An atomistic description of alloys and core shells nanoparticles
Lasse K. Sørensen, Anton D. Utyushev, Vadim I. Zakomirnyi, Hans Ågren
JJournal Name
An atomistic description of alloys and core shells nanoparticles
Lasse K. Sørensen, ∗ a Anton D. Utyushev, b , c Vadim I. Zakomirnyi, a , b , c and Hans Ågren a , d , e Using the extended discrete interaction model we investigate the tuneabilty of surface plasmonresonance in alloys and core-shell nanoparticles made from silver and gold. We show that thesurface plasmon resonance of these alloys and core-shell particles to a large extent follow Vegard’slaw irrespective of the geometry of the nanoparticle. We show the evolution of the polarizabilitywith size and demonstrate the highly non-linear behaviour of the polarizability with the ratio of theconstituents and geometry in alloys and core-shell nanoparticles, with the exception for nanorodalloys. A thorough statistical investigation reveals that there is only a small dependence of thesurface plasmon resonance on atomic arrangement and exact distribution in a nanoparticle and thatthe standard deviation decrease rapidly with the size of the nanoparticles. The physical reasoningfor the random distribution algorithm for alloys in discrete interaction models is explained in detailsand verified by the statistical analysis.
Like for plasmonic nanoparticles in general, there is a great dealof interest in bimetallic, or alloyed, small nanoparticles dueto their potential applications in a number of technological ar-eas, like bioimaging , biomedical plasmonic based sensors anddevices , and in heterogeneous catalysis. For instance, the posi-tion of plasmon resonances can be adjusted over a wide rangeof wavelengths by varying the composition of the constituentmetals, making bimetallic particles an interesting proposition foruse in surface plasmon enhanced imaging. The recent advance-ment in their synthesis and characterization have thus openedup possibilities to produce such alloyed or bimetallic nanopar-ticles for particular purposes and applications. This goes espe-cially for bimetallic particles formed by noble metal elements likeplatinum, gold , silver and copper. Like for the correspondingmonometallic plasmonic particles there are requirements on theirdesign with respect to crystallographic structure, shape and sizeand - for certain applications- surface functionalization. In addi-tion to what is required for monometallic particles, there is alsoneed to characterize the element composition and the internal a Department of Theoretical Chemistry and Biology, School of Engineering Sciences inChemistry, Biotechnology and Health, Royal Institute of Technology, Stockholm, SE-10691, Sweden. b Siberian Federal University, Krasnoyarsk, 660041, Russia. c Institute of Computational Modeling, Federal Research Center KSC SB RAS, Krasno-yarsk, 660036, Russia. d Federal Siberian Research Clinical Centre under FMBA of Russia, 660037, Kolomen-skaya, 26 Krasnoyarsk, Russia. e College of Chemistry and Chemical Engineering, Henan University, Kaifeng, Henan475004 P. R. China. ∗ Corresponding author: [email protected] elementary distribution of the particles and the homogeneity ofthe particle population. Making alloys or core-shell structures isreally the only way to get a decent blue-shift in plasmon reso-nance frequency since geometry alterations always gives intensered-shift. As in other areas of nanothechnology the synthesisand characterization procedures can be greatly boosted by de-sign strategies based on theoretical modelling . The bimetallicnature and the new parametric dimensions that appear for suchparticles with respect to the monometallic ones, pose special re-quests not easily met by traditional classical plasmonic models.This goes especially for â˘AIJsmallâ˘A˙I bimetallic nanoparticles forwhich the use of dielectric constants of bulk materials, makes itimpossible to take into account structural differentiation for thedielectric response. In particular when the small particles aremixed with different elements and with large surface to volumeratios and thus when the mean free path of the conduction elec-trons need to be considered. Here approaches are called for thatare more precisely can relate to the discrete atomic structure ofthe nanoparticles to the dielectric and plasmonic properties. Un-fortunately pure quantum approaches are still only applicable forthe very small particles, leaving a size region − nm unattain-able by either classical and quantum theory. Discrete interactionmodels have the inherent capacity to deal with size effects downto the atomic level. In a recent work we presented an extendeddiscrete interaction model (ex-DIM) to simulate the geometric de-pendence of plasmons in the size range of − nm where theClausius-Mossotti relation is replaced by a static atomic polar-izability to obtain the frequency-dependent dielectric function. The static atomic polariziability was modeled as the sum of threesize-dependent Lorentzian oscillators and, with Gaussian chargedistributions and atomic radii that vary with the coordination
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Journal Name, [year], [vol.] ,, a r X i v : . [ phy s i c s . op ti c s ] J u l umber. The frequency-dependent Lorentzian oscillators dependon the plasmon length along the three Cartesian directions us-ing the concept of the plasmon length as defined in the work ofRinge et al. . In this way we extended previous discrete inter-action models to make possible description of the polarizabilityof nanoparticles with different size , shape and composition, andtake account of the dependence of the polarizability of the surfacetopology or structure of the metallic nanoparticles. This extensionof the DIM models serves as a necessary step in order to modelbimetallic particles. The purpose of the present work is to usethe extended DIM model to explore the appearance of plasmonicexcitation in alloyed nanoparticles, taking alloys between goldand silver as example, and explore how their plasmonic proper-ties evolve with respect to concentration of the two components,how size and shape of the full particles modify the properties,how these features can be related to the corresponding proper-ties of the monometallic particles containing either element andto compare with experimental findings that now are available.
The extended Discrete Interaction Model (ex-DIM) , is a fur-ther development of the DIM model where significant im-provements in the description of the surface topology, geometricdependence and parameterization of the SPR(s) are introduced.The ex-DIM model has a special applicable edge for systems in the − nm size range, where quantum mechanical models can-not be applied, due to the scaling of these methods, and wherethe concept of a bulk dielectric constant used in classical modelsbreaks down. The − nm size range is, however, a very im-portant region since this is the size region where the onset of theSPR(s) is seen and where the nanoparticles are still small enoughto be used in bio-medical applications. Due to the inability ofextrapolating data from quantum mechanical methods and clas-sical methods into the − nm size range the ex-DIM must beparameterized directly from experimental data. We will in Sec.3.1 briefly show how a new element easily can be parameterizedfor the ex-DIM and in Sec. 3.3 describe how alloys with a giveninitial distribution is implemented.The ex-DIM model is a discrete structure model where eachatom is represented by a Gaussian charge distribution and en-dowed with a polarizability and a capacitance which govern theinter atomic interaction. The Lagrangian is written in the usualform as the interaction energy E minus the charge equilibrationconstraint expressed via the Lagrangian multiplier λ : L [ { µ , q } , λ ] = E [ { µ , q } ] − λ ( q tot − N ∑ i q i ) , (1)where N is the number of atoms, q i is the fluctuating charge as-signed to the i -th atom, and q tot is the total charge of the nanopar-ticle. The interaction energy E [ { µ , q } ] in this way captures alldifferent types of interactions involving fluctuating dipoles µ ,charges q and an external field and described in greater detailin ref. .The surface topology is captured by a coordination number, as defined by Grimme , and is assigned to each atom. The coordi-nation number f cn modifies the atomic polarizability through thescaling of the radius α ii , kl ( ω ) = (cid:18) R i ( f cn ) R i ,bulk (cid:19) α i , s , kl L ( ω , P ) (2)and likewise for the capacitance c ii , kl = δ kl f c with f c = c i , s (cid:20) + d R i ( f cn ) R i ( ) (cid:21) L ( ω , P ) . (3)In Eqs. (2) and (3), R i ,bulk is the bulk radius of the atom, R i ( f cn ) the coordination number scaled radius , α i , s , kl the staticatomic polarizability , d = a scaling factor and L ( ω , P ) a size-dependent Lorentzian. The polarizability and capacitance of al-loys will in this way not only have a spacial dependence from thediscrete structure from the interaction but also a small one fromthe modified surface atoms.The geometric dependence of the SPR is determined by thesize-dependent Lorentzian L ( ω , P ) L ( ω , P ) = N ( L x ( ω , P x ) + L y ( ω , P y ) + L z ( ω , P z )) , (4)where each Lorentzian depends on the plasmon length P i in thegiven direction ω i ( P i ) = ω a ( + A / P i ) , (5)and in this way cluster size dependence and complicated geo-metrical shapes, with up to at least three SPRs, can be simulatedfor solid particles. Since the atomic radius for different atomsis slightly different the plasmon length P i is not a constant evenwhen the same discrete structure is used for alloys though thechange is only in the difference between the atomic radii of theconstituents. ω a and A are the only fitted parameters in the ex-DIM model .The isotropic polarizability is determined from the fluctuatingcharges, q , and dipoles, µ which are determined by inversion ofthe relay matrix. In this way all SPRs are presented to-gether in the same spectrum.
A fundamental problem of the geometric models is the need for adiscrete structure in the simulations. In molecular physics the ge-ometry of known molecules is usually tabulated and can be readin directly. This, however, is not the case for metallic nanopar-ticles where neither the discrete structure or even the numberof particles are known. Only the overall geometric shape of thecluster with dimension on the nm length scale with some errorbars are known and these provide no information about the inter-nal discrete structure. Geometry optimization is also not helpfulsince geometry optimization is NP-hard and therefore not feasiblefor clusters with thousands of atoms.
We will therefore herebriefly discuss the influence of discrete structure and a pragmaticyet accurate approach in which a geometry is easily generated forboth pure metals and for alloys.
Journal Name, [year], [vol.] , ig. 1 Linear fit of the experimental data compared with the re-calculated clusters using the fitted parameters. The number of atoms inthe clusters here vary from 135-7419 with sizes from 1.45-6.18 nm. From systematic investigations on small silver and gold nano par-ticles it is evident that there is a clear trend in the evolutionof the surface plasmon resonance with size on the nanometerscale.
Yet for any given particle size all measured particlesvary in both the discrete structure and particle number. Since thediscrete structure differences does not lead to an extreme broad-ening of the plasmon peak for larger clusters the discrete structuredifferences instead show that having the exact discrete structure,like in molecular physics, is not of extreme importance for theSPR. This implies that having the same discrete lattice structure,for which the method has been optimized with, should suffice forall sizes.For smaller structures the sensitivity to the discrete structure isexpected to be larger which is also seen in the parameterizationof gold in Fig. 1. This greater variation in the SPR is due to thegreater percentage variation in the number of particles for a fixedplasmon length and that with fewer atoms comes an increasingsensitivity to the placement of the individual atoms in the dis-crete structure. For clusters with a plasmon length of 1.85 nmwe, in this case, see the greatest variation in the surface plasmonresonance, from 2.45-2.67 eV, but we here also have the great-est percentage variation in the particle number since the numberof atoms range from 141-249. For larger clusters we see a muchsmaller variation in the surface plasmon resonance. For examplewe see that particles with a plasmon length of 5.92 nm the sur-face plasmon resonance only varies from 2.41-2.42 eV and eventhough the particle number ranges from 6051 to 7011.The parameterization of gold data taken from experiments ofgold clusters in solution in the 3.6-17.6 nm size region have beenused.
The parameterization was performed by fitting the ex-perimental data to the inverse plasmon length and afterwardsfinding an optimum frequency for a set of different clusters. Fromthis ω a and A of Eq. 5 could be fitted in the same way that thesilver parameters was obtained. We here find ω a = and A = in atomic units. Furthermore for the static po-larizability we use α = au and set the broadening γ = along with the surface and bulk radii of r = and r = ,respectively.While the ex-DIM is parameterized from spherical clusterswithin a − nm size range and a limited frequency interval itstill remains valid in a much broader frequency range as shown bycalculations on nanorods and nanocubes . The large frequencyrange of the ex-DIM model is possible because any red or blueshift due to geometric distortions from a sphere can be describedby the interaction between the atoms in this model and no exter-nal data is required for this . The ex-DIM model is therefor notlimited in the frequency range by the parameterization range un-like classical models which are limited by the experimental rangefor which the dielectric constant have been measured. Like for the pure metals the discrete structure of alloys is not pos-sible to obtain from experiment and contain an added complica-tion since the unit cells of the metals in the alloy will differ. Thelatter problem of not knowing the lattice parameters of the alloycan usually be overcome using the empirical law from Vegard a A − x B x = ( − x ) a A + xa B (6)where the alloy lattice parameter a A − x B x is approximated by aweighted mean of the two constituents A , B lattice parameters a A and a B ,respectively. For core shell structures connecting the lattice of the core withthe shell is not a simple problem for models using a discrete struc-ture and there does not seem to be a simple way to connect twoperfect lattices with different lattice parameters without havingto distort these lattices at the boundaries. In order to overcomethis we have optimized gold and silver using the same lattice con-stants which is only possible due to the very little difference inthe lattice constants for these two metals. We see the usage of thesame lattice constants as a pragmatic approach for this particulartype of core-shell structures and not a general solution.
Since the exact placement of the constituents in an alloy cannotbe predicted and differ from cluster to cluster we have chosen torepresent this using a simple random selection of elements basedon an initial probability distribution. This means that there willbe both a random spacial distribution of elements along with asmall variation in the ratio between the elements due to the ran-domized drawing of the elements. In this way no two clusters willbe exactly alike and a statistical analysis of the slightly expectedbroadening of the SPR for alloys can be analyzed in terms of thevariation in the ratio and spacial distribution of the constituentsand error bars for the alloys can be assessed. Two random distri-butions for a sphere and disc structure are shown in Fig. 2.
Even though Vegard’s law initially was formulated in order to es-timate the lattice parameters of alloys it has often been extended
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Journal Name, [year], [vol.] ,, ig. 2 Schematic representation of alloy sphere (a) and disc (b) nanopar-ticles taken from a random distribution of Au (yellow) and Ag (grey)atoms. include all properties of alloys P A − x B x = ( − x ) P A + xP B (7)where P is any property. We are here particularly interested inexamining if Vegard’s law also holds true for the progression ofthe position of the SPR with mixing of the constituents as indi-cated in other studies for alloys and to some extentcore shells or if this is geometry dependent as experiments onnano discs predict. Since the polarizability or extinction crosssection differs for Au and Ag it would also be of interest to see ifthe strength of the response to an external field also follow Veg-ard’s law since the strength of the response determines strengthof the local electric field from the SPR. Here previous resultsshows a non-linear dependence for the response for sphericalclusters.
Before this we will perform a statistical analy-sis to determine how the error varies with spacial and constituentdistribution along with size of the cluster for a spherical alloynanoparticle in order to get an estimation of the error bars whenperforming a single calculation.
The aim of the statistical analysis is to examine the effect of ran-domly drawing elements and give estimations for error bars sinceit will not always be possible to make a full statistical analysisfor every cluster. Secondary goals are to ensure that the modelbehaves as would be expected from a random drawing of a finitenumber of elements with a given probability, to analyse how largethe error is for classical models using a simple linear combinationof the constituents along with how the standard deviation varieswith the size of the cluster.From the fitting of Au in Fig 1 we would expect that the stan-dard deviation of an alloy would decrease with size since theimportance of the individual placement of an atom would de-crease and less sensitivity to variations in the actual distributionof the constituents should be seen. Secondly from a statisticalperspective we would expect that the standard deviation for al-loys where one of the constituents is very dominating is lowerthan one where the distribution is more even.In Table 1 we present a statistical analysis for a set of clustersranging from 249 to 6051 atoms or 1.85 to 5.79 nm in size withthree different distributions in order to show both the size and distribution dependence of the standard deviation of the SPR andextinction cross section. Due to computational resources we re-duce the number of sampling points with the size and width ofthe sample though significant changes with increasing number ofsampling points is not expected and all trends are easy to see.Due to the larger standard deviation in the SPR seen for smallersamples the frequency range sampled in Table 1 is not the samethough the interval between each frequency was constant and setat 0.0001 au. Since the CPU time set for each size, above 1553atoms, is the same the number of samples for each distributionvaries. We have in Table 1 added extra digits on both the stan-dard deviation σ and the averages µ in order to better illustratethe trends.As expected we see that the standard deviation in the numberof Ag atoms increases with both size and evenness in the distri-bution. For the SPR we also see, as expected from Fig 1, thatthe standard deviation for the SPR decreases with size and in-crease with evenness in the distribution. At around 3000 atomsthe standard deviation is below 2 nm and therefore below theaccuracy which can be expected from any calculation. From anatomistic perspective we therefore also see that the classical wayof treating an alloy without any resolution at the atomic leveldoes not introduce any significant error for the SPR of larger sys-tems calculated using classical methods. Furthermore wesee that with good size correction the classical methods for alloyscan safely be extended to small systems down to 4-5 nm. For al-loys where on of the constituents is very dominating, above 90percent, even smaller systems can be safely simulated. There is,however, one outlier in the data in Table 1 namely the 675 atomscluster with distribution of
Ag and
Au where the stan-dard deviation for the SPR is 11.2nm where a standard deviationof around 1.5nm would be expected from the trend. This outlierappears because many of the spectra for this 675 atoms clustershows a double peak, which the maximum alternates between,thereby causing a very broad distribution and hence large stan-dard deviation.In order to analyse and illustrate the effect of the random dis-tributing on a fixed number of constituents and the variation ofthe constituents due to the random drawing of the constituentswe have chosen to focus on the 1553 atoms cluster with differentdistributions. The 1553 atoms cluster shows a clear differencein the standard deviation of both the number of Ag atoms andthe SPR for the different distribution and a sufficient number ofsamples could easily be collected as seen in Table 1. In Figs. 3-5the three different distributions from Table 1 of the 1553 atomscluster is plotted.In Fig. 3 we clearly see that the SPR follows the change awayfrom the average number of Ag atoms ∆ µ Ag with a red shift for anegative shift of ∆ µ Ag and blue shift for a positive shift of ∆ µ Ag as would be expected. For the clusters with the same number ofconstituents we also see that the spacial distribution of the atomsalso matters though less than the variation in ∆ µ Ag since we havea very nice and centered peak.Comparing the three distributions in Figs. 3-5 we clearly seethat the Ag 90 Au 10 distribution in Fig . 3 is significantly morenarrow and peaked both along ∆ µ Ag and λ . This can also be seen Journal Name, [year], [vol.] , able 1 Statistical data from the sampling of Ag and Au alloys of different sizes and distributions showing the average, standard deviation, minimumand maximum values found for the number of Ag atoms in a cluster, the SPR and Extinction cross section per atom. Extra digits have in some casesbeen added in order to better illustrate trends. Distribution Samples Ag SPR [nm] Extinction cross section [nm /atom]atoms % Ag % Au µ σ min max µ σ min max µ σ min max90 10 9099 224 5 205 242 348.4 1.92 342.6 356.5 0.2486 0.0109 0.1914 0.2866249 50 50 9099 124.4 7.9 96 157 392.9 7.85 368.9 424.6 0.1678 0.0091 0.1331 0.199430 70 9099 74.7 7.2 49 101 431.9 10.3 402.5 460.2 0.1496 0.0064 0.1263 0.177390 10 8810 607 8 578 639 350.8 11.2 332.4 367.1 0.2149 0.0075 0.1956 0.2368675 50 50 9099 338 13 290 385 410.7 4.91 395.9 427.8 0.1542 0.0038 0.1408 0.169730 70 9099 203 12 159 254 448.2 4.26 430.3 462.5 0.1416 0.0028 0.1305 0.154190 10 8134 1398 12 1355 1436 359.7 1.25 356.0 364.5 0.2405 0.0030 0.2288 0.25101553 50 50 5564 777 20 709 852 412.3 2.99 405.0 421.9 0.1620 0.0028 0.1520 0.172530 70 6194 465 18 403 535 450.6 2.79 440.2 459.2 0.1444 0.0019 0.1363 0.151190 10 2732 3030 17 2976 3090 367.3 0.96 363.6 370.4 0.2368 0.0020 0.2297 0.24363367 50 50 1634 1684 28 1598 1784 423.1 1.71 417.2 429.0 0.1673 0.0018 0.1622 0.172330 70 2621 1009 26 919 1121 457.5 1.60 452.0 462.6 0.1529 0.0011 0.1488 0.156790 10 1159 5446 23 5372 5515 370.6 0.49 369.0 372.2 0.2575 0.0017 0.2515 0.26246051 50 50 746 3024 38 2900 3145 424.2 1.30 419.6 428.2 0.1719 0.0013 0.1682 0.176230 70 809 1818 33 1710 1921 459.4 1.20 456.2 464.0 0.1546 0.0009 0.1516 0.1575 Fig. 3 Statistics data for a 1553 atom alloy cluster with a random dis-tribution with a probability distribution of 10 Au 90 Ag.Fig. 4 Statistics data for a 1553 atom alloy cluster with a random dis-tribution with a probability distribution of 50 Au 50 Ag. Fig. 5 Statistics data for a 1553 atom alloy cluster with a random dis-tribution with a probability distribution of 70 Au 30 Ag. from the projection onto the Counts axis where it is clear thatthe less even the distribution is the more clear and narrow thepeak becomes. The significantly smaller standard deviation forthe SPR of the Ag 90 Au 10 distribution compared to the Ag 50Au 50 and Ag 30 Au 70 distributions is also clearly visible fromthe projection of the distribution onto the λ axis.The extinction cross section per atom in Table 1 shows no realvariance with size but only with distribution where the extinctioncross section increases with the amount of Ag in the alloy. Thoughthe difference in the extinction cross section between Ag 30 Au 70and Ag 50 Au 50 distributions is significantly smaller than couldbe expected from Vegard’s law. We will analyse this observationin more detail in Sec. 4.3 when we look at the evolution of the po-larizability as a function of the constituents. We here also see theexpected trend where the standard deviation decreases with sizethough remain rather small throughout. We here note that sincethe broadening of the Lorentzians in Eq. 4 is not fitted only thetrend and not the absolute values of the extinction cross section Journal Name, [year], [vol.] ,,
Au where the stan-dard deviation for the SPR is 11.2nm where a standard deviationof around 1.5nm would be expected from the trend. This outlierappears because many of the spectra for this 675 atoms clustershows a double peak, which the maximum alternates between,thereby causing a very broad distribution and hence large stan-dard deviation.In order to analyse and illustrate the effect of the random dis-tributing on a fixed number of constituents and the variation ofthe constituents due to the random drawing of the constituentswe have chosen to focus on the 1553 atoms cluster with differentdistributions. The 1553 atoms cluster shows a clear differencein the standard deviation of both the number of Ag atoms andthe SPR for the different distribution and a sufficient number ofsamples could easily be collected as seen in Table 1. In Figs. 3-5the three different distributions from Table 1 of the 1553 atomscluster is plotted.In Fig. 3 we clearly see that the SPR follows the change awayfrom the average number of Ag atoms ∆ µ Ag with a red shift for anegative shift of ∆ µ Ag and blue shift for a positive shift of ∆ µ Ag as would be expected. For the clusters with the same number ofconstituents we also see that the spacial distribution of the atomsalso matters though less than the variation in ∆ µ Ag since we havea very nice and centered peak.Comparing the three distributions in Figs. 3-5 we clearly seethat the Ag 90 Au 10 distribution in Fig . 3 is significantly morenarrow and peaked both along ∆ µ Ag and λ . This can also be seen Journal Name, [year], [vol.] , able 1 Statistical data from the sampling of Ag and Au alloys of different sizes and distributions showing the average, standard deviation, minimumand maximum values found for the number of Ag atoms in a cluster, the SPR and Extinction cross section per atom. Extra digits have in some casesbeen added in order to better illustrate trends. Distribution Samples Ag SPR [nm] Extinction cross section [nm /atom]atoms % Ag % Au µ σ min max µ σ min max µ σ min max90 10 9099 224 5 205 242 348.4 1.92 342.6 356.5 0.2486 0.0109 0.1914 0.2866249 50 50 9099 124.4 7.9 96 157 392.9 7.85 368.9 424.6 0.1678 0.0091 0.1331 0.199430 70 9099 74.7 7.2 49 101 431.9 10.3 402.5 460.2 0.1496 0.0064 0.1263 0.177390 10 8810 607 8 578 639 350.8 11.2 332.4 367.1 0.2149 0.0075 0.1956 0.2368675 50 50 9099 338 13 290 385 410.7 4.91 395.9 427.8 0.1542 0.0038 0.1408 0.169730 70 9099 203 12 159 254 448.2 4.26 430.3 462.5 0.1416 0.0028 0.1305 0.154190 10 8134 1398 12 1355 1436 359.7 1.25 356.0 364.5 0.2405 0.0030 0.2288 0.25101553 50 50 5564 777 20 709 852 412.3 2.99 405.0 421.9 0.1620 0.0028 0.1520 0.172530 70 6194 465 18 403 535 450.6 2.79 440.2 459.2 0.1444 0.0019 0.1363 0.151190 10 2732 3030 17 2976 3090 367.3 0.96 363.6 370.4 0.2368 0.0020 0.2297 0.24363367 50 50 1634 1684 28 1598 1784 423.1 1.71 417.2 429.0 0.1673 0.0018 0.1622 0.172330 70 2621 1009 26 919 1121 457.5 1.60 452.0 462.6 0.1529 0.0011 0.1488 0.156790 10 1159 5446 23 5372 5515 370.6 0.49 369.0 372.2 0.2575 0.0017 0.2515 0.26246051 50 50 746 3024 38 2900 3145 424.2 1.30 419.6 428.2 0.1719 0.0013 0.1682 0.176230 70 809 1818 33 1710 1921 459.4 1.20 456.2 464.0 0.1546 0.0009 0.1516 0.1575 Fig. 3 Statistics data for a 1553 atom alloy cluster with a random dis-tribution with a probability distribution of 10 Au 90 Ag.Fig. 4 Statistics data for a 1553 atom alloy cluster with a random dis-tribution with a probability distribution of 50 Au 50 Ag. Fig. 5 Statistics data for a 1553 atom alloy cluster with a random dis-tribution with a probability distribution of 70 Au 30 Ag. from the projection onto the Counts axis where it is clear thatthe less even the distribution is the more clear and narrow thepeak becomes. The significantly smaller standard deviation forthe SPR of the Ag 90 Au 10 distribution compared to the Ag 50Au 50 and Ag 30 Au 70 distributions is also clearly visible fromthe projection of the distribution onto the λ axis.The extinction cross section per atom in Table 1 shows no realvariance with size but only with distribution where the extinctioncross section increases with the amount of Ag in the alloy. Thoughthe difference in the extinction cross section between Ag 30 Au 70and Ag 50 Au 50 distributions is significantly smaller than couldbe expected from Vegard’s law. We will analyse this observationin more detail in Sec. 4.3 when we look at the evolution of the po-larizability as a function of the constituents. We here also see theexpected trend where the standard deviation decreases with sizethough remain rather small throughout. We here note that sincethe broadening of the Lorentzians in Eq. 4 is not fitted only thetrend and not the absolute values of the extinction cross section Journal Name, [year], [vol.] ,, s to be interpreted here for the ex-DIM. Since Vegard’s law assumes linearity hence a linear energy scalefor the SPR should be used λ Vegard ( x , R ) = ( − x ) λ Au ( R ) + x λ Ag ( R ) . (8)We will in the following use eV and not nm, even if nm is theprevalent choice of unit, since only the former of the two is linearenergy unit.We will here examine if any non-linearity effects in the SPR canbe induced by the geometric structure of the alloy clusters as ob-served experimentally by Nishijima et al. on nano discs. The ex-perimentally observed a very large red shift in the spectra they re-produced in theoretical predictions using FDTD where the plasmafrequency ω p and relaxation time τ was extracted by insertingthe experimental data into the Drude model. We will here ex-amine two spherical clusters with different radii, a nanorod and anano discs though with different relative dimensions as the onesused experimentally by Nishijima et al. . Another possibility ofmixing metals is by making core-shell structures. For the core-shell structures we will also examine two geometrical structuresnamely spheres and rods where we show both Au core and Agshell along with Ag core and Au shell in order to examine if thereis significant difference in which metal is the core and shell. Sincethe atoms in ex-DIM are discrete the steps in the distribution ofthe metals for core shell structures cannot be divided into equalsteps as it can for alloys. Furthermore the use of spherical struc-tures having a percentage wise large core will result in an atomi-cally thin surface and the very large core should therefore be in-terpreted with care. In appendix 5 we plot all spectra from whichdata have been extracted.From Figs 6-8 we show Vegard’s law for position of the SPR ofalloys with different geometries and distributions. We have in Fig6 chosen a random alloy from those sampled in Sec. 4.1 and notthe average since not all distributions and geometries have beensampled. The same random choice goes for the nanorod and nan-odisc alloys in Figs. 7 and 8. One should therefore keep in mindthe standard deviation along with the minimum and maximumfor the SPR shown in Table 1 when interpreting these results.Comparing the 1553 and 6051 atoms spherical clusters in Fig 6we see that changing the size of the sphere does induce any non-linearity nor does the change in geometry which can be seen bycomparing all Figs from 6 to 8. This is in line with other exper-imental and theoretical works except that by Nishijima etal. . In Fig. 8 there appears to be some systematic non-linearityin the position of the SPR. By analysing the spectra in Fig. 17in appendix 5 we see that the non-linearity comes from the ap-pearance of a shoulder for the pure Ag cluster which turn into adouble peak with
Au mixed in and finally the shoulder be-comes the dominant peak with − Au in the alloy. Thisbehaviour is not expected to be seen for all discs and the devia-tion up to 0.2 eV seen is much below the very significant red shiftobserved by Nishijima et al. . At the right had side of Figs 6-8the nm scale is also shown. Since the energy range used is small
Fig. 6 Position of SPR for spherical Au/Ag alloy nanoparticles from1553 atoms (stars) and 6051 atoms (circles) with different percentagedistribution of Au and Ag along with dashed lines showing Vegard’s law.Fig. 7 Position of SPR for Au/Ag alloy nanorod with 2315 atoms(squares) with different percentage distribution of Au and Ag along witha dashed line corresponding to Vegard’s law. the nm scale will also be almost linear but not completely. Thiscan explain why some have observed weak non-linear trends inthe SPR since this will automatically appear if the nm and not theeV energy scale is used.The core-shell structure with an Au core and Ag shell shown inFig. 9 also follows Vegard’s law while the Ag core Au shell core-shell structure shows a small dip of up to 0.2 eV when the numberof atoms in the Ag core exceeds 50 and below 80 percent. In thisregion we, for this cluster, see a large broadening of the spectraas seen in Fig. 18
For pure metals the polarizability per will decrease proportionalto the inverse plasmon length and, in the ex-DIM with the currentbroadening factor, approach an asymptotic bulk limit of approxi-mately 115 au as seen for Au in Fig. 10. The total polarizabilityin spheres is therefore in general proportional to the static polar-izability and the inverse plasmon length.Since all alloys and core-shell structures obeyed Vegard’s law,
Journal Name, [year], [vol.] , ig. 8 Position of SPR for Au/Ag alloy nanodisc with 4033 atoms witha dashed line showing Vegard’s law.Fig. 9 Au-Ag and Ag-Au core-shell nanospheres with different sizes ofcore and shell as a function the percentage of Ag atoms along with dashedlines showing Vegard’s law.Fig. 10 Polarizability per atom for Au as function of inverse plasmonlength. Fig. 11 The polarizability at the maximum of the SPR of a 6051 atomsAu/Ag alloy spherical cluster as a function of the percentage of Ag inthe cluster along with a dashed line showing Vegard’s law. except for a small deviation for the nanodisc alloy and the Ag coreAu shell as seen in Figs. 8 and 9, and seeing that the polarizabilityper atom is proportional to the static polarizability it could easilybe assumed that the polarizability also would obey Vegard’s law p Vegard ( x , R ) = ( − x ) p Au ( R ) + xp Ag ( R ) , (9)where p is the polarizability. This, however, is in general not thecase as seen in both experiment and theoretical studies. For the Au and Ag sphere alloys a very characteristic dip inthe polarizability, as seen in Fig. 11, is observed when addingAg to an Au cluster despite the fact that Ag has a higher staticpolarizability. As seen from Fig. 11 there is a minimum with40 percent Ag in the alloy and the polarizability for 30 and 50percent Ag is very close. The closeness for the polarizability with30 and 50 percent Ag explains why the extinction cross sectionper atom in Table 1 for these distributions are very close and the90 percent Ag significantly higher. For the spheres we see exactlythe same trend as seen in other studies.
For the rod alloy, seen in Fig. 12, we observe a perfect linearcorrelation of the polarizability with the distribution for the lon-gitudinal SPR. The longitudinal SPR of the rod alloy is the onlyalloy geometry, that we have tried, where the polarizability fol-lows Vegard’s law and the only geometry where there is no lossin response to the external field in comparison to Vegard’s law.The nanodisc alloy, seen in Fig. 13, shows a very large dip inthe maximum polarizability when having 10-30 percent Au in thealloy. The dip in the maximum polarizability is again related tothe appearance of a shoulder and double peak as seen in Fig. 17where the peak is significantly broader than for the rest of thenanodisc alloys.For the core-shell structures shown in Fig. 14 there is no obvi-ous trend and larger variations from small changes in the distribu-tion is seen. The large dip is seen for the large Ag core and a thinAu shell where the number of atoms in the core is just below 70percent. This is again caused by a large broadening of the spectraas seen in Fig. 18. As the only one that we have found does theAu core Ag shell nanoparticle have a few sizes of core and shell
Journal Name, [year], [vol.] ,,
Journal Name, [year], [vol.] ,, ig. 12 The maximum value of the polarizability as a function of percentof Ag in a 2315 atoms Au/Ag alloy nanorod with an aspect ratio .Fig. 13 The maximum polarizability as a function of the percentage ofAg in Au/Ag alloy nanodisc with 4033 atoms with a dashed line showingVegard’s law. Fig. 14 The maximum polarizability as a function of the percentage ofAg/Au and Au/Ag spherical cluster with 1553 atoms along with a dashedline showing Vegard’s law. where the polarizability is above that predicted by Vegard’s lawthough nothing systematic. We here present a way of creating accurate calculations on alloysusing a discrete atomic structure model with good estimation oferror bars. This was achieved by using a simple random numbergenerator and distribution for the constituents in our extendeddiscrete interaction model (ex-DIM) in the initial placement ofatoms in the alloy. We lay out the physical reasoning of why it isappropriate to use a perfect lattice for all sizes of both pure met-als and alloys which is backed up by a statistical analysis of thealloys from where error bars can be estimated from. We see thetrend in the error follows that expected trend from the fitting ofthe size dependence of Au clusters and statistics where the stan-dard deviation of the surface plasmon resonance (SPR) increasesinversely with size and evenness of the constituents in the alloy.Much of the motivation for creating alloys or core-shell struc-tures is motivated by the ability of also blue shifting the SPR andhaving a chemically less reactive surface while still having signif-icant response to the impinging light. Another motivating factwas the experimental report by Nishijima et al. on the breakingof Vegard’s law for the SPR simply from geometrical alter-ations. We have here examined three Au-Ag alloys with differentgeometrical structures namely spheres, a rod and a disc and havefound no deviations from Vegard’s law for the SPR, using a linearenergy unit, that is not readily explained by the estimated errorbars or from the emergence of a double peak. For spheres this is inline with other experimental and theoretical predictions. For the discs we were not able to reproduce the experimentalfindings by Nishijima et al. even if the atomic interaction inthe ex-DIM should be able to simulate the polarization of the in-dividual atoms. The idealized structure of alternating Ag and Auatoms giving a closed shell pair structure of Ag and Au envisionedby Nishijima et al. as an explanation of the large red shift of theSPR will of course never be seen by a random distribution andwe have not observed any extremely red shifted outliers in our Journal Name, [year], [vol.] , ata. For spherical core-shell structures the Au core Ag shell alsoshows agreement with Vegard’s law while the Ag core Au shellshows a red shift of up to 0.2 eV compared to the expected fromVegard’s law for large Ag cores though these peaks showed muchlarger broadening and these was no systematic behaviour in theevolution of the SPR with variations of the size of the core couldbe observed.While the SPR, in all cases tested here, closely follow Vegard’slaw this was not the case for the maximum value of the polariz-ability which shows a great dependence on the geometry of thenanoparticle. For the alloys there in general was a slight broad-ening of the spectra leading to a lower maximum value of the po-larizability. This showed up systematically for the spherical alloyswhere there was a minimum in the polarizability at 40 percentAg in the Au/Ag alloy irrespective of size. The nanorod structurewas the only alloy which followed Vegard’s law and the polar-izability was not below that predicted by Vegard’s law. For thenanodisc we, in this case, saw a sudden dip in the polarizabilitydue to having a double peak for alloys containing 75-90 percentAg. The polarizability for the core-shell structure showed a veryunsystematic nature and it would therefore be hard to predict thepolarizability for these.From an atomistic perspective we see that the classical way oftreating an alloy without any resolution at the atomic level doesnot introduce any significant error for the SPR of larger systemscalculated using classical methods. Provided that the sizecorrection for the dielectric function is good then there shouldbe no problems for classical methods in simulating nanoparticlealloys down to the 4-5nm or even smaller depending on the dis-tribution of elements since at these sizes we still see very smallstandard deviation in our calculations.
Conflicts of interest
There are no conflicts of interest to declare.
Acknowledgements
H.Å. and V.Z. acknowledge the support of the Russian ScienceFoundation (project No. 18-13-00363). L.K.S acknowledges thesupport of Carl Tryggers Stifetelse, project CTS 18-441. The sim-ulations were performed on resources provided by the SwedishNational Infrastructure for Computing (SNIC) at NSC under theproject "Multiphysics Modeling of Molecular Materials", SNIC2019/2-41.
Appendix A: Complete curves for alloys and core-shellnanoparticles
We here show the complete curves for all alloys and core-shellnanoparticles. For the spherical alloys seen in Fig. 15 we seethat the shape of the spectra is not altered by the mixing of Auand Ag and Vegard’s law for the SPR is obeyed. The dip in thepolarizability as discussed in Sec. 4.3 and plotted in Fig. 11 forboth spheres is also visible in Fig 15. The fact that the width doeschange can be caused by the fact that the broadening factor usedfor Au and Ag is the same.For the nanorod alloy in Fig. 16 both the longitudinal and
Fig. 15 Spherical alloy Au-Ag nanoparticle from 1553 (top) and 6051(bottom) atoms with different distributions.
Journal Name, [year], [vol.] ,,
Journal Name, [year], [vol.] ,, ig. 16 Optical spectra for different distributions of Au/Ag nanorod alloywith 2315 atoms and aspect ratio of 5.4.Fig. 17 Alloy disc from 4033 atoms with diameter = 9nm, and height= 4nm. transverse SPR are clearly visible. From Fig. 16 it is seen thatVegard’s law is obeyed for both the SPR and polarizability for thelongitudinal SPR. For the transverse only the SPR follows Vegard’slaw. The nanorod alloy is the only nanoparticle, we have found,where Vegard’s law is obeyed for both the SPR and the polarizabil-ity and also the only nanoparticle alloy where the polarizability isnot below Vegard’s law.The longitudinal and transverse SPR is also visible for the nan-odisc alloy in Fig. 17. We her clearly see what appear to besome systematic deviation from Vegard’s law in Fig. 8 is due tothe appearance of a shoulder for the pure Ag cluster which turninto a double peak with Au mixed in and finally the shoul-der becomes the dominant peak with − Au in the alloy.The shoulder and double peak also explains the sharp drop inthe maximum value for the polarizability in Fig. 13. So while asmall and systematic deviation from Vegard’s law is observed herethe deviation is far from that observed by Nishijima et al. wherethe red shift in the of both Au and Ag is below the pure Aupeak. For the core-shell spectra in Fig. 18 no real pattern emerges inthe progression from pure Au to pure Ag clusters since both themaximum polarizability and the FWHM changes rapidly. Due tothe atomistic nature of the ex-DIM having smaller steps in some
Fig. 18 Optical spectra of Au core and Ag shell (top) and Ag core andAu shell (bottom) spherical nanoparticles with 1553 atoms. of the areas where there is a rapid change can be difficult.
Notes and references
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