Influence of Scalar-Relativistic and SpinâOrbit Terms on the

Article Cite This: J. Phys. Chem. C XXXX, XXX, XXX−XXX

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Influence of Scalar-Relativistic and Spin−Orbit Terms on the Plasmonic Properties of Pure and Silver-Doped Gold Chains M. H. Khodabandeh,† N. Asadi-Aghbolaghi,† and Z. Jamshidi*,†,‡,§ †

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Chemistry & Chemical Engineering Research Center of Iran (CCERCI), Pajohesh Boulevard, 17th Km of Tehran-Karaj Highway, P.O. Box 1496813151, Tehran, Iran ‡ Chemistry Department, Sharif University of Technology, Tehran 11155-9516, Iran § Amsterdam Center for Multiscale Modeling Section Theoretical Chemistry, VU University Amsterdam, De Boelelaan 1083, 1081 HV Amsterdam, The Netherlands S Supporting Information *

ABSTRACT: The unique plasmonic character of silver and gold nanoparticles has a wide range of applications, and tailoring this property by changing electronic and geometric structures has received a great deal of attention. Herein, we study the role of the quantum properties in controlling the plasmonic excitations of gold and silver atomic chains and rods. The influence of relativistic effects, scalar as well as spin−orbit, on the intensity and energy of plasmonic excitations is investigated. The intensity quenching and the red shift of energy in the presence of relativistic effects are introduced via the appearance of d orbitals directly in optical excitations in addition to the screening of s-electrons by mixing with the occupied orbitals. For the linear gold system, it will be demonstrated that by increasing the length the relativistic behavior declines and the contribution of d orbitals to the plasmonic excitations evidently decreases. Furthermore, silver atoms are doped in gold chains and rods (with two different arrangements) to realize how gold−silver interactions decrease the relativistic effects and enhance the intensity of collective excitations. Finally, to strengthen the plasmonic behavior of gold, the elongation of chain and doping with suitable atoms such as silver (with the classical plasmonic behavior) can be introduced as the manipulating ways to control the influence of scalarrelativistic and spin−orbit effects and, consequently, reinforce the plasmonic properties. materials. In the coinage metal group, the interplay of the filled localized d-shell with the half-filled delocalized s-shell is crucial; however, relativistic effects explain the singularity of gold in comparison to the other elements of group IB.20 Furthermore, the optical properties of gold nanostructures are affected by enhancing the degree of s−d hybridization and the screening of the s-electrons by the d electrons that can lead to quenching of the intensity of plasmonic excitations in comparison to silver. Extended studies by Idrobo et al.21 highlighted the contribution of d-electrons in accurate evaluation of electronic transition energies. Therefore, relativistic corrections as the most important term that affect the energy of the d-shells should be considered. Moreover, precise ab initio calculations on Au2 and AgAu dimers22,23 have indicated the importance of spin−orbit (SO) coupling for an excellent agreement of the low-energy absorption bands with the experiment. Castro et al.24 showed the relevance of spin−orbit coupling in the energies and intensities of the absorption spectra

1. INTRODUCTION Light−matter interaction properties of nanostructures and nanoclusters are a topic of great fundamental interest.1 The plasmon excitation as a collective oscillation of valence electrons of metals with pseudofree electrons, which may be tuned by changing either physical and chemical properties of nanomaterials, has well-established applications. Tailoring this optical characteristic has potential applications in photophysics,2,3 optical data storage,4−8 ultrasensitive sensing (chemical and biological),9−11 and energy conservation and storage.12,13 The optical properties of silver and gold nanosystems have long been of great fundamental interest,14−19 as the spectra of their collective oscillations with their sharp and distinct peaks are well known and well described by classical electrodynamic models. The classical models can be applied for large nanosystems; however, for smaller metal clusters, due to the emergence of the quantum effects, this does not provide an accurate description of absorption spectra. Therefore, the quantum-mechanical description of the plasmonic phenomena attracted much attention with the goal to understand and link the intrinsic differences of electronic structures to the feature of plasmonic spectra and make a contribution to the design of new © XXXX American Chemical Society

Received: December 14, 2018 Revised: March 12, 2019

A

DOI: 10.1021/acs.jpcc.8b12045 J. Phys. Chem. C XXXX, XXX, XXX−XXX

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The Journal of Physical Chemistry C

the NiAl surface) and others theoretical calculations.25,39−42 Geometry optimizations for symmetric pure and bimetallic chains were done, and their UV spectra were compared with those of nonoptimized structures (see Figure S1 in the Supporting Information). Overall, the trend is similar and a slight blue shift (less than 0.2 eV) was observed. This minor deviation can be related to the shorter calculated distance (2.63−2.73 Å) in comparison to the fixed 2.89 Å experimental distance. The time-dependent density functional theory (TD-DFT) within the adiabatic local density approximation (ALDA)43−46 was employed using the statistical average of orbital potentials (SAOP) exchange-correlation functional47,48 in combination with all electron triple-ζ double-polarized Slater-type orbitals.49 The chains with an even number of atoms have been selected to avoid the open-shell calculations. The accurate description of the virtual orbitals involved in the valence excitations is important for calculating optical properties,50 and the SAOP functional is famous as the functional that exhibits the correct asymptotic behavior.51,52 The excitation energies Ωn and the oscillator strengths f n are calculated by solving the following eigenvalue equation using Davidson’s algorithm

of gold nanostructures (they applied plane wave basis sets and spin−orbit-corrected pseudopotentials) and revealed the clear influence of this term, especially by reducing the dimensionality. The unique plasmonic properties of nanorods and chains arise from the confinement of electrons in one dimension (1D), which leads to two bands that correspond to a sharp longitudinal peak in low-energy region and a broad transverse band with high-energy excitations. The fabrication of gold chains on NiAl(110)25 led to theoretical studies that estimated the existence of collective excitation in one dimension during absorption of light. Coinage metal chains and nanorods with distinctive plasmonic excitations can be a good case study to find the importance of relativistic and spin−orbit quantum effects in the plasmonic behaviors. However, in small quantumlike systems, the transition of an electron between discrete energy levels results in distinct peaks. Therefore, classifying different types of electronic excitations and identifying which excitations are the origin of the collective oscillation may be difficult. Bernadotte et al.26 proposed an approach based on scaling the correlation of electrons during a quantum chemical calculation of excitations and showed the difference between single-electron and plasmon excitations in molecular chains and clusters. The nonclassical low-intensity optical response of gold nanoclusters in comparison to high-intensity classical-like plasmonic properties of silver led to the experimental and theoretical studies that modified the optical properties of gold by changing the composition and the chemical configuration.27−29 The driving motivation for this improvement is the extensive application of gold nanoparticles in biosensing,9,10,30 cancer therapies,31−33 and drug delivery.34 Accordingly, one of the most promising methods is the use of one element (gold or silver) to modify the electronic properties of the other. The bimetallic silver/gold nanostructures have been recently synthesized and provided highly efficient optical materials, which differ from those of the monometallic nanoclusters,35−38 and their plasmonic properties are controlled by changing the ratio of silver to gold. In this work, first of all, we try to understand the influence of quantum effects, scalar-relativistic (SR) contributions and spin− orbit (SO) coupling, on the intensities and energies of plasmonic excitations; the goal is an accurate description of this phenomenon for gold and silver and understanding the effect of system size. Furthermore, we consider the bimetallic silver/gold chains and rods to determine how interactions of gold and silver atoms in alloys with different Ag/Au ratios decrease s−d mixing and increase s−s transition (as the result of decreasing SR and SO effects), which enhance the intensity of collective excitations. Moreover, we focus on the plasmonic excitations of two different AgAuAg and AuAgAu alignments of bimetallic chains and rods in comparison to pure Agn and Aun ones.

ΩFI = ωI2 FI

(1)

where Ω is a four-indices matrix with elements Ωiaσ,jbτ; the indices consist of products of occupied-virtual (ia and jb) Kohn− Sham (KS) orbitals, whereas σ and τ refer to the spin variable. The eigenvalues ω2I correspond to squared excitation energies, whereas the oscillator strengths are extracted from the eigenvectors FI. The Ω-matrix elements can be expressed in terms of KS eigenvalues (ε) and the coupling matrix K Ωiaσ , j τ = δστδijδab(εa − εi)2 + 2λ (εa − εi) K iaσ , j τ b

b

(εb − εj)

(2)

The elements of the coupling matrix K in the adiabatic local density approximation (ALDA)53 are given by ÅÄÅ 1 K ijσ , klτ = d r d r′ φiσ (r)φjσ (r)ÅÅÅÅ ÅÇ |r − r′| ÉÑ Ñ + f xcστ (r, r′, ω)ÑÑÑÑφkτ (r′)φlτ (r′) ÑÖ (3)

∫ ∫

where φ is the KS orbital and f στ xc (r,r′,ω) is the exchangecorrelation kernel. The SAOP XC functional was applied in the self-consistent field calculations. λ is the scaling factor introduced by Bernadotte et al.26 In their research, the excitation energies were determined in dependence of the scaling parameter λ, which was varied from 0 (no exchange correlation) to 1 (full exchange correlation) in steps of 0.1. In the following TD-DFT calculations, λ is 1. By turning off the coupling matrix K (λ = 0), we get a spectrum at Kohn−Sham level (KS spectrum), where the excitation energy corresponds to the KS eigenvalue difference and the oscillator strength is calculated from dipole transition moments between KS orbitals. In addition, the plasmon analysis according to Bernadotte et al. implementation has been done by calculating the excitation energies with scaling λ from 0 to 1 with step 0.1.26 It makes a smooth shift between the noninteracting and the interacting electronic environment. In another world, we can recognize the plasmonic transition(s) from single-particle excitations and consider the effect of relativistic term on this smooth shift.

2. COMPUTATIONAL DETAILS Quasi-one-dimensional metallic chains with well-defined plasmonic excitations in comparison to the higher-dimensional systems received particular interest in recent theoretical studies. This simple artificial system provides insight into the role of intrinsic transition-metal properties in the plasmonic excitations. In this work, we generally focus on the longitudinal excitons of linear systems. Thus, the pure and bimetallic gold and silver chains with a fixed bond length of RAg−Ag = RAu−Au = 2.89 Å were chosen as the model of ID structures. These interatomic distances stem from the experimental distance (gold chains on B


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Figure 1. Absorption spectra of linear (a) silver and (b) gold chains with increasing the length at scalar-relativistic (SR) level of theory (convoluted with a Gaussian function of full width at half-maximum (FWHM) = 0.2 eV).

Figure 2. Absorption spectra of (a) silver and (b) gold nanorods grown along the ⟨110⟩ direction (with 32, 50, and 68 atoms) at LB94 scalar-relativistic (SR) level of theory convoluted with a Gaussian function of FWHM = 0.2 eV.

matrix vector, p is the moment operator, and c is the speed of light. In the following sections, two types of structures have been used for pure and alloyed systems: linear chains with the even number of atoms ranging from 6 to 24 and 32-, 50-, and 68-atom nanorods constructed from a 5-fold ring which grows along the ⟨110⟩ direction. To obtain reliable energies for the rod systems, the structural relaxation was done. Computational points of all of the above structures were also employed for alloyed systems, (AunAgm), in both linear chain and rods. The replacement of Au by Ag to make the typical alloys will be explained in further sections (in literature, different alloying patterns have been applied for other aims).28,36,41,57,58

The scalar-relativistic (SR) effect, eq 4, and fully relativistic case (including the spin−orbit coupling via self-consistent approach), eq 5, were considered using the zeroth-order regular approximation (ZORA) formalism.52,54−56 ij yz ZORA c2 ZORA ZORA HSR ΦSR = jjjV + p 2 pzzzΦSR 2c − V { k ZORA ZORA = ESR ΦSR

ij yz c2 HZORA ΦZORA = jjjV + σ ·p 2 σ ·pzzzΦZORA 2c − V k { i c2 j ZORA ZORA )ΦZORA = jjjV + p 2 = (HSR + HSO p 2c − V k yz c2 + 2 σ ·(∇V × p)zzzΦZORA = EZORA ΦZORA 2c − V {

(4)

3. RESULTS AND DISCUSSION The absorption spectra of 1D linear chains and rods as the simple model were recorded for pure and bimetallic Ag/Au. The recognizable plasmon peak in absorption spectra and the low computational cost due to high symmetry make these structures suitable for investigation of the nature of plasmonic oscillations and the separation of the aforementioned peak from single-

(5)

where the potential V contains the nuclear field and the electron Coulomb and exchange-correlation potentials. σ is the Pauli spin C


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Figure 3. Total (tot) and partial (-d and -s) density of states (DOS) of Ag8 and Au8 chains with (SR) and without (NR) considering the scalarrelativistic effects.

Figure 4. Computed spectra of pure Agn and Aun chains with (a) n = 8, (b) n = 16, and (c) n = 24 at three levels; nonrelativistic (NR), scalar-relativistic (SR), and fully relativistic (SR + SO) (convoluted with a Gaussian function of FWHM = 0.2 eV).

particle excitations. Figures 1 and 2 show the absorption spectra of neutral closed-shell Ag6−24 and Au6−24 linear chains and Agn and Aun (n = 32, 50, and 68) rods, respectively. To achieve some of the goals mentioned in the previous section, the modeling of rod forms was done by growth along the ⟨110⟩ direction. The aforementioned linear chain and rod models of Ag clusters display sharp low-energy longitudinal and broad high-energy transverse plasmon peaks, which survive for decreasing size. In the case of gold structures, these plasmon peaks are distinguishable only for large sizes (Au18−24); see Figures 1 and 2. This characteristic optical behavior has been reported at real-time

TD-DFT level of theory for sodium and potassium chains as well by Yan and Gao.59 It should be noted that the peaks on the right side of spectra shown in Figure 1 (especially Figure 1a) are arising from the combination of ∏ → ∏ transitions. Some of them are so-called transverse plasmonic excitations, and others are related to d−s or d−p interband transitions. The overlap between energy locations of these transitions (i.e., ∏ → ∏) makes the transverse plasmonic excitations an unsuitable choice to manipulate in linear chain or rod systems with small cross section. Thus, we focus on the longitudinal plasmonic behavior for the rest of the article. As this peak is generated by the D


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ism has modified the former results. The stabilization of s orbitals and destabilization of d orbitals are seen for both Ag and Au atoms. It is known that the relativistic contraction (εrel/εnr: the ratio of s-orbital energy in the presence and absence of relativistic effects) of the 6s shell of Au is much larger than that of the 5s of Ag.62 These level displacements have the consequences that the orbitals containing predominantly d and s characters are found clearly closer together in gold than in silver chains. Therefore, d−s band separation changed and 5d orbitals contributed strongly to HOMO hybridization of gold. The fully relativistic ZORA implementation via a selfconsistent formalism is the second step of relativistic calculation that also considers the spin−orbit effect. For the gold atom and its chain, spin−orbit coupling is particularly important and causes the energy splitting of dπ and dδ, which alters the molecular orbital ordering. Geethalakshmi et al.23 reported for Au2 such orbital rearrangement as the result of spin−orbit interaction and makes the dπ3/2 * (as a spinor) with predominantly 5d character, being the HOMO (see Figure 2 in ref 50). After including spin−orbit effects, the contribution of the d-character to high-energy sσ1/2 spinors increased relative to similar orbitals found in a scalar-relativistic calculation. As shown in Figure S2, spin−orbit coupling has a minor effect on orbital energy and d−s hybridization of silver orbitals whereas it has a major effect on Au systems. According to the given description, both parts of relativistic effects, scalar and spin− orbit parts, are responsible for a notable screening of s-electrons by d-electrons in Au, which although is negligible in Ag. In this part, the computed absorption spectra of Ag8 and Au8 chains with different formalisms (up to 4 eV) are compared (Figure 4a). For the Ag8 chain, the first allowed excitation in the presence of scalar-relativistic (SR) effects occurs at 1.32 eV with a large oscillator strength of f = 1.62. These values agree well with the theoretical results of Guidez and Aikens.63 Ignoring the relativistic effect in the TD formalism gave this excitation somewhat more intensity than the relativistic one, with energy and oscillator strength of 1.30 eV and 2.06, respectively. Although the scalar-relativistic term affected the ground-state orbital energy (see Figure S2), the energy of the first allowed excitation did not change significantly (less than 0.02 eV in Figure 4a). As discussed in the previous section, the relativistic term slightly quenches the oscillator strength of longitudinal plasmonic peaks relative to nonrelativistic by enhancing the dcharacter of the orbitals involved in the transition. It is similar to what was found by Yan and Gao for longitudinal peaks using nonrelativistic real-time propagation calculations along with scalar-relativistic pseudopotentials.64 It should be noted that the relativistic term , followed by d-contributions, have a substantial quenching influence on transverse plasmonic peaks above 4 eV,64 which is not discussed in our work. The optical absorption spectrum of Au8 is more complicated (Figure 4a), and the number of excitations in the low-energy region (up to 2 eV) is more than for Ag8. The longitudinal plasmonlike excitation in the Au8 system has a lower intensity and appears at a lower energy in comparison to the Ag8 chain. Figure 4a clearly shows that the Au8 spectrum has been severely affected by relativistic effects and turning this effect off enhances the intensity clearly and longitudinal peaks are recorded at a higher energy. In contrast, the spin−orbit coupling splits the absorption peaks and decreases their intensities (see Figure 4a). It is worth noting that the splitting of the d orbitals and the increment of ds hybridization due to spin−orbit effects increase the number of alterations inside the occupied and/or virtual

transition from the highest occupied molecular orbital (HOMO) to the lowest unoccupied molecular orbital (LUMO) (and in a more precise statement, this peak was seen to originate from a linear combination of ∑ → ∑ transitions) by elongation of chains, the number of electrons increases and also HOMO−LUMO energy gap decreases. As a result, the intensity of the peak increases and its position is shifted to red. Moreover, the significant increment of longitudinal mode intensity as a function of the chain length proves the collective oscillation nature of this excitation. Of course, as noted by Piccini et al., it is more proper to call such excitations at the linear system with the larger thickness (such as rods) plasmons.39 However, for gold systems, the number of high-energy broad interband transitions makes the distinction between collective and single excitations challenging (as can be seen in Figures 1b and 2b for small chains and rods). In the following sections, we consider the relativistic effects as an important key parameter to realize the mentioned optical challenges, followed by computation of the d-contribution in plasmonic transitions, and then examine the presence of silver atoms beside gold atoms in alloy systems as a chemical way to control the longitudinal plasmonic behavior. 3.1. Scalar-Relativistic and Spin−Orbit Effects. Gold and silver have nearly the same crystal parameters, and there is one electron each in their valence shells, which results in the similar chemical behavior; in contrast, they have different plasmonic properties. The longitudinal and transverse plasmon peaks of silver are sharp and distinctive, which, of course, are more complicated for gold. Relativistic effects (scalar and spin− orbit coupling) are undeniable factors in causing above discrepancies of group IB elements.21,60,61 Figure 3 displays this effect on the total and partial (-s and -d) density of states (DOSs) of Ag8 and Au8 (and also Figure S2 on electronic configurations and their energy levels). Moreover, Figure 4 displays the absorption spectra of Ag8,16,24 and Au8,16,24 chains at the nonrelativistic (NR), scalar-relativistic (SR), and spin−orbit (SO) levels of theory. It should be emphasized that in the SO level of theory the spin−orbit coupling is considered in addition to the SR part in a self-consistent fully relativistic ZORA formalism. First, the nonrelativistic behavior of the aforementioned systems was briefly studied to see the change due to different levels of relativistic effects. According to Figures 3 and S2, the nonrelativistic part of the electronic configuration of Ag and Au atoms (NR-Ag and NR-Au) and chains (NR-Ag8 and NR-Au8), the 6s electrons of Au and the 5s electrons of Ag strongly participate in the highest energy occupied band. Also, at this level of theory, the lowest energy unoccupied band, which contains the 6s character in Au (5s character in Ag), is found well above the energy of the 5d (4d) derived orbitals. More details can be found in Figure 3, which shows the total and partial density of states (DOSs) of Ag8 and Au8 chains with and without relativistic effects. A similar pattern is clearly seen for DOSs, and in both systems, the s- and d-DOSs do not penetrate each other. It is anticipated, at the nonrelativistic level, that Ag8 and Au8 (also for larger chains) will show the same behavior in electronic transitions. For instance, the electronic spectra of Ag8 and Au8 have the same shape. As depicted in Figure 4, the first allowed longitudinal excitation, which is attributed to the electronic structure, to s−s transition appears at nearly the same energy. Of course, the results can be prominently different by considering the relativistic terms. As can be found in Figures 3, 4, and S2, the DFT calculation based on the scalar-relativistic ZORA formalE


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to the longitudinal band explains the similar energy and intensity of this transition at different levels of theory. On the other hand, for a small gold chain, Au8, as can be anticipated, there is an obvious contribution of d-character at the scalar-relativistic (≈55%) and spin−orbit (≈72%) levels of theory. The detailed investigation shows the d → s nature of the transition for most of the low-energy excitations. Thus, the relativity causes the interband transitions to occur in the lower energy region, near to the intraband (s → s) plasmonic transitions. It should be noted that the high-energy occupied s level in gold is actually the hybridization of s and d; therefore, these levels are named ds instead of s. Thus, for a gold chain, the contribution of d → s transitions considerably quenches the ds → s excitations and makes broad peaks with low intensity. The correlation between the increasing size and d-character of plasmonic excitations for Aun is plotted in Figure 5. This figure

spinors. These relocations in the smaller system with two gold atoms, Au2, have been reported, such as relocation between the highest sσg and dπg after including the spin−orbit effect.23 Therefore, unlike the results for the Ag8 chain, the spin−orbit coupling has a large effect on the electronic excitations of neutral Au8 via quenching oscillator strength and increasing the interband excitations in the low-energy region. It should be noted that for closed-shell systems the effect of spin−orbit coupling in many of the molecular properties is not as large as for the optical properties.56 Relativistic effects and spin−orbit coupling have been considered for the large, Ag16,24 and Au16,24, systems. As can be seen in Figure 4b,c, increasing the length of chains consequently decreased the effect of relativistic and spin−orbit terms on the intensity of longitudinal peaks. This is a slightly different pattern compared to that in the work of Castro et al. (ref 24) for linear chains. To realize and justify these effects, the electronic structures and contribution of orbitals in the transition will be analyzed in the next section. 3.2. Effect of d-Contributions. In the previous section, the aforementioned quenching of oscillator strength in the presence of relativistic effects is introduced via the appearance of d orbitals directly in the optical excitations in addition to the screening of s-electrons by mixing with HOMOs. The same behavior has also been reported in nonlinear Agn (n = 2−8) clusters by Idrobo et al.21 The evaluation of d-contribution in the excitation can help us to describe quantitatively the effect of relativity on noble-metal clusters. To obtain the contribution of the d-electrons in the specific region of optical spectra, we estimated the percentage of the d-character in the transitions as follows21 Ω < Ec

%d =

∑n n

fn ∑ov |Fnov|2 |⟨d|ϕo⟩|2 Ω < Ec

∑n n

fn

Figure 5. Correlation between d-contribution (black line) and intensity (red line) for one of the prominent longitudinal excitations of the Au6−24 chain at scalar-relativistic (SR) level.

contains the intensity (oscillator strength) and %d-character for one of the prominent longitudinal excitations of Au6−24. Reducing the d-character and enhancing the oscillator strength with elongation of the chain have been shown quantitatively for plasmonic excitations in the first region of the spectrum, and the trend of increasing intensity with size has been explained. As an example, the involvement of d orbitals in the most prominent plasmonic excitations (see the most intense stick spectra in Figure 4) decreases. Their values are 41, 13, and 3% for the corresponding oscillator strengths of f = 0.32, 1.95, and 4.66 for Au8, Au16, and Au24 chains, respectively. The d-band mixing in longitudinal plasmon excitations is negligible in Au16 as well as in longer gold chains and has an insignificant effect on the damping of these excitations. Furthermore, the partial DOSs associated with the mentioned structures are provided in Figure S3. However, the separation between s- and d-DOSs is not clearly visible by increasing the length and led to the conclusion that considering the dcontribution of the all excitations below 2 eV (Table 1) and also one of the prominent longitudinal excitations (Figure 5) through eq 6 is a valuable index to realize the mentioned effect. A decrease in d-character by increasing the length can be attributed to the reduction of the appearance of relativistic effects in the optical behavior of elongated chains. This can be verified by comparing the energy variation of the highest d levels in Au8 and Au16 in the presence and absence of relativistic NR effects: ESR d − Ed . For Au8, eliminating the relativistic effects NR made a quite large change in energy of the d orbitals (ESR d − Ed SR NR ≈ 0.17 eV) in comparison to Au16 (Ed − Ed ≈ 0.02 eV). Thus, amplification of longitudinal plasmonic behavior in longer gold chains results from reduction of the aforementioned two effects:

× 100 (6)

2 where f n is the oscillator strength of excitation at energy Ωn, |Fov n | is the weight factor of occupied-virtual ov orbital pairs, and ⟨d| ϕo⟩ is the d-projection of the occupied orbitals |ϕo⟩. All transitions are weighted by their oscillator strengths f n and summed over all of the weighted transitions up to a certain energy cutoff Ec. The summation over oscillator strength in the denominator is the normalization factor. The d-character in the low-energy excitations (below 2 eV) for Ag8,16 and Au8,16 at different levels of theory is shown in Table 1.

Table 1. Percentage of the d-Character (d%) for Excitations below 2 eV (According to Equation 6) structure

NRa

SRb

SR + SOc

Ag8 Au8 Ag16 Au16

0.03 0.20 0.01 0.02

1.04 55.21 0.04 21.68

1.05 71.74 0.04 49.14

a

Nonrelativistic effects. bScalar-relativistic effects. cFully relativistic effects.

On the one hand, for a small silver chain, Ag8, including scalarrelativistic and spin−orbit coupling effects, the contribution of d-character in excitations (below 2 eV) is less than 1.1%, and it is actually zero for NR calculation. So, all allowed excitations (in this range) are s → s transitions. As noted above, the spinpolarized calculation does not affect the low-energy excitation and also the intensity. The negligible contribution of d-character F


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Figure 6. Computed spectra of (a) AuAgAu and (b) AgAuAg with 8-atomic chains and (c) AuAgAu and (d) AgAuAg with 16-atomic chains, recorded at the fully relativistic (SO + SR) level of theory (convoluted with a Gaussian function of FWHM = 0.2 eV).

damping of d-screening and the decrease of the size of relativistic effects. It can be anticipated that inserting the dopant elements such as silver atoms with highly stable d-shell (in comparison to highenergy relativistic d-shell of gold) can be an option for variation or maybe adjustment of plasmonic oscillations in gold base chains. The bimetallic nanoparticles with controllable core and shell dimensions have been studied extensively due to plasmonic excitation energies that can be tailored by changing the ratio of gold and silver29,65,66 3.3. Doping Gold Chains and Rods with Silver. As discussed above, relativity has the greatest influence on the variation of d-contribution in electronic excitations of Au systems and consequently on the nature of plasmonic oscillations or single-particle excitations. It is anticipated that reducing the d-character during excitations would strengthen the plasmonic behavior of Au chains. Therefore, it can be assumed that the presence of excess pseudofree electrons with snature lowers the d-contributions in the final longitudinal excitations. It seems that this aim would be achieved by inserting elements with free s-electrons and smaller relativistic effects than gold. As the silver atom fulfills this and has the most similar chemical properties to gold, we try to replace gold atoms with silver and study the effect on the plasmonic properties. Moreover, the similarity of crystalline characteristics for these two metals in bulk provides the possibility to study equal structures of different compositions without considering the influence of strong structural variations. Figure 6 shows the calculated spectra of the new alloyed chains with two types of dopings in the presence of scalar-relativistic and spin−orbit effects (SO + SR). The computed spectra of 8- and 16-atom alloys at the SR level of theory have been represented in Figure

S4. Two different types of alloying have been done by replacing gold atoms with silver in the center, AuAgAu, or on either side of the chain, AgAuAg. Also, then the plasmonic excitations of 8and 16-atomic chains and 68-atomic rods with different ratios of silver/gold atoms were investigated and compared to those of pure ones. The ratios of silver/gold atoms, NAg/NAu, in both types of alloys were chosen to be 1/3, 1/1, and 3/1. The high and low limits of longitudinal plasmonic intensity are related to the pure Ag and Au chains, respectively. As shown in Figures 6 and S4, the presence of silver atoms in gold chains (with any ratio) increases the longitudinal peaks, which agrees with the theoretical results that have been reported for bimetallic nanorods by Lozano et al.28 Of course, the variation in the number and position of Ag atoms in gold chains resulted in different enhancements. For the same ratio, NAg/NAu (i.e., 1/1), a slightly more enhancement of longitudinal peaks appears for AuAgAu (Figure 6a,c) in comparison to AgAuAg (Figure 6b,d). Detailed observations of Figures 6 and S4 show that the spin-polarization term (SR + SO) has more impact on the optical spectra of AgAuAg systems (Figure 6b) than on those of AuAgAu (Figure 6a). The less affected spectra of AuAgAu under SR + SO and almost the similar pattern of these spectra to the pure Ag chain in lowenergy longitudinal regions (i.e., up to about 2 eV) can be observed by comparing Figures 6 and S4. These observations sparked the idea of tailoring the chain composition and modifying the plasmonic property by quenching of spin−orbit and scalar-relativistic effects. The following discussions are more focused on this issue. The change of NAg/NAu ratio from 1/3 to 1/1 increases the plasmon peak intensity in both 8- and 16-atomic alloys. The intensity of the aforementioned peak decreases from NAg/NAu of G


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Table 2. d-Contribution (in %) of the Excitations below 2 eV at Scalar-Relativistic (SR) Level for Pure and Alloyed AunAgm with 8Atomic Chains and 16-Atomic Chains (in Parenthesis) structure Ag8 (Ag16) Au8 (Au16) AgAu6Ag (Ag2Au12Ag2) Ag2Au4Ag2 (Ag4Au8Ag4) Ag3Au2Ag3 (Ag6Au4Ag6) Au3Ag2Au3 (Au6Ag4Au6) Au2Ag4Au2 (Au4Ag8Au4) AuAg6Au (Au2Ag12Au2)

NAg/NAu

%d

1/3 1/1 3/1 1/3 1/1 3/1

1.04 (0.04) 55.20 (21.68) 34.80 (13.87) 29.62 (17.44) 25.98 (7.80) 41.67 (10.03) 7.10 (3.94) 4.09 (1.70)

%dAu

34.68 (13.68) 29.41 (17.30) 24.14 (0.12) 41.61 (10.00) 6.83 (3.92) 3.53 (1.66)

1/1 to 3/1 for 8-atomic alloys at both relativistic levels (especially, including spin−orbit; see Figure 6a,b). In contrast, it increases the aforesaid peak for the 16-atomic alloy by moving from 1/1 to 3/1 (see blue and green lines in Figures 6 and S4). A detailed consideration of d-contribution can explain the above observation. The percentages of s- and d-characters (of allowed excitations up to 2 eV) at the SR and SR + SO levels for (AuAgAu)8,16 and (AgAuAg)8,16 alloys are reported in Table 2. Typically, the AuAgAu chains with different NAg/NAu ratios show lower mixing of d orbitals in comparison to AgAuAg (and especially in the presence of spin−orbit effects). For instance, for 8-atomic AuAgAu alloy, the d-contributions for the ratios of NAg/NAu = 1/1 and = 3/1 are 7.1 and 4.1% and for AgAuAg are 29.6 and 26.0%, respectively (in Table 2). Although for 8-atomic AuAgAu alloy, the ratio of NAg/NAu = 1/3 (at SR level) shows the higher d-contribution, 41.7%, in comparison to 34.8% for the same ratio of AgAuAg alloy, which is interesting. The inclusion of the spin−orbit term changed the order, and the d% for NAg/ NAu = 1/3 of 8-atomic AuAgAu and AgAuAg are 52.1 and 62.1%, respectively. The above claim originates from an inspection of Table 2 for 16-atomic alloys. Generally, for the defined 16-atomic AuAgAu alloy, the d-contributions in plasmonic excitations are significantly less than for AgAuAg one. For example, in the longitudinal plasmonic region for Au2Ag12Au2 and Ag6Au4Ag6, the obtained d-contributions are 1.7 and 7.8%, respectively. A mere inspection of the population in Table 2 shows that scontribution in excitations of AuAgAu cases (1/1 and 3/1) comes mostly from both gold and silver in contrast to AgAuAg, where it considerably belongs to silver atoms. Thus, it implies that the longitudinal peak in AuAgAu cases, especially 1/1, has originated from a collective oscillation of s valence electrons of both Ag and Au atoms. In other words, the relativistic effects could not appear in the longitudinal (plasmonic) excitation of this kind of composition. It can be seen that the collective oscillations in 1/3 composition of both 8-atomic AuAgAu and AgAuAg extend almost on the Au part (with more %sAu). To generalize our discussions, we considered the d- and scontributions of pure and alloyed structures of 16-atomic chains in Table 2. Additionally, Figures 7 and 8 show the computed spectra of the equal ratio of NAg/NAu (1/1) for 8- 16- and 24atomic chain and 68-atomic rod systems at the SR level. It is known that the energy gap of a chain reduces by increasing its length. Thus, the longitudinal modes shifted to red by about 0.8 and 0.2 eV by elongation of the chain from 8- to 16-atomic and from 16- to 24-atomic, respectively. Furthermore, this peak in Au16, unlike a smaller chain, is separated from broad transverse higher energy peaks (in the region 1.5−2.5 eV) and exhibits the obvious enhancement in intensity, which proves the pseudocollective oscillation nature of this excitation. As can be seen in

%dAg

%s

%sAu

%sAg

0.12 (0.18) 0.22 (0.14) 1.85 (7.68) 0.06 (0.02) 0.27 (0.03) 0.52 (0.05)

90.6 (85.01) 41.09 (70.78) 78.59 (75.37) 64.22 (87.85) 68.07 (91.98) 54.40 (78.12) 84.84 (83.55) 89.18 (83.85)

58.85 (38.23) 20.11 (36.89) 13.14 (75.03) 49.96 (24.40) 42.50 (41.92) 37.20 (21.09)

19.74 (37.15) 44.12 (50.96) 54.94 (16.72) 4.44 (54.05) 42.34 (41.64) 51.99 (62.77)

Figure 7. Computed spectra for the same ratio (NAg/NAu = 1/1) of (a) 8-atomic, (b) 16-atomic, and (c) 24-atomic Agn, AuAgAu, AgAuAg, and Aun, structures at scalar-relativistic (SR) level convoluted with a Gaussian function of FWHM = 0.2 eV.

Figure 8. Electronic spectra of Ag68, Ag18Au32Ag18, Au18Ag32Au18, and Au68 rod structures (at LB94 scalar-relativistic (SR) level) convoluted with a Gaussian function of FWHM = 0.2 eV.

H


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Figure 9. (a) s-DOSs and (b) d-DOSs of different compositions of 68-atomic rods (Ag68, Ag18Au32Ag18, Au18Ag32Au18, and Au68) at scalar-relativistic (SR) level.

Figure 10. (a) Variation of the squared excitation energies of the first three intense peaks of Ag8 and Au8 upon scaling the electronic interaction. The curves with a large slope clarify the plasmonic character of the excitations. The curves with a small slope clarify the single-particle excitations. (b) Variation of the squared excitation energies of the longitudinal plasmon peaks of Ag8, Au2Ag4Au2, Ag2Au4Ag2, and Au8 chains upon scaling the electronic interaction. Excitation energies are depicted in units of the HOMO−LUMO gap (Δ, eV) for related systems.

appears with higher energy and intensity than that in Au68. Manipulation of the optical behavior of Aun systems by alloying with silver is the suggestion of the previous section. Figure 8 displays the absorption spectra of 68-atomic AgAuAg and AuAgAu rods. Like that of the linear chains, the alloying intensified the longitudinal plasmonic peak with respect to pure gold. The intensity of plasmonic excitations for the equal ratio of Ag/Au increased clearly. However, in detail, the rod form exhibits different behavior compared to linear chains. In 68-atomic alloys, the AgAuAg composition shows a more intense plasmonic peak relative to the AuAgAu one. This result has also been shown by Lozano et al. for 37-atomic alloyed rods.28 Comparing the partial DOS of the aforementioned structures (Figure 9) explains this plasmonic behavior. As can be seen in Figure 9, s-bands for two types of alloys are almost similar, although different alloying leads to different d-DOSs. The d-bands in AgAuAg in comparison to those in AuAgAu are localized further away from the Fermi level. This fact causes the d-band transitions to occur in energies far away from longitudinal plasmonic s-band transitions. 3.4. Scaling Coulomb Kernel. Finally, the analysis of plasmon excitations can also be carried out by means of scaling

Figure 7, the plasmonic excitation for Au16 appears at 0.74 eV (f = 3.0), whereas the corresponding value for Ag16 is about 0.81 eV (f = 4.5). It was mentioned in the previous section that the relativistic effects followed by d-screening and d-mixing are responsible for the low intensity of a Au chain. However, it should be noted that the d-contribution in the region of plasmonic peaks for Au16 (d% = 21.7) is significantly lower compared to that for Au8 (d% = 55.20). As shown before, Figure 5 emphasizes the reduction of d-contribution with elongation for plasmonic excitation of Aun chains. For different ratios of AuAgAu and AgAuAg alloys, the d-contributions in Table 2 are less than 10.0 and 17.4%, respectively. For the same portion of Ag/Au, in both types of alloys, the d% are not similar, Au4Ag8Au4 ≈ 3.9% and Ag4Au8Ag4 ≈ 17.4%, and Figure 7 shows the sharp intense plasmonic excitations for both types of alloys with respect to pure gold. As can be seen in Figure 2, for the chosen 32-, 50-, and 68atomic rods of silver and gold, different absorption patterns appear. Elongation of selected silver rods from 32- to 68-atomic enhanced the longitudinal peak in absorption spectra. Like those of the linear chains, the spectra of Aun rods have different broad peaks near to each other in contrast to Agn rods, especially for n = 68. Figure 2 shows that the plasmonic peak in the Ag68 rod I


Article

The Journal of Physical Chemistry C the coupling matrix K (in eq 2) suggested by Bernadotte et al.26 for identifying the plasmon transition. Furthermore, we try to clarify the effect of relativistic term and alloying on this kind of excitations. Herein, we revisit the 8-atomic chain considered in earlier sections and the λ scaling has been done for it as it is hard to distinguish the plasmonic modes from other single-electron transitions for such small chains. Figure 10a has plotted the variation of energies of the first three most intense transitions (in the less than 2 eV domain) at the nonrelativistic and scalarrelativistic levels of theory for pure Ag8 and Au8, independent of the scaling parameter. In this figure, the plasmonic nature of excitations is characterized by a strong sensitivity of the excitation energy with respect to λ (comparing the bold large points with small points for single-electron excitations) because the evolution from the noninteracting electronic medium (λ = 0) to the fully interacting medium (λ = 1) is mainly characterized by the electric restoring forces encoded in correlation kernels of eqs 2 and 3. The strong λ-dependency of plasmonic excitation is clear for nonrelativistic silver and gold and exhibited the nearly similar tendency with the of λ (see the curves that are displayed by the bold hollow circle and triangle for Ag8 and Au8, respectively). However, by turning the relativistic term on the dependency of gold plasmonic excitation from silver was getting started (the filled red triangular curve in comparison with the filled black circle one). The detailed consideration shows that after λ = 0.5, when the gold system fully interacts with the relativistic environment, the plasmonic behavior of related transition (the filled red triangular curve) becomes weak. Therefore, it seems that the slope of the plasmonic curve that comes from λ-scaling is another suitable index (beyond the d- and s-electron contributions) to perceive the relativistic effect on plasmonic properties. Moreover, such analysis was done for bimetallic 8-atomic chains with a similar ratio of Ag and Au in the presence of relativistic effects. Figure 10b shows the plasmonic λ-scaling curves for the aforementioned bimetallic chains. As can be seen, a different plasmonic character was demonstrated from the promising trend in the slope of excitation energy with respect to the scaling factor. However, for higher values of λ (>0.5), the characteristic features of their plasmonic excitation are more conspicuous, and the bimetallic chains showed the plasmonic behavior more than the gold chain. This result is from their higher slope in comparison with the gold one. Also, considering the curves in Figure 10b shows the strength of plasmonic in bimetallic chains with a different ordering. Thus, it can be said that the results obtained by the latter analysis by λ-scaling agree with the obtained results from the former analysis using the dcontribution index. For instance, the higher slope of λ-scaling analysis for Ag8 and Au2Ag4Au2 chains confirmed their less dcontribution and its effect on quenching the plasmonic excitations.

reduced the relativistic effect, decreased the s−d mixing, and increased s−s transitions, enhancing the plasmonic intensity. Therefore, to modify gold plasmonic behavior, changing the chain length and composition (by doping with silver) was found as the manipulating approach, which controls the appearance of scalar-relativistic and spin−orbit effects. We expect that this method of controlling the plasmonic properties paves the way for a variety of novel applications such as the development of spintronic devices, quantum optical technology, and electron transport in single molecules. In future studies, we will try to extend this perspective to investigate the effect of bond nature on the plasmonic properties.

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ASSOCIATED CONTENT

S Supporting Information *

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.jpcc.8b12045.

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Comparison of the spectra of optimized and nonoptimized 16-atomic chains for pure and alloyed systems and also their internuclear distances; schematic view of the molecular orbitals in the active space generated from the linear combination of the atomic (a) 4d and 5s for Ag8 (b) 5d and 6s for Au8 obtained at SR and SR + SO levels; total and partial density of states (DOSs) of 16- and 24atomic chains with and without considering the relativistic effects; computed spectra of 8- and 16-atomic AuAgAu and AgAuAg chains at SR ZORA-DFT level (PDF)

AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected], [email protected]. ORCID

Z. Jamshidi: 0000-0003-0976-1132 Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS This work was supported by the Chemistry and Chemical Engineering Research Center of Iran and Iran National Science Foundation. We acknowledge the developer group of Software for Chemistry & Materials (SCM) and computing resources of VU University of Amsterdam. The authors would like to acknowledge Erik van Lenthe for valuable help. Also, M.H.K. wishes to acknowledge Dr. S.H. Navabi for helpful discussions.

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REFERENCES

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