Plasmon excitations in mixed metallic nanoarrays

We study the plasmonic properties of arrays of atomic chains which comprise noble (Cu, Ag, and Au) and transition (Pd, Pt) metal atoms using time-dependent density-functional theory. We show that the response to the electromagnetic radiation is related to both physics, the geometry-dependent confinement of sp-valence electrons, and chemistry, the energy position of d-electrons in the different atomic species and the hybridization between d and sp electrons. As a result it is possible to tune the position of the surface plasmon resonance, split it to several peaks, and eventually achieve broadband absorption of radiation. Mixing the arrays with transition metals can strongly attenuate the plasmonic behaviour. We analyze the origin of these phenomena and show that they arise from rich interactions between single-particle electron-hole and collective electron excitations. The tunability of the plasmonic response of arrays of atomic chains, which can be realized on solid surfaces, opens wide possibilities for their applications. In the present study we obtain guidelines how the desired properties can be achieved.


Introduction
The optical properties of nanosystems are highly sensitive to their size, shape, and the chemical composition, and can dramatically differ from those of their bulk cousins. Over the past decade, a wide variety of plasmonic structures based on gold and silver have been fabricated to manipulate the light absorption at the nanometer scale for novel applications and basic research of physical phenomena. [1][2][3][4] These applications rely to some extent on the ability to tune the nanoparticle plasmon resonances, which has played a crucial role in stimulating the current interest in nanoplasmonics. In order to be able to intelligently tune the system properties, including the absorption spectrum, it is essential to understand its dependencies on the physical and chemical parameters.
An early theoretical study on the collective excitations in a few-atom sodium clusters was carried out using Time-Dependent Density Functional Theory (TD-DFT) by Kummel et al. 5 who showed that collective excitations exist even for very small clusters. More recent work addresses the mechanism of the collective excitations and field enhancement in higher dimension clusters using TD-DFT. [6][7][8] Ma et al. studied the sensitivity of plasmon resonance in Au nanoparticles and their dimers as a function of the particle size and the inter-particle distance. 9 Studies of the plasmon excitations in two-dimensional planar Na structures 10,11 or MoS 2 nanoflakes 12 reveal the importance of dimensionality in the formation and development of the plasmon peaks.
Scanning Tunneling Microscope (STM) experiments by Nilius et al. 13 demonstrated the development of 1D band structure in Au chains on NiAl(110) when the number of atoms in the chain exceeds 10. Inspired by this experimental finding, theoretical calculations have also predicted the presence of collective plasmon modes in a few-atom chains of several metallic elements: Na, 14 Ag, 15 and Au 16 (for over-review, see, e.g., Ref. 17 ). Experimental observation of such collective excitations requires the chains to be grown on a substrate that does not quench them rapidly. While there are theoretical indications that the NiAl(110) surface does not affect the electronic properties of the Au chains, 18 its metallic nature precludes a short lifetime for any plasmon excitation.
On the other hand, it is possible to grow Au chains and wires on semiconductor substrates such as Si(557), 19 Ge(001) 20 and quartz 21 which may be amenable for capturing the plasmon effects (especially if the substrate bandgap is much larger than the chain-plasmon and other excitation energies of interest). In a previous study of some of the present authors 22 the role of transition metal (TM) doping in the generation of plasmon modes in single Au chains was examined. It was found that doping leads to several new excitations in the absorption spectrum reflecting changes in the potential around the TM atom and collective effects of the "localized" (TM d-) and delocalized (s) electrons. Indeed, the mutual effects of both electronic subsystems may be nontrivial, in particular leading to a change of the spectral function of the localized electrons 23 or to local electronic resonances around the dopant atom, as was shown experimentally in the case of Pd-doped Au chains on NiAl(110). 24 This complexity opens the door to new opportunities for tuning the optical properties by combining noble and transition metal nanostructures.
Besides changing the composition of nanostructures, the optical spectra can be tuned by changing the size, shape, and geometry. In the present study, we consider an arrangement of small atomic chains and arrays. The optical properties of both pure and "mixed" arrays of homonuclear chains are calculated with TD-DFT. In order to understand optical properties, it is essential to analyze the nature of different features in the excitation spectra. This analysis is also the first step in systematic tuning of nanostructures in order to achieve desired properties such as strong resonance peaks or spreading of the absorption intensity over certain wavelength regions. In the analysis we use Transition Contribution Maps (TCMs) to identify how the individual Kohn-Sham (KS) transitions collectively contribute to the given photoabsorption peaks. 25,26 The excitation spectra and TCMs can be constructed from both Casida calculations 27 and the better scaling time-propagation TD-DFT. 28 Our results and analyses show a strong interplay between the physical (geometry-dependent electron confinement) and chemical (element-dependent electronic structure) effects parallel to that between the single-particle and collective excitations. For example, we show that in mixed arrays TM atom chains may quench the plasmon mode dwelling in noble metal chains.
The organization of the paper is as follows. In Section 2, we describe the detailed geometries of the nanoarrays and our computational methods employed in TD-DFT calculations and in their analysis. In Section 3.1, we analyze the plasmons formed in pure atomic arrays. While our results are consistent with previously published atomic systems, 11,17,29 we provide a detailed understanding via analysis with TCM. The TCMs visualize the quantum effects within the array, such as the dependence of the plasmon frequency on array size. Additionally the pure systems are the foundation on which to extend these methods to more challenging mixed systems which require quantum mechanical analysis. In Section 3.2, we show that the plasmon in mixed arrays can either be maintained or destructively quenched by the second chain.
2 Systems studied and methods for their modelling First, atomic wires comprising noble (Au, Ag, Cu) or transition (Pd, Pt, Ni, Fe, Rh) metal atoms with length varying from 2 to 19 atoms are constructed. Then, arrays consisting of up to eight chains are assembled to form pure or mixed arrays of homonuclear chains (see Figure 1b for a homonuclear array). To mimic the supported nanostructures, a square (simple cubic) lattice is assumed for planar (rod-like) arrays and the bond length is set to 2.89Å, as measured in experimentally realized Au chains on the NiAl(110) substrate. 13 The obtained spectra were insensitive to changes in the bond length within experimental error.
The electronic structures of the nanostructures are calculated within DFT using the solid-state modified GLLB-SC exchange-correlation potential 30  The optical response of the atomic wires and arrays is calculated using the LCAO time propagation (TP) TD-DFT code 36 with weak δ -pulse perturbation. 40 In this approach, the electron wave functions are evolved after an external electric (in dipole approximation) δ -pulse along the long wire axis. The time-dependent induced density provides the time-dependent dipole moment from which the dynamical polarizability and the ensuing frequency-dependent photoabsorption spectrum are determined. The photoabsorption spectrum with Gaussian broadening of σ = 0.07 eV is sufficiently obtained using a time step of 10 as for a total propagation time of 30 fs.
The induced density matrix between occupied and unoccupied KS states and the corresponding dipole matrix elements allows the photoabsorption spectrum to be decomposed into contributions from individual discrete electron-hole transitions with well-defined relative weights. 28  The great value of TCMs is that they visualize clearly whether there are low-energy electronhole transitions which collectively form a plasmonic excitation at ω (red spots below the ω probe line). 28,41,42 The excitation energy comprises the energies of individual transitions and the electron-electron interaction contribution described by TD-DFT. 43 Negative contributions may cause damping of the plasmonic excitation (blue spots, typically above the line ω = ε u − ε o ). Moreover, a change from negative (blue) to positive (red) in the contribution of a transition close to the probe line can result in plasmon splitting or fragmentation as will be demonstrated for mixed arrays. 28,44 Similar to the photoabsorption decomposition into different electron-hole contributions, the induced electron density can be decomposed into partial densities corresponding to the different electron-hole transition contributions. These real-space representations show their value in the following discussion. More theoretical and and practical details about these analysis tools can be found in a recent article. 28 Some of the nanowire systems were also calculated using Gaussian 03 45 with a B3PW91 hybrid functional 46-48 and a LanL2DZ basis set. 49 The main features of the absorption spectra were consistent with the LCAO-TP-TD-DFT calculations. 50 All the results presented and analyzed in this paper are obtained with the LCAO-TP-TD-DFT method.

Results and Discussion
In this Section, the optical responses of pure and mixed arrays are considered. The effects of the finite size and s(p)-d hybridization of electronic states between similar and different types of atoms on the valence electron structure and thereby on the plasmonic response will be discussed.
To date, there has been a lively discussion on whether these strong absorption modes in small nanosystems or molecules are plasmons or whether they are single-particle excitations. 7,29,42,43,51-54 For example, Piccini et al. 29 concluded based on the fact that only one KS transition contributes to the strong absorption peak of atomic Au chains that the excitation is singleparticle-like in nature. On the other hand, Bernadotte et al. 42 showed that some of the excitations in nanostructures are collective by studying the wave vector dependence of excitations and scaling the electron-electron Coulomb interaction. Although many excitations have single-particle nature, it was concluded that the strongest absorption peaks in atomic Au chains around 1 to 2 eV be-long to the collective, plasmonic excitations. After all, a pure single electron excitation depending on Coulomb strength would be a only an indication of self-interaction error. Such aspects can be visualized with TCMs for differentiating collective excitations from single-particle excitations in molecular systems, such as the atomic nanoarrays considered in this paper. Although features of plasmons from macroscopic materials are known to emerge in molecular systems, 53 our systems are far from the size of nanoparticles showing clear surface plasmon resonance with induced densities restricted on the particle surface. 43 The intense absorption modes from collective excitations in molecular systems could be described as molecular plasmons to distinguish them from conventional plasmons. However, for short we use the term plasmon interchangeably.

Molecular plasmons in pure arrays
Previous studies of single Au chains revealed that increasing the chain length above approximately 8 to 10 atoms generates a strong absorption mode. 16,22,29 Increasing the number of atoms further intensifies the modes and causes a slowly saturating redshift. In this work, we studied similar  29 The jump in the plasmon energy is largest from one to two Au chains. This is due to the increase of the valence electron confinement and the average density between the atom chains as seen in Figure 1a inset; classically the plasma frequency increases with the increasing electron density.
Slightly surprisingly the plasmon energy does not increase but slightly decreases between the planar arrays of three to four Au chains. Thereafter, the plasmon energy increases again, although rather moderately and monotonically, up to the array of 8 Au chains. The non-monotonic behavior for the smallest arrays, which does not arise classically, is due to the quantum mechanical confinement of the average electron density between the Au chains and the evolution of the nodal structure of occupied state valence electron wavefunctions perpendicular to the Au chains. "Subbands" cor- The weak electron spillout is expressed in the small work function of the triple chain array relative to the other sizes, and shown in the density of states with respect to the vacuum level in Figure   S2. Opening more subbands with edge nodes in larger arrays does not have as pronounced of an influence and the change to the work function is minor. Thus the plasmonic frequency increases modestly and monotonically up to arrays of eight chains.
In a similar manner, the electron spillout from the 2 x 2 x 14 nanowire is larger than that from the planar 4 x 14 array and the plasmonic frequency redshifts from the array to the nanowire. The transverse nodes in the nanowire are between two Au on each face so that the plasmonic frequency is even lower than that of the 3 x 14 array. However, the electron density corresponding to the lowest subband has a tendency to confine in the middle of four Au chains resulting in a plasmonic frequency larger that that for the 2 x 14 array.  Both the TCM of the strong absorption peak in the double chain array in Figure 3a and the TCM for the single chain in Figure S3 show one dominant electron-hole transition contribution. The plasmon resonance energy, ω, is significantly larger than the KS eigenvalue difference of the corresponding transition. This shift is a result of the strong influence of the electron-electron interaction taken into account in TD-DFT, and obtained as a gradual evolution in the scaling method. 42,54 In this respect, these resonances can be attributed as molecular plasmon excitations despite being comprised of only one dominant transition.  In the above TCMs for molecular plasmonic excitations, the main contributing KS transitions appear as red spots below the probe line corresponding to the plasmon resonance energy, ω = ε u − ε o . These transitions remain weakly illuminated when probing the higher energy resonances.
We present the TCM of the double chain Au array at an excitation peak of 2.80 eV as an example in Figure 3b. Higher harmonic molecular plasmons also contribute to this absorption peak and are circled. These electron-hole transitions occur between states where the difference in the number of longitudinal nodes is three. Similar higher order molecular plasmons have been identified in atomic chains by Bernadotte et al. using the λ -scaling method. 42 The TCM clearly shows that the photoabsorption peak has contributions from both plasmons below the probe line and singleparticle excitations on the probe line. The induced density in the inset of the figure is a product of the mixing and constructive interference between the dipole moments of each contribution. This mixing of plasmonic and single-particle transitions offers a possibility of generating hot electrons at the dense d-states. There are rather few states and possible transitions between them, and so hot electron generation is not expected to be an intense process in double chains. Arrays of several atomic chains would be more effective.

Molecular plasmons in mixed arrays
In this sub-section, we focus on mixed, or coupled, arrays comprising two chains each fourteen atoms long. By mixing chains of different elements, the position of the surface plasmon resonance can be tuned or the molecular plasmon fragmented into several peaks with broadband photoabsorption. For example, as seen in Figures 5(a,d) the plasmon peak of 14Ag-14Cu double chain is situated halfway between the plasmon peak of pure Ag and pure Cu double chains. However, when the Cu chain is joined with the Au chain, the plasmon peak is shifted toward that of the pure Au double chain ( Figures 5(b,e)), and when the Ag chain is joined with the Au chain, the plasmon peak is practically at that of the pure Au double chain (Figures 5(c,f)). In striking contrast, when a Au chain is joined with a TM (Pd, Pt, Ni, Fe, Rh) chain, the plasmon modes are highly suppressed and several peaks form as seen in the case of the 14Au-14Pd double chain in Figure 6.
The behavior of the molecular plasmon in the double chains comprised of two different noble metals, which are shown in Figure 5

Au-Au
Au-Pd Pd-Pd The effect of mixing arrays with homonuclear chains on the average electron density can be explained using the differences in the decay length of the valence electron density and the ionization potentials. The valence electron density decay length of Cu is smaller than Ag and Au, which are approximately equal. 55 The ionization potentials of Au is much larger than Cu and Ag, which are approximately the same. Combining these properties, we can expect that the valence electron density between the chains in the Cu-Ag mixed array is approximately equal to the average of the valence electron densities of the pure Cu double chain array and the pure Ag double chain array.
Therefore the plasma frequency for the mixed Cu-Ag double chain, which is proportional to the square root of the average electron density, is located at the midpoint between the plasmon of the Cu-Cu and Ag-Ag double chains. In the case of the Cu-Au double chain, the larger ionization potential of Au reduces the electron spillout from the Cu chain in comparison with the Cu-Cu double chain. However, since the decay length of the electron density of the Cu chain is smaller and the interchain distance is fixed, the resulting electron density between the Cu and Au chains does not rise very much from the average between between two Cu and two Au chains. Therefore the plasma frequency is shifted modestly from the midpoint between the Cu and Au double chain values. Finally, in the case of the Ag-Au double chain the decay length of the electron densities are similar for both chains. Then the larger ionization potential for Au can compensate the smaller valence electron contribution from the Ag chain by reducing the electron spillout from the Ag chain and as a result the plasma frequency for the Ag-Au double chain is nearly the same as for the Au-Au double chain.
We will now turn to the more detailed description based on TCMs which are shown in Figure 7 for Cu-Cu, Cu-Au, and Au-Au double chain arrays. In each case, the plasmon excitation has one dominant KS transition between delocalized sp-states (see the KS orbitals in Figure 3a) and the strong dipolar induced density is similar to the pure Au double chains. In this picture series, the shift of the plasmon energy is related to the energy of its constituent dominating KS transition.

Conclusions
We have analyzed the optical properties of multiple pure and mixed noble/TM metal planar nanoarrays of fixed length and varying width. The absorption peak moves to higher (visible light) energies when the width of the array increases, but the developing subband structure, due to the quantum confinement in the finite atomic arrays, produces a non-monotonic shift. The electron spillout is strongest in the smaller arrays, and clearly expressed in a redshift in the absorption mode between arrays with three and four Au chains, and their respective work functions.
The strong absorption mode is produced by low-energy transitions between sp-delocalized states with large dipole moment, and TCMs provide insight into their plasmonic nature. The contributions to the absorption mode fall below the ω probe line, which is present even in a single Au chain, indicating molecular-plasmonic character of the mode. As the width of the nanoarray is increased, the number of transitions originating from the newly opened subbands becomes more numerous. The increased number of transition contributions and developing nodal structure in atomic arrays is instructive in understanding the behavior of larger nanoparticles. Although nodal structure is more complex in larger nanoparticles, it is evident that the plasmons of larger nanoparticles have contributions from an increasing amount of KS transitions and some of which may fragment the plasmon. 28,44 The