzed the their projections on the density of states. We found that the d xz orbital response were different from the d xy and d yz ones due to to an unbalanced charge distribution in the structure. To the best of our knowledge, for the first time, we reported such anisotropic behaviour. Previous theoretical works obtained the permittivity of alloyed structures by

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interpolating and fitting models using the pure samples as starting points. Consequently, no anisotropy could be found 18,20,41 . Also, using first principle techniques, optical characterization of bimetallic structures were described as the a mean average of the three Cartesian directions 4,19,21,22 . Likewise, this approach potentially hide any perspective of observing the anisotropic behavior shown. The present work shed light into the discrepancies present in the experimental and theoretical results found in the literature for Ag 0.50 Au 0.50 alloys. Moreover, one could obtain a birefringence by replacing Ag atoms with more massive elements, or with more pronounced relativistic effects and different electronegativities. The analysis of the non-linear optical properties could be performed, and we expected it to reveal anisotropic responses to the second-harmonic generation and third-order susceptibility 26 .

Computational Details
Plane-waves-based DFT calculations were performed to obtain the optical and electronic properties of Ag-Au bimetallic structures, using the Quantum ESPRESSO package 32,42,43 . We employed the For the electronic occupation, we adopted the Marzari-Varderbilt cold smearing with a Gaussian spreading of 0.01 Ry on the Brillouin zone integration, taking into account the metallic occupation character of the systems 48 . The density of states (DOS) calculations required a denser mesh of k-points.

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Therefore, we used the non-self-consistent calculation with 32 × 32 × 32 k-points with no shift, and an gaussian broadening of 0.1 Ry.

Permittivity
The wavelength-dependent permittivity (ε(ω)) was evaluated using the independent-particle approximation (IPA) through the SIMPLE code. 23 At the IPA level, the excitonic interaction is neglected.
However, it is still suitable to obtain the optical properties of metallic systems through this method. 33,36 The advantage of using the SIMPLE code relies on the use of optimal basis functions based on the Shirley integration approach, allowing for a very dense k-points sampling with affordable computational cost. 36,49 The interband transitions contribution to the optical permittivity for a negligible transferred momentum of the photons is given by 23 where the summations are over the wave-vectors k in the first Brillouin zone, and over valence (n) and conduction (n ′ ) band states with energies E kn and E kn ′ , respectively. P k,n→n ′ is the dipole matrix element denoting the transition between valence and conduction band states. The states occupation is given by the Fermi-Dirac distribution f kn(n ′ ) , and η is an empirical interband broadening to reproduce experimental scattering processes. The intraband transitions contribution to the optical permittivity is given by a Drude-like term dependent on the plasma frequency ω d , 23 where γ is the empirical intraband broadening. To calculate the intraband and interband optical permitivity we employed a highly dense mesh with 72 × 72 × 72 equally spaced k-points, with the threshold for the optimal basis set to 0.01bohr 3 . The intraband and interband broadenings were chosen to be 0.002 Ry and 0.013 Ry, respectively.

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Partial charge analysis Figure 4 shows isosurfaces of charge obtained from the DFT calculation and processed using the VESTA software. 50 The isolevel was set 0.1 for all the structures to allow for comparison. The integration in the charge density shown in Figs. 4 was also done using VESTA, with the results obtained with the projwfc.x software of Quantum ESPRESSO package 32 . We integrate the charge projection along the showed directions for a slice of the unit cell ranging from 20% to 40% of the cell's volume in that direction.
We choose to include isosurfaces with the level between 2% to 7% of the total cell charge, where we spot the charge anisotropy among planes.
The charge distributed across the unit cell was also analysed via the Bader and Voronoi methods. The Bader analysis consists in separating the charge in regions using minimum density surfaces. The analysis was done employing the Critic2 51 post-processing code that uses the Yu-Trinkle integration method 39 .
The results of the Bader partial charge analysis were presented in the Supplementary table 1