Anomalously Soft and Stiff Modes of Transition-Metal Nanoparticles

Article pubs.acs.org/JPCC

Anomalously Soft and Stiff Modes of Transition-Metal Nanoparticles Marisol Alcántara Ortigoza,*,†,‡ Rolf Heid,*,‡ Klaus-Peter Bohnen,*,‡ and Talat S. Rahman*,† †

Department of Physics, University of Central Florida, Orlando, Florida 32816, United States Karlsruher Institut für Technologie, Institut für Festkörperphysik, D-76021 Karlsruhe, Germany

‡

ABSTRACT: We propose an explanation for the enhanced low- and highenergy tails of the vibrational density of states (VDOS) of nanoparticles (NPs) with respect to their bulk counterparts. Density functional theory calculations of the frequency and eigenvector of each mode allow us to identify radial breathing/multipolar and nonradial tidal/shear/torsional vibrations as the modes that populate such tails. These modes have long been obtained from elasticity theory and are thus analogous to the widely studied and observed pulsations in variable stars. The features particular to the VDOS of NPs are rationalized in terms of the charge density distribution around low-coordinated atoms, the quasi-radial geometric distribution of NPs, force constant variations, degree of symmetry of the nanoparticle, discreteness of the spectrum, and the confinement of the eigenmodes. Our results indicate that the high- and lowenergy tails of the VDOS may be a powerful tool to reveal information about the chemical composition and geometric structure of small NPs. In particular, the size of the conf inement gap at the low-frequency end of the VDOS and the extent by which the high-frequency end surpasses the bulk limit signal whether a NP is bulk-like or non-bulk-like and the extent to which it is disordered.

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investigations have found that elasticity can describe and predict radial modes and even dipolar and quadrupolar ones. Moreover, as far as radial modes are concerned, recent experiments of Juvé et al.11 using time-resolved spectroscopy further confirmed the validity of elasticity theory even for nanoparticles of less than 1 nm. The reason for which this approach remains valid at the nanoscale as well as at the astronomical scale is not coincidentaland we shall apply it to discuss the displacement pattern of our NPs. Briefly, the general continuous elasticity theory can take into account the essentials needed to reproduce the displacement patterns and frequencies of any finite body, including the effects of size, isotropy/ anisotropy, shape, and so forth, as long as reliable stress tensors and proper boundary conditions be provided.4 Thermodynamic properties and the interactions of phonons with other ’ons’ inherently depend on the energy spectrum under consideration. For this reason, novel features in the vibrational density of states (VDOS) of NPs have for more than four decades intrigued the material-science community.5,12−26 In particular, compared to their corresponding bulk counterpart, the VDOS of NPs has an enhanced population of both low- and high-frequency modes,13−15,19−25,27,28 the latter being even stiffer than the stiffest mode in the corresponding bulk materialsuper-bulk-band modes, even for NPs that are expected to have reached a bulk-like inner structure. For example, modes above the bulk band have been observed for Fe NPs as large as 10 nm,19 which counterintuitively display an

honons in bulk and surfaces have proved to be consequential for the explanation of several disparate phenomena in condensed-matter physics such as optoelectronics, superconductivity, heat transport, and diffusion.1,2 With dramatic developments in nanoscience leading to the introduction of an ever-increasing number of nanoparticles (NPs) of a variety of shapes, sizes, and compositions, two questions naturally arise: Are the proposed geometric structures3 for NPs dynamically stable? And what thermodynamic properties do they predict for the NPs? The answer to the latter question, in fact, was to a large extent available since 1907, on the basis of more than one century of elasticity theory and Einstein’s explanation to the breakdown of the Dulong-Petit law (see ref 4 and references therein). Yet, until 1972 Hoare and Pal5 obtained the 3N-6 discrete spectrum of NPs of polytetrahedral and D5h symmetry to investigate with two-body potentials how the non-Debye-like vibrational spectrum of NPs influences their thermodynamics properties. Soon after, Baltes and Hilf6 also obtained the discrete spectrum of a 2.2 nm spherical NP by using the solutions of the elastic equations of motion provided by Lamb4,7 (1881), in order to explain the low-temperature specific heat of Pb NPs. There is by now enough evidence that elasticity theory can explain key generalities about the vibrational and thermodynamic properties of NPs.4 This indicates that the physics describing vibrational modes remain practically unchanged with varying size, from nanoparticles to astronomical objects.4 Starting from the experiments of Del Fatti et al.,8 Palpant et al.,9 and Hodak et al.10 for Au and Ag nanoparticles with diameters between 4 and 120 nm, many optical spectroscopy © 2014 American Chemical Society

Received: October 7, 2013 Revised: April 11, 2014 Published: April 18, 2014 10335

dx.doi.org/10.1021/jp409956e | J. Phys. Chem. C 2014, 118, 10335−10347

The Journal of Physical Chemistry C

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icosahedra, Mark’s decahedra, or irregular) of Ag NPs (∼1.5−4 nm) and the presence of pure radial or torsional modes and whether these modes are localized at few atoms or are collective. Still, revealing the influence of the geometric structure on charge density demands electronic structure calculations. First-principles calculations, in particular, are essential in order to obtain with sufficient accuracy the charge density (and thereby the force constants and eigenvectors) of transition metals with strong covalent bonding, magnetic properties, or heterogeneous composition, because of the quantum nature of the balance between kinetic and potential energy that ultimately minimizes the total energy. An early ab initio study of core−shell Ag27Cu7 NP, for example, has already established that the charge density distribution between neighboring atoms of two species correlates with the length of the bond as long as the species type remains the same.34 For example, the geometry and composition of Ag27Cu7 NP render some (core−shell) Cu−Ag bonds stronger than any (inner) Cu−Cu bond (note that the reverse conclusion can be reached for the bonds in the L12 AgCu bulk alloys34 and for Cu adatoms adsorbed on a Ag surface35), notwithstanding that these Cu−Ag bonds are slightly longer than the Cu−Cu ones. In short, as we shall see, the charge density distribution, rather than the bond lengths, is suitable for tracking the origin of the high- and low-frequency modes, since it does signal the strength of the bonds and thus the magnitude of the force constants. In the course of preparation of this manuscript, Calvo reported the VDOS of AgxAu1−x NPs of 1−4 nm via manybody potentials derived from the tight-binding method.36 One of the focuses is the dependency of the VDOS on (1) Ag concentration, (2) segregation versus alloying, and (3) the swapping of species in core−shell clusters.36,37 Even more recently, Sauceda et al.38 also reported the VDOS of Ag, Au, and Pt NPs of 0.8−3.5 nm with icosahedral, decahedral, and FCC structures using many-body Gupta potentials. They characterize the dependence of the VDOS on the morphology of their NPs. Importantly, the results of Narvaez et al.,22 Calvo,36 and Sauceda et al.38 for NPs of 0.8−3.5 nm also support the validity of elasticity theory concerning the fundamental mode and the lowest frequency radial mode. We shall thus analyze them in the context of our work in our discussion. In this work, we present results of a first-principles investigation of the VDOS of a set of small NPs (∼1 nm) on the basis of charge density distribution, force constants, and vibrational eigenvectors. This set consists of 34- or 38-atom NPs whose geometry and composition has been taken from previous calculations34,39,40 or guided by experimental characterization of larger NPs.41−44 While these assumed geometries may not necessarily be those of the lowest-energy isomers, they allow us to focus on factors that influence the VDOS of dynamically stable NPs. Our chosen set of NPs, consisting of 3 bulk-like and 3 non-bulk-like geometries, allows us to compare weak and strong covalent bonding, zero and finite magnetic moment, and core−shell versus layered structures, making our conclusions applicable for a broad range of ∼1-nm-sized transition-metal NPs. In our quest, we have also used related thin films and fictitious NPs featuring bulk force constants as benchmarks for insights into the VDOS of NPs. Revealing the origin of the features in the VDOS for our set of NPs in addition provides the basis for understanding similar features that are also captured by larger NPs. Beyond that, we demonstrate how the VDOS of these small NPs can serve as

average lattice parameter slightly, but measurably, larger than that of bulk Fe. Very early molecular dynamics calculations by Dickey and Paskin for bulk-like NPs using a Lennard-Jones potential showed that the larger the surface-to-volume ratio, the more the lowest-frequency VDOS is enhanced.13 Soon after, the calculations of Hoare and Pal5 yielded that small icosahedral and tetrahedral NPs display characteristically high-frequencies and they attributed it to a breathing mode. Much later, on the basis of semiempirical-potential calculations, Narvaez et al.22 found that the structure plays an important role in the features of the VDOS obtained for Ag NPs of ∼1.5−4 nm. Specifically, NPs with icosahedral geometry or with multigrains were found to have more super-bulk-band modes and a larger VDOS at the low frequency end than bulk-like NPs. This correlation is quite general and hence important. Nevertheless, the structure of the NP alone cannot account for the enhanced VDOS at low and high frequencies. Later on, semiempirical-potential calculations of the VDOS for bulk-like single-grain20,21 and multigrain27 Ag NPs of 2−4 nm found that the high-frequency end of their VDOS is also enhanced with respect to that of the bulk. Regarding the localization of the modes, Dickey and Paskin13 reported that the low-frequency modes derive from the vibrations of surface and edge atoms. Then, by analyzing the eigenmodes obtained by Lamb for a homogeneous isotropic sphere,7 Tamura et al.29 obtained similar results, and also found that higher-frequency modes corresponded to inner atoms. Hoare and Pal5 showed that the low-frequency modes were associated with the increased softness of the potential. Later on, Kara and Rahman20 also ascribed the low-frequency modes to the loss of neighbors and softening of force constants and associated the high-frequency modes with the shortening of nearest neighbor distances and stiffening of force constants. As informative as the latter calculations were, their account leads to the paradox that while atoms at the surface of a NP have reduced coordination and thus the largest shrinkage, they are not the ones contributing the most to the highest frequency modes. The solution is that neither reduced coordination nor bond-length shrinkage map trivially into the force-constant matrix. Therefore, analyzing the eigenvector of each vibrational mode becomes critical. In this regard, we must also mention that elasticity theory has been applied to investigate the surface relaxation in nanoparticles by Tamura et al.29 Specifically, contraction has been modeled by a denser surface but the calculation yielded an overall softening. It has been suggested that, while the result is correct, the model is not, since it is the electron density via the force constants that must be modeled, and not the atom density.4 Finally, for 34-atom Ag−Cu NPs displaying a variety of shapes, Yildirim et al.25 noted a correlation between increasing number of Cu atoms and enhancement of the VDOS at high frequencies which was assigned to shorter Cu bonds, but as the harmonic oscillator model indicates, the mass disparity between Cu and Ag may strongly influence the particulars of the VDOS. Unraveling the origin of the high- and low-frequency tails in the VDOS thus requires analysis of (a) the displacement patterns associated with each mode, (b) the role of the relative mass of the constituents for heterogeneous NPs, and (c) the electronic charge density distribution. Information about (a) and (b) above can be obtained from elasticity theory7,29−32 and lattice-dynamics calculations, which may be based on semiempirical potentials.33 For example, Narvaez et al.22 identified the relationship between the structure (crystalline, twined 10336



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symmetric position forming three-atom short-bonded Pt chains. As a result, some nearest neighbor Pt−Pt bonds at the surface are as short as 2.65 Å, whereas others are basically broken since they become as long as 3.8 Å (the calculated bond length in bulk Pt is 2.80 Å). Dynamical Stability. We have obtained the eigenfrequencies and eigenvectors of the 3N-6 eigenmodes of the six NPs, containing either 34 or 38 atoms. Note that the six modes that are not considered correspond to the zero-frequency and rigid translations and rotations. We find that, although the structure of our NPs is not necessarily that of the lowest energy isomers, they are dynamically stable, except for Pt18Fe20. The latter was found to be unstable against a vibrational mode that involves its pure-Fe (001) facets, a result that may speak to trends in the morphology of L10 PtFe NPs.45 We shall come back to this unstable NP in the discussion. In the rest of this section we report results only for the five NPs that are dynamically stable. Vibrational Density of States. The calculated vibrational spectrum of our NPs can be appreciated via the VDOS shown in Figures 1 and 2. As expected for nm-sized systems,6,7,12,30 we find at the low-frequency end a few-meV gap followed by welldefined discrete peaks (both observable even after the Gaussian smoothing). The low cutoff frequency, the fundamental frequency, is marked with an asterisk (*) in Table 2. As highlighted by the vertical lines in Figures 1 and 2, each NP exhibits modes whose frequency is above the vibrational band limit of the related bulk systems, the super-bulk-band modes. Our Pt−Fe NPs show a correlation between structural order (maximum fraction of equivalent atoms) and the width of the VDOS. Namely, NPs with non-bulk-like ordering (pIh7) have broader spectrum than those with bulk-like ordering (L10). The former have more super-bulk-band modes, some of which are in fact stiffer than the stiffest super-bulk-band mode of NPs with bulk-like ordering (fcc and L10), as shown in Table 2 and Figures 1 and 2. They also have more modes at the lowfrequency end, e.g., below 5 meV, than NPs with bulk-like ordering (L10), some of which are softer than the softest mode

a descriptor of their geometric structure and composition information not easily uncovered otherwise.

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RESULTS Geometric Structure. Table 1 displays the top-view of the relaxed structure of the specific NPs of interest here: Ag27Cu7, Table 1. Atomistic Structure (top-view), Composition, and Structural Properties of the Particles Analyzed in This Work

Cu38, Pt27Fe7, Pt17Fe17, Pt20Fe18, and Pt18Fe20. The side view of each (except Pt18Fe20) is also displayed in Table 2. We note that after structural relaxation the initially assumed symmetry of all of them is preserved, except for that of Pt17Fe17, which deviates from (17,17)pIh7 toward a structure of lower symmetry: The Pt atoms at its surface shift from the fivefold

Table 2. Atomistic Structure (side-view), Frequency ν (in meV) and Displacement-Pattern Type of the Modes Populating the Low- and High-Frequency Ends of the VDOS of Each NPa

The types are ”tidal” (T), ”shear” (Sh), or a combination of these when the vibration is localized at outer atoms and tangential (see text). T also means that the vibrations create one or more ”mass density” poles at the surface alternating expansion and contraction, whereas Sh means merely that the vibrations perturb bonds of outer atoms in a direction perpendicular to the bonds, including torsional displacements. ”Radial” (R) implies that the radial bonds in the NP are perturbed in the mode. They can be either the ”breathing” (B) of inner atoms or a nonradially symmetric ”multipolar” (M) vibration involving inner atoms. The parentheses in front of the frequency indicates the number of modes represented by the corresponding frequency (or frequency range) and displacement pattern (type). The star (*) indicates the low-end cutoff frequency (see text).

a

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(20 and 23 meV, respectively). Not surprisingly, most modes whose frequency is below the bulk-band limit of the heavy species correspond to vibrations in which the heavy atoms play a major role and, vice versa, most modes whose frequency is above the bulk-band limit of the heavy species correspond to vibrations in which the light atoms play a major role. Another interesting feature in Figures 1 and 2 is the presence of a dip in the VDOS for the NPs with core−shell structure, Ag27Cu7 and Pt27Fe7, each of which has the lighter atoms in the core. The dip is around the bulk cutoff frequency of the heavier atoms and is particularly pronounced in Pt27Fe7. Vibrational Displacement Patterns. Describing the vibrational displacement pattern of the 3N-6 modes of each of these five NPs is a demanding task. This is not only because the number of modes is large, but also because the displacement patterns themselves are rather intricate. Only a few modes could be classified in a relatively simple way, such as the ”inner-atom radial breading mode” or ”outer-atom shear rotational mode”. For that reason, we provide animated simulations of the eigenvectors of each mode elsewhere.46 One remaining drawback is that the complexity of the modes of NPs is such that one or two views (top and side, for instance) of the NPs vibrating in a particular mode may not be enough to visualize the displacement pattern thoroughly. A mode may seem ”tidal” from one perspective, shear−torsional from another, and be unintelligible from yet another perspective. Therefore, in addition to the animations, we provide in Table 2 a simplified characterization of the highest and lowest modes, which correspond to the super-bulk-band modes and the first low-frequency peak in the VDOS, respectively, both of which are the subject of these work. In our classification, ”tidal” refers to ”mass density variations” produced by an atomic vibration that is not only multipolar (e.g., monopolar, dipolar, quadrupolar, etc.) but also localized at the outer atoms, whereas the classification ”multipolar”, although similar to ”tidal”, is reserved for multipolar vibrations involving inner atoms as well. The classification ”shear”, in turn, means that the vibrations are tangential to the surface, perturb outer atoms in a direction perpendicular to their bonds, but are not multipolar. A shear mode may be either unidirectional or torsional. In our classification, ”radial” vibrations are those which perturb the bonds in a radial direction of the NP but do not necessarily have radial symmetry. For example, while radial-breathing modes do have a radial symmetry, radial-multipolar modes do not. Such details, however, will not be discussed for each mode but are specified for those in Table 2. Analysis of the displacement patterns show that the highestfrequency modes of each NP involve subshell atoms, the stiffest always being a radial breathing mode of the inner atoms against the static outer atoms (see Figure 3a,b). Low-frequency modes, in turn, are tangential, mainly involving shell atoms, the lowest ones being shear (either unidirectional or torsional) and tidal (either dipolar or quadrupolar, see, e.g., Figure 3c,d). Yet, in all NPs the fundamental mode is primarily shear-torsional (Sh) but with a tidal (T) component, particulary for Ag27Cu7, whereas the lowest frequency of Pt27Fe7 is purely torsional (see ref 46). Electronic Charge Density Distribution. Visualization of the charge density distribution of Pt17Fe17 and Pt27Fe7 NPs (Figures 4 and 5) show, just as for Ag27Cu7,34 a conspicuous charge density accumulation between shell and subshell atoms in the radial direction that is significantly larger than the charge density between tangential bonds and even larger than that

Figure 1. VDOS of (a) Cu38 NP and (b) Ag27Cu7 NP.

Figure 2. VDOS of (a) Pt20Fe18 NP and (b) Pt27Fe7 NP and Pt17Fe17 NP.

of the bulk-like NP. On the contrary, we do not find a correlation between the low/high cutoff frequencies and composition, i.e., in terms of the mass ratio. For example, in the range from 0 to 5 meV, Pt17Fe17 has more soft modes than similarly structured Pt27Fe7 despite the higher content of Pt in the latter (see Table 2). It also has more soft modes than the bulk-like Pt20Fe18 with not too different composition. Moreover, of the three Pt−Fe NPs considered here, the one with the least amount of Fe, Pt27Fe7, displays the highest frequency (44.7 meV). For our Ag−Cu and Pt−Fe NPs, on the other hand, the stoichiometry happens to relate to the VDOS and can be estimated from the integrated normalized VDOS. Specifically, one can see from Figures 1b and 2a,b that the fraction of the heavy species (Pt or Ag) is approximately proportional to the integral of the normalized VDOS up to their bulk-band limit 10338



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DISCUSSION

Features of the Vibrational Density of States. In Figures 1 and 2, our calculated VDOS for the set of NPs considered here show features not often highlighted in atomistic simulation and extremely difficult to assess47 by some spectroscopic measurements: a few-meV gap followed by well-defined discrete peaks. Still, the importance of the discreteness of the spectrum in relation to the limits of the Debye model (applicable only for macroscopic object) was first discussed by Max Planck.48 The VDOS of the NPs scales neither quadratically (as in single-element bulk systems) nor linearly (as previous calculations6,20,25 have shown for NPs) with the frequency. This can also be seen in the VDOS of AgxAu1−x NPs reported by Calvo36,37 and the Au, Pt, and Ag NPs reported by Sauceda,38 also showing that the VDOS of NPs smaller than 2.5 nm does not display the quadratic dependence assumed in Debye’s model or the linear one. Such attributes are to be expected for nm-sized systems because of the conspicuous vibrational confinement for small NPs. The VDOS of our NPs also exhibit modes above the respective bulk bands, the super-bulk-band modes, in agreement with previous calculations.5,21,22,37 It is worth mentioning nevertheless that many-body Gupta potentials do not yield modes above the experimental and ab initio bulk band (∼195 and 153 cm−1 for Pt and Au, respectively49) for bulk-like NPs except for Pt38 (207 cm−1).38 Still, first-principles calculations of the VDOS of Pt38 yield a considerably higher maximum frequency (228 cm−1).50 One could argue that comparison with the bulk limit using the same Gupta potential is in order. Unfortunately, even when the vibrations of the bulk are available from the same many-body Gupta potentials, the outlook is unclear. Namely, on one hand Gupta-potential calculations indicate that the highest-frequency mode of bulklike Au NPs is ∼27% higher than the corresponding bulk limit (110 cm−1) even for NPs as large as 4 nm. Surprisingly, this highest frequency does not increase for smaller NPs down to 1 nm (only within the error bar of the calculation). On the other hand, in this Gupta-potential calculation the highest frequency of each bulk-like NP is lower than that of the bulk band

Figure 3. Schematic representation of (a) the highest-frequency mode (radial) for the pIh7 core−shell structures (some shell atoms have been removed to reveal the interior of the NP); (b) the highestfrequency mode (radial) for the fcc Cu38 NP (some shell atoms have been removed to reveal the interior of the NP); (c) a low-frequency (tidal) mode for the pIh7 core−shell structures; (d) a low-frequency (tidal) mode for the fcc Cu38 NP. In each figure, the atoms of the NPs are at a maximum of the distortion and the arrows schematize the restoring forces.

among inner atoms. A similar trend is observed in the charge density distribution of our bulk-like nanostructures, particularly for Pt20Fe18 (Figure 6). For Cu38, the charge accumulation around the bonds between surface and ”subsurface” atoms is not as conspicuous as in core−shell NPs, but can still be appreciated clearly in Figure 7b,d. We note that in this NP the charge accumulation also occurs tangentially among surface atoms. These surface atoms thus participate in eigenmodes of Cu38 whose frequency is as high as 29.7 meV, slightly larger than the cutoff frequency of bulk Cu (see ref 46).

Figure 4. Three-dimensional (a,b) and two-dimensional (c,d) density profiles of Pt17Fe17 depicting the bonds with largest charge density. The upper panel of c and d display the 2D profile together with the structure, as in Figure 7. 10339



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Figure 5. Two-dimensional density profiles of Pt27Fe7 showing (a,b) the bonds formed by Fe(core) (red balls) and Pt(shell) (gray balls) atoms and (c) the bonds formed between Pt(shell) atoms. The upper panels display the 2D profile together with the structure, as in Figure 7.

Figure 6. Three-dimensional (a) and two-dimensional (b,c) density profiles of Pt20Fe18, as explained in Figure 4.

obtained in experiment and via ab initio calculations49 and than the highest frequency of a 1 nm Au NP obtained from ab initio calculations (162 cm−1).84 Our first-principles calculations show that the size of the confinement gap, and the extent to which the high end of the VDOS surpasses the bulk limit, depends on structural details. For example, there is substantial difference between the VDOS of bulk-like ordered NPs and that of our core−shell (pIh7) NPs. The former do not display frequencies much beyond the bulk-band limit: the maximum frequency is ∼20% higher than the corresponding bulk limit, whereas core−shell NPs display modes whose frequencies are up to ∼50% higher, which is in agreement with the findings of Narvaez et al.22 Thus, assuming that the VDOS of even larger bulk-like NPs (2−4 nm) would diverge less from that of the bulk, we anticipate that bulk-like ordering in currently synthesizable NPs can be identified by a VDOS whose high-energy tail surpasses the bulk-limit modestly. Regarding the low-frequency end, the more symmetric the NPs (with fewer inequivalent atoms) the larger the confinement gap. Our results are also in agreement with the calculations by Narvaez et al.22 for Ag bulk-like NPs of ∼2 nm.

They found that while the gap is particulary low (∼0.5 meV) for NPs with grain boundaries and surface disorder, it may be as large as 2 meV for bulk-like structures (note that the present bulk-like NPs display larger gaps (∼4 meV) because they are smaller, ∼1 nm). To understand this behavior, one first needs to realize that the confinement gap exists regardless of the structure and size, since its origin is only related to finiteness. Second, one should notice that disorder, non-bulk-like coordination, and grain boundaries imply elongation/shrinkage of the bonds (sometimes even bond breaking) and/or atoms of marked low coordination. Since these structural characteristics are more prevalent and significant in grain boundaries and disordered or non-bulk-like NPs than in bulk-like ones, the former necessarily display more bonds that are broken or bonds weaker than those found in bulk-like NPs. These are the source of modes of pronouncedly low frequency, which ultimately render gaps smaller than those found in bulk-like NPs. EAM calculations22 and many-body Gupta-potential calculations38 have reported that the period of the lowest-frequency mode of metal NPs shows a slight shape dependence for NPs of 1−4 nm. However, regardless of the shape, the period of this 10340



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consistently display higher frequencies than bulk-like NPs and a larger amount of low-frequency modes, e.g., below 5 meV, in agreement with our results. Sauceda et al.38 also found that bulk-like NPs display a narrower VDOS than non-bulk-like NPs. Given the stark structural differences, the question arises whether there are grounds to expect a priori that the shape of the phonon DOS of small NPs would relate to that of the corresponding bulk material. To answer this question, we analyze in the following the localization and the displacement patterns of all modes of each NP,46 and their relation to the charge density distribution and, in the case of heterogeneous NPs, the masses involved. On the Localization of the High- and Low-Frequency Modes. We find that the highest frequency of each NP is a super-bulk-band mode that involves subshell atoms (Figure 3a,b). Low-frequency modes, in turn, involve mainly the shell atoms (Figure 3c,d). We note that this mode localization could, in principle, be considered similar to that found for larger single-grain single-element NPs. 21 However, there are exceptions. For example, the highest frequency of Pt20Fe18 and the rest of the super-bulk-band mode of Cu38 involve not only inner atoms but also shell atoms. A similar feature can be seen in the displacement patterns reported by Sauceda et al.38 for the highest-frequency mode of Au NPs of ∼2 nm. Furthermore, one of the lowest-energy modes (6.8 meV) of Cu38 largely involves inner atoms. Moreover, while contraction/elongation of the inner Cu atoms gives rise to the stiffest mode of Ag27Cu7 (32 meV), a mode having exactly the same displacement pattern, but in which some Ag atoms also participate, results in a frequency of only 13 meV. Localization of the Highest Frequency Mode: Charge Density Distribution. The most obvious explanation for the localization of the high-energy modes can be obtained from the charge density distribution. Core−shell structuresAg27Cu7, Pt17Fe17, and Pt27Fe7show a charge density accumulation between shell and subshell atoms in the radial direction that is larger than that among inner atoms (Figures 4, 5, and Figure 10 of ref 34). This feature seems to be irrespective of whether the bonds between Cu atoms in the bulk are typically stronger than the Ag−Cu bonds in the (L12) Ag3Cu or AgCu3 bulk alloys34 or, conversely, whether the bonds between Fe atoms in the bulk are weaker than the Pt−Fe bonds in the (L10) PtFe alloy. The same trend is observed for the charge distribution in our bulk-like nanostructures (for Cu38, the charge accumulation around the bonds between surface and ”subsurface” atoms is not as conspicuous as in core−shell NPs since the charge accumulation also occurs among surface atoms). The existence of this high charge density in the shell−subshell region is expected to be present in covalently bonded NPs in general, since it arises because of the low coordination of periphery atoms. The underlying reason is that the charge corresponding to the ”dangling” bonds of the outer atoms redistributes to strengthen the remaining bonds (such as between periphery atoms and their NN in the radial direction). One can consider, for example, that while the cohesive energy of bulk Cumainly gained through its bonding with 12 NNis ∼3.5 eV, the binding energy of a Cu adatom on Cu(111)mainly gained through its bonding with only 3 NNis ∼2.7 eV. The relatively large charge density between shell and subshell atoms does enhance the force constants in the expected fashion. For instance, in the case of Ag27Cu7, the bond-projected force constants between inner Cu−Cu bonds are at most 25% larger

Figure 7. Two-dimensional (2D) density profiles of Cu38 depicting the bonds with largest charge density. The upper panels display the 2D profile together with the structure as a guide for the location of the atoms involved.

mode holds for the most part to the linear size-dependence obtained from continuous elastic theory. Calvo37 did not find noticeable lower fundamental frequencies in non-bulk-like NPs. Instead, for a fixed size, 3 nm NPs (N = 923), Calvo found a strong dependence on the nanoparticle shape (cuboctahedral and icosahedral).37 For these already sizable NPs (of ∼3.0 nm), however, the outer shape is not expected to affect the main features of the VDOS above certain frequency (∼7 meV). The size sharpens the details of the shape of the VDOS. For example, if one analyzes Figure 3 of ref 19, Figure 1a of ref 21, and Figure 3 of ref 38regardless of the shape, the main features of the VDOS converge even for NPs composed of 500 atoms (∼2.5 nm). So, the dependence is in fact on the internal structure: The VDOS of FCC bulk-like NPs strongly differs from that of non-bulk-like ones. In fact, by analyzing their VDOS plots, one can also see that non-bulk-like NPs 10341



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components is large enough and/or the two species are synthesized to form core−shell NPs in which the lighter atoms are in the core. First of all, we propose to pay attention not only to the value of the highest frequency, but also to the density of states above the bulk limit. Note that although Ag27Cu7, with only 20% Cu, has stiffer frequencies than Cu38, the VDOS decays quickly at lower energies. This is an indication of the low Cu content. The same applies for Pt27Fe7: although it has the highest top frequency among all other PtFe NPs, its VDOS is lower than that of Pt18Fe20 and Pt27Fe7 in the range 18−31 meV. Therefore, in general, a low VDOS at frequencies beyond the bulk cutoff frequency of the heavier species is indicative of a low fraction of the lighter one. In other words, it is not the top f requency but the integrated VDOS above the bulk limit of the heavy species that in small and segregated NPs can provide a measure for the content of the light species. Alternatively, the fraction of the heavy species (Pt or Ag) is approximately given by the integral of the normalized VDOS up to their corresponding bulk-band limit. The latter approximation may be useful for characterizing the composition of NPs because it is not coincidental, though it cannot be generalized. It works because (1) the integral of the VDOS is proportional to the number of particles N; (2) the mass ratio is large enough so that the vibrational spectra of their respective bulk phase do not overlap much; (3) even when the spectra overlap considerably, as in the case of bulk Ag and Cu, if the two species are segregated in a core−shell NP (as in Ag27Cu7), only a few of the low frequency modes involve the inner species, relatively light atoms, because these are confined in a small volume. In fact, we infer that the marked dips at intermediate frequencies for Ag27Cu7 and Pt27Fe7 indicate features related to their segregated core−shell structure since no conspicuous dip or gap appears for the bulk-like Pt20Fe18 or the (partially disordered) pIh7 Pt17Fe17. One may conclude that small and segregated NPs, whose lighter atoms are confined in the core and constitute a small fraction of the total number of atoms (whereas the heavier atoms constitute the shell, are more numerous and often low coordinated), do not support many modes in which either (1) both species participate, (2) light atoms sustain low-frequency modes  because they are few and confined  or (3) heavy atoms sustain high-frequency modes. However, in cases such as that of AuAg NPs,37 the separation of the contribution from each species to the VDOS is not very clearly seen in core−shell NPs with small content of Ag because, although the latter atom is lighter than Au, Ag−Ag bonds are significantly weaker than Au−Au bonds. As a result, for example, both metals have approximately the same bulkband limit (∼155 meV) and thus the contribution of each species to the VDOS overlaps almost totally. Allowed Eigenvectors: The Ultimate Factor Determining the Spectrum. Charge-density and mass-ratio arguments explaining the VDOS are still contingent on the actual eigenvectors involved. Up to this point, one can say that the charge-density and mass-ratio analysis (1) pinpoint the source of super-bulk-band modes and (2) give a rationale for the space localization of high- and low-frequency modes. However, this line of reasoning will hold ground and thus be useful only if there exist eigenvectors that embody the above underlying causes. The next questions are thus whether the eigenvectors conform with the ideas exposed so far and whether one can draw some basic generalities about the eigenvectors accounting for the most conspicuous features of the VDOS, i.e., the lowand high-energy tails. In the rest of this section, we describe the

than those of bulk Cu, whereas those between shell(Ag)− subshell(Cu) atoms are up to 100% larger than those of bulk Cu. Note that this is in striking contrast to the trend in other Ag−Cu systems, in which the Ag−Cu bonds are weaker than Cu−Cu bonds.34,35 The charge density analysis above provides a rationale for the existence of super-bulk-band modes. There arise, however, complexities and subtleties when analyzing the VDOS of heterogeneous NPs. The combination of ”light” and ”heavy” atoms in a NP can either conceal the bond stiffening caused by the shell−subshell charge-density enhancement or combine with it to augment the blue-shift of the VDOS. An example of the first scenario is the Pt20Fe18 NP. Specifically, although the shell−subshell charge density accumulation is fairly prominent among Pt atoms in this NP, their relatively large mass does not allow the enhanced Pt−Pt force constants to be reflected in the high-energy tail of its VDOS (Figure 2a). In fact, its highest frequency (only 33 meV) involves mainly Fe atoms. On the Correlation between Composition and the VDOS. Heterogeneous NPs bring forth the question of whether and how their VDOS signals their composition. The harmonic oscillator model alone, for example, implies that for NPs composed of elements with contrastingly different masses, as those analyzed by Yildirim et al.,25 the modes involving lighter atoms populate the high-energy tail of the VDOS (regardless of alloying effects, the smallness of a nanostructure, typical bond-lengths among the species, etc.). It is thus obvious that increasing the content of lighter atoms must blue-shift the high-energy tail of the VDOS or at least enhance it. We find nevertheless that the opposite behavior may also take place. For instance, increasing the content of Fe from 26% to 50% (Pt27Fe7 → Pt17Fe17) contributes to the VDOS in the middle energy range (∼20 meV) but does not shift the high-frequency end to higher frequencies or enhance significantly the VDOS at the high-energy range (Figure 2b). It is also noteworthy that lighter-atom content and/or bond lengths cannot explain why of all four PtFe NPs, the one exhibiting the stiffest mode of all is the one with the lowest Fe content, Pt27Fe7. One could argue that the non-bulk-like pIh7 structure leads to this result when considering the L10-type NPs. Still, that could not explain why increasing the Fe content from 26% to 50% while keeping the same structure and symmetry (Pt17Fe17) lowers the topmost mode from 45 to 43 meV. An example in which a larger content of lighter atoms does render a higher uppermost frequency is that of our L10-type (layered structure) PtFe NPs. Namely, Pt18Fe20 (the unstable NP, see ref 45) has a top frequency of 36 meV, while Pt20Fe18 has a top frequency of 33 meV. Although one could have guessed this result qualitatively on the grounds that (1) Fe is much lighter than Pt and/or (2) Fe makes shorter bonds than Pt, the way the Fe content works in this case is not necessarily directly related to these features. The reason is that the 33 meV mode of Pt20Fe18 and the 36 meV mode of Pt18Fe20 actually correspond to displacement patterns fundamentally different from each other. A different approach to establish the composition of a NP based on its VDOS was proposed by Calvo and Balbuena. They asserted that the fraction of a given element in a heterogeneous NP can be readily identified from the element-specific VDOS.17 This information, however, is not always readily available from experiment. Clearly, the effect of increasing the content of lighter atoms is not necessarily straightforward. Despite the above, we find that it is possible to characterize the composition of small NPs if the mass ratio of its 10342



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Contribution from the Conspicuous Discretization of the Vibrational Spectrum. In order to understand the features in the VDOS of a NP, one must note that its discretization is a phonon confinement effect, which, although present in all real systems, becomes very pronounced only at the nanoscale. In fact, such features were already apparent in the modes at the Γ̅ point of the Brillouin zone in the work by Allen et al.51 on the evolution of the phonon dispersion of thin films, as a function of thickness. These are nonpropagating modes and thus the corresponding Γ̅ -point VDOS mimics the confinement effect on the VDOS of NPs. Allen et al. showed that 5-layer and even 3-layer fcc thin films yield a finite number of phonon-dispersion branches that at the Γ̅ -point already span the entire energy range of the bulk vibrations. The discretization effect, however, is washed out because the total VDOS of thin films is inevitably continuous since vibrations can still propagate and disperse in directions parallel to the surface. As a result, the VDOS enhancement at the low-energy end is not as conspicuous as that in NPs. Nevertheless, a q-resolved VDOS (at Γ̅ and other high symmetry points to take into account a finite number of force constants between in-plane NN) based on their dispersion curves would also show sharp peaks at high and low energies since the frequencies are discretized. In order to better appreciate how discretization reflects in the VDOS, we have plotted in Figure 8a the vibrational modes at Γ̅ of 3 × 3 5-

displacement patterns of the highest- and lowest-frequency eigenmodes (we refer the reader to the simulations of each mode provided in ref 46). Displacement Pattern of the Highest Frequency Modes. Analysis of the displacement patterns show that the highest frequency of each NP is a breathing mode of subshell atoms (Figure 3a,b and ref 46). This result is precisely in accord with the charge density consideration and implies that radial modes of subshell atoms are stiffer than any mode in the bulk because they correspond to in-phase perturbations of the regions of highest charge density. One can say that high-energy modes, including super-bulk-band ones, take place when a NP yields eigenmodes that stretch bonds (sometimes almost longitudinally) individually stronger than those in the bulk. In the case of heterogeneous NPs, these eigenmodes are of course expected to involve the motion of the lightest atoms. It is the threedimensional confinement of the vibrations that gives rise to modes which, strictly speaking, have no counterpart in surfaces, and certainly not in the bulk. Displacement Pattern of the Lowest Frequency Modes. The lowest-frequency modes, in turn, mostly correspond to tidal (quadrupolar or dipolar) or torsional modes of shell atoms (see Figure 3c,d and Table 2). In all cases the fundamental frequency is primarily shear-torsional (Sh) but with a tidal (T) component. Similar results can be obtained from elasticity theory. In particular, the solutions to the equation of motion of an isotropic and homogeneous sphere provided by Lamb,7 and applied by Tamura29 to NPs, correspond to some of the lowfrequency modes presented here. However, given the shape anisotropy of our NPs, they are better represented by the results from elasticity presented by Love30 for a homogeneous sphere rotating and/or acted upon by external deformation forces, factors that map into an effective ellipticity. Still, our simulations allow us to appreciate a richer spectrum of vibrations that are not one-to-one with the vibrations of a sphere or an ellipsoid because of the evident deviations of our nanoparticles from spherical or ellipsoidal and the complex charge density redistribution caused by the loss of neighbors. Furthermore, our simulations show that, in general, lowfrequency vibrations are so because the motion of the involved atoms does not perturb radial bonds of high charge density and, in general, most bonds are left invariant. They are lowest if the motion is either perpendicular to the stretched bonds (shear) or along tangential bonds that are enlarged by the nature of the structure or weakened because of the radial charge distribution (tidal) and, as expected in heterogeneous particles, if the involved atoms are those with largest mass. Understanding the Features of the VDOS of Nanoparticles. We have seen that low-frequency modes necessarily are tied to the existence of relatively small force constants and these, in turn, to the low coordination of surface atoms.21 On the other hand, high-energy modes, imply stretching of bonds (sometimes almost longitudinally) individually stronger than those in the bulk. However, to explain the enhanced VDOS at low energies observed in NPs,13−15,19,20 one has to take into account also that the normalized VDOS at low frequencies is necessarily enhanced because of the discretized vibrational spectrum of NPs. In other words, NPs have a finite and relatively small number of vibrational modes, and thus the f raction of ”low-frequency” modes becomes comparable with the fraction of modes at higher energies with decreasing NP size. Something similar happens with the super-bulk-band modes.

Figure 8. VDOS of (a) a 3 × 3 5-layer Cu and Ag (001)-films at the Γ̅ point; and (b) Cu38 NP and fictitious 32-atom and 96-atom Cu NPs with bulk force constant.

layer Cu and Ag (001) films. We see that in both cases one also obtains discrete peaks accounting for the few (132) modes of 45 atoms contained in a 3 × 3 supercell. VDOS of Real NPs vs Fictitious Bulk-Force-ConstantGoverned NPs. To completely isolate the contribution of vibrational confinement from that of low coordination to the presence of peaks at the low-energy end of the VDOS of NPs, we consider the VDOS of fictitious NPs whose vibrations are 10343



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the context of astrophysics (and oceanography) as the p-modes of white-dwarf stars (so-called because pressure is the principal restoring force56−61). In turn, low-frequency vibrations that have also apparently no counterpart in the bulk or on surfaces but are related to the so-called oscillatory g-modes of white dwarfs (those in which gravity is the principal restoring force56−61), which have been widely studied and characterized with the help of continuum mechanics. Continuum mechanics has been successfully applied to Earth30,62,63 and the luminosity pulsations of variable stars56−61 for about 100 years.4 However, notice that while within the latter branch of physics, the vibrational modes have been well characterized and visualized,56 the same level of characterization and visualization is more challenging for nanoparticles. Only within the last 15 years have optical spectroscopy experiments been able to detect radial8,10,11,64−68 modes. More recently, dipolar69,70 and quadrupolar70,71 modes have been studied. However, ellipsoidal9,72,73 nanoparticles that introduce an axial anisotropy have been analyzed less intensively. Still, in all the above studies, the vibrational modes have been successfully modeled by the solutions of the general equations of elasticity, in particular, those obtained by Lamé, Euler, Bernoulli, Rayleigh, Kelvin, Lamb, Pochhammer, Jeans, Love, and others.30,31 The reason for which both NPs and stars of fluid mass have related oscillations is not coincidental.4 On the one hand, the radial or quasi-radial symmetry of both the NP atomic structure and the electronic charge density make an ellipsoid a good model for our nanoparticles in a first approximation. For modeling the fluid region of a star, the equations of motion are the same as those of elasticity, but the stress tensor is replaced by the pressure and/or the gravitational tensor and, naturally, the eigenmodes of a solid and fluid spheres are related.74 Importantly for the present study, modeling the pulsations of stars requires including the effect of rotations and other external forces that break the spherical symmetry and give rise to an axial anisotropy that is equivalent to the shape anisotropy of our NPs.30 Thus, although many of our modes are not necessarily described by the elastic spherewidely studied for nanoparticlesyet, they are well-known and characterized in rotating variable stars. Moreover, the fact that stars are approximately homogeneous but have at the surface a layer of different composition (partial ionization surface56) is also an important factor that makes them comparable to our inhomogeneous core−shell NPs.

governed by bulk force constants in all three directions. To model such NPs, we have simply taken a supercell containing 32 and 96 bulk Cu atoms (i.e., all with coordination 12 and fcc bulk bond lengths) and calculated their Γ̅ -point modes. As expected, Figure 8b demonstrates that one also obtains noticeable peaks at the low-frequency end notwithstanding that the modeled NP does not undergo the force-constant softening attached to low coordination. Figure 8b also confirms that the contribution of those low-frequency peaks to the VDOS decreases with increasing number of atoms. The same arguments apply to the high-frequency tail. Namely, only a few super-bulk-band modes in our relatively small NPs are responsible for the salient peaks above the bulk limit, but their contribution to the VDOS will decrease for larger NPs. Connection between the Vibrational Modes of NPs and Macroscopic Systems. Comparison with Surface Modes. It is important to note that the eigenvectors defining the two extremes in the VDOS are not that alien to widely studied surface phonons. Perhaps the reason they have not been identified as such is because, for in-plane infinite systems, surface modes are better classified as vertical, longitudinal, and shear-horizontal, and not as radial or nonradial. Another factor that probably has kept the phonons of NPs unveiled is that not much attention has been paid to the vibrational modes of freestanding thin-films themselves, as one often uses them merely to simulate semi-infinite surfaces. With regard to the highfrequency breathing modes, a radial mode like those reported here would be equivalent in a surface to a purely vertical vibration of the atoms of the second layer at the Brillouin zone center, which happens not to be an eigenmode of flat surfaces. Nevertheless, our calculations at the Γ̅ -point for Ag(001) and Cu(001) films indicate the highest frequency mode is a vertical mode throughout the slab, except the outermost layer. Interestingly, in these modes the layers vibrate against each other. Notice thus that such a vibrational mode would be the surface analogue of a breathing modeas one can appreciate in the layered NP, Pt20Fe18 (see animations in ref 46). As a matter of fact, in both experiment and calculations, surfaces whose atoms are strongly covalently bound and which undergo a strong inward relaxation display super-bulk-band modes at the zone boundaries. Ru(0001) is incidentally the most remarkable case since the super-bulk-band modes are visibly present at Γ̅ .52 The space localization of these modes is not clear, however. Even for Ag(001), recent measurements53 have been able to detect modes at the bulk limit, apparently second-layer vertical modes.54 Furthermore, experiments and calculations have shown that Cu stepped surfaces also display super-bulk-band modes localized at the step and precisely at the second steppedlayer.55 Although the actual displacement patterns of the lowfrequency modes of these NPs may not help a great deal in relating them to surface modes, it is even easier to explain them in terms of surface phonons. These NP modes correspond to the softest mode in flat metal surfacesthe S1 shear-horizontal mode, in which, just as in NPs, most bonds are left invariant (since S1 is strongly localized at the surface layer) and the motion of the surface atoms is actually not parallel to any NN stretched bond. As an acoustic mode, as the wave-vector approaches zero, the bond-length perturbations die off and hence the frequency tends to zero. Variable Stars. As mentioned in the Introduction, although the breathing-type modes have no analogue in bulk systems, they are well-known and have been characterized in detail in

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SUMMARY We have presented first-principles electronic-structure and vibrational-dynamics calculations of ∼1 nm transition-metal nanoparticles to lay down simple and transparent arguments of how the charge density distribution in NPs, the eigenvectors, and the relative masses in heterogeneous NPs are the elements shaping key generalities of the VDOS of NPs. The chargedensity distribution driven by the low coordination of surface atoms NPs affords a rationale as to why the highest frequency mode is radial, while the lowest frequency modes are tidal-like. The former correspond to eigenmodes that involve a longitudinal stretching of bonds that are individually stronger than those in the bulk because of the dangling bonds of lowcoordinated atoms. In our NPs these are subshell modes. Nonetheless, the fact that flat and stepped surfaces display second-layer vertical super-bulk-band modes is indicative that as long as symmetry and composition allows it, subshell radial 10344



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COMPUTATIONAL DETAILS We perform density-functional-theory periodic supercell calculations within the pseudopotential approach.77 We treat the exchange-correlation functional in the generalized gradient approximation (GGA) using the functional by Perdew−Burke− Ernzerhof (PBE).78 The maximum kinetic energy of the plane waves used to describe valence states is set to 400 eV. To simulate isolated NPs, we place them in a cubic unit cell of at least 18 × 18 × 18 Å3, large enough to guarantee that NPs in periodic images of the supercell interact negligibly, as shown by convergence tests in which we used supercells of dimensions up to 25 × 25 × 25 Å3. Integrations inside the Brillouin zone are performed over 1 k-point. Minimization of the total energy of the NPs and related films as a function of atomic positions is achieved by reducing their Hellmann−Feynman forces77 below 10−3 Ry/au. We obtain the real-space force constants, vibrational frequencies, and eigenvectors of Ag27Cu7 within the density functional perturbation theory52,79 (implemented in the mixed-basis scheme80) as a benchmark test for calculations using the finite-displacement method (implemented in the Vienna ab initio simulation package81 (VASP) and using the projector-augmented-wave-method pseudopotentials82), which was the method of choice for all other NPs. The charge density distribution is analyzed with an exponential scaling function in order to highlight the strengthening/softening of the bonds. The VDOS is obtained from the discrete set of eigenfrequencies by broadening these with a Gaussian function 0.5 meV wide. The displacement pattern of each vibrational mode is derived from the eigenvectors. The XCrySDen software83 was used to obtain a schematic representation of the structure of the NPs, to visualize their charge density profiles, and to perform animated simulations of the vibrational modes (see ref 46).

modes are expected to render the highest-frequency modes of neutral NPs. In contrast, low-frequency modes, that correspond to tidal or torsional eigenmodes, have a small component along the strongest bonds. We have demonstrated that the low- and high-frequency tails of the VDOS of NPs necessarily seem enhanced because NP vibrations are highly confined and thus the finiteness of the number of modes of a NP becomes quite conspicuous. Moreover, the density of low-frequency modes seems particularly enhanced with respect to the bulk, surfaces, and thin films partly because the fraction of low-frequency modes becomes increasingly important as the length within which the modes are confined becomes smaller. We have shown that it is not necessarily possible to draw a general trend for the VDOS as a function of composition on the basis of mass (or bond-lengths) alone since the eigenmodes, and thus the VDOS, sensitively depend on the details of the structure. We demonstrate that the VDOS is also not readily delineated by the bonds rendering stiff/soft force constants because the VDOS is contingent on the eigenvectors allowed by the particular atomic configuration. We call attention to the fact that the vibrational modes of our small NPs have no counterpart in bulk systems, yet for some of them there is a correspondence with luminosity pulsations observed in variable stars.56 Radials, dipolar, and quadrupolar modes have long been measured experimentally (see, e.g., refs 8−11) and obtained in continuous elastic body approximation calculations,6,7,29,30,73 but here first-principles calculations of internally structured particles confirm and call attention for the first time that the nature of the vibrations in discrete systems at the nanoscale is also the same as that of luminosity pulsations in the astronomical-object scale and, not surprisingly, described also by continuous mechanics. Our results suggest that the singular vibrational dynamics of small NPs (∼1 nm) have several implications of technological importance. The high- and low-energy tails of the VDOS, for instance, reveal information about the atomic structure and composition of small NPs. Namely, the size of the confinement gap at the low-frequency end and the extent to which the highfrequency end of the VDOS surpasses the bulk limit distinguish whether a NP has a bulk-like ordering or not. One consequence of the latter is that NPs with non-bulk-like ordering have broader spectrum than NPs with bulk-like ordering. Also, for bimetallic NPs with a sufficiently large species mass ratio, the fraction of the heavy species is approximately given by the integral of the normalized VDOS up to their bulk-band limit. Moreover, the VDOS of NPs displaying species segregation (core−shell structure) with inner lighter atoms exhibit a dip around the bulk cutoff frequency of the heavier atoms. Finally, once it has been noticed that the absence of long wavelength modes makes the VDOS decidedly non-Debye-like (neither linear), it becomes clear that the low-energy end (either wide confinement gaps or, conversely, sharp peaks) will dominate the mean squared displacement and thermal properties in general of NPs in a nontrivial manner, which is completely different from that known for pure bulk systems. One may also ask as to how the discretized and confined vibrations of NPs interact with electronic excitations or adsorbate vibrations. The presence of relatively large size- and shape-dependent confinement energy gaps at the low-energy end of the VDOS can be expected to promote the diffusion of adsorbates.2,54,75,76

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AUTHOR INFORMATION

Corresponding Authors

*E-mail: [email protected]. Phone: +1-407823-2325. Fax: +1-407-823-5112. *E-mail: [email protected]. *E-mail: [email protected]. *E-mail: [email protected]. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS This work was supported in part by grants from DOE-USA (DE-FG02-07ER46354). M. Alcántara Ortigoza is indebted to S. Stolbov for numerous insightful discussions. The authors thank L. Baker for critical reading of the manuscript. The computations were performed in the Stokes Advanced Research Computing Center (Stokes ARCC) at the University of Central Florida and in the Karlsruher Institut fü r Technologie, Institut für Festkörperphysik.

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REFERENCES

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