Formation of collective excitations in quasi-one-dimensional metallic nanostructures:Size and density dependence

Amy Cassidy,1 Ilya Grigorenko,2 and Stephan Haas1

1Department of Physics and Astronomy, University of Southern California, Los Angeles, California 90089, USA2Theoretical Division T-11, Center for Nonlinear Studies, Center for Integrated Nanotechnologies, Los Alamos National Laboratory,

Los Alamos, New Mexico 87545, USA�Received 7 November 2007; revised manuscript received 16 April 2008; published 4 June 2008�

We investigate theoretically the formation of collective excitations in atomic scale quasi-one-dimensionalmetallic nanostructures. The optical response of the system is calculated within the linear-response theory andthe random-phase approximation. For uniform nanostructures a smooth transition from quantum scaling ofsingle-particle excitations to classical plasmon scaling is observed, depending on the system’s length andelectron density. We find crucial differences in the scaling behavior for quasi-one-dimensional and three-dimensional nanostructures. The presence of an additional modulating on-site potential is shown to localizeelectrons, leading to the response of the system that is highly sensitive to the number of electrons at lowfillings.

DOI: 10.1103/PhysRevB.77.245404 PACS number�s�: 73.20.Mf, 71.45.Gm, 73.22.�f, 78.67.�n

The creation, amplification, and control of plasmon exci-tations in metallic nanostructures promise an extreme useful-ness in near-field scanning microscopy and single moleculedetection.1,2 The recent advances in the fabrication and con-trol of low-dimensional nanostructures at atomic resolution�for example, see Refs. 3 and 4� make possible the control ofthe interaction of these systems with electromagnetic radia-tion on a quantum-mechanical level. For example, clusters ofAu atoms, arranged in linear chains on a NiAl�100� surface,3

promise to be good candidates for the purpose of a local-fieldenhancement, as well as for other applications, including op-tical media with a negative refractive index,5 and subwave-length focusing of electromagnetic radiation.6 Thus, it be-comes significant to examine the fundamental aspects of thelight-matter interaction of nanoscale systems with coherentelectromagnetic radiation, in particular, the necessary condi-tions for the formation of collective response in metallicnanostructures.

It is known that in small nanostructures, one can observeboth collective modes and single-particle excitations.7,8

Since, for many situations, strong local-field enhancement innanostructures is due to the collective plasmon excitation,9 itbecomes a fundamental question: how many electrons areenough to create collective �plasmon� response in a metallicnanostructure, and what is the role of the nanostructure’sgeometry in the formation of the collective response?

Because of the significant inherent finite-size energy gapsin atomic scale systems, it is essential to perform a detailedquantum-mechanical analysis that identifies the relevant en-ergy and length scales for the different types of excitationmodes. In this paper, we study the nature of electronic exci-tations in quasi-one-dimensional metallic nanostructures in-vestigating the smooth transition between single-particle andcollective response as a function of the system’s size, elec-tron density, and on-site potential.

In the present approach, the excitation spectrum of thesystem is determined using linear-response theory within therandom-phase approximation �RPA� in the long-wavelengthlimit q→0. We study how the characteristic excitation ener-

gies scale with the system’s size and electron density.First, we find that for a fixed number of electrons Nel and

variable length L of a quasi-one-dimensional system, there isa smooth transition from multiple single-particle excitationsto a single dominant collective plasmonlike resonance. Forrelatively small length, the observed L−2 scaling is attributedto the single-particle transitions. It can be qualitatively ex-plained in the framework of the quantum particle in a boxpicture. We denote this regime as “quantum.” On the otherhand, for larger sizes of the system, the plasmon frequencyscales as L−1/2 that is consistent with the classical plasmonresonance frequency scaling �p��n, where n�Nel /L is theelectron density in quasi-one-dimensional systems. We de-note this regime as “classical.” We denote the size at whichthe single-particle excitations converge into a single plasmonpeak as the critical size Lcr.

Second, we investigate how the critical size Lcr scaleswith the electron density. It is known8 that for three-dimensional nanostructures, the formation of the collectiveresponse occurs for smaller system sizes with increasing ofthe electron density. This is intuitively understandable sincefor higher electron densities, there are more electrons in thenanostructure that can participate in the collective oscilla-tions. In our simulations, we find that for a fixed size L of aquasi-one-dimensional system, for lower electronic densitiesthe plasmon resonance is recovered, whereas for higher den-sities the response of the system is found to be consistentwith quantum single-particle excitations. This observation isin dramatic contrast with the optical response of three-dimensional nanostructures. We give a qualitative explana-tion of the observed effect based on the different scaling ofthe Fermi velocity in quasi-one-dimensional and three-dimensional quantum systems.

Furthermore, we study the formation of the collective re-sponse in spatially inhomogeneous systems. We suggest thatsuch systems can be assembled with a similar technique, asin Refs. 3 and 4, but using different species of atoms. A localpotential is introduced at alternating atomic sites, and theeffect of this modulation on the optical response is examined.

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A simple model of consecutive quasi-infinite wells is pro-posed to analyze the excitation spectrum and system re-sponse as a function of the number of electrons. Finally, weinvestigate the changes in the electromagnetic response forquantum systems, which undergo a transition from extendedplane waves to localized wave functions, as the strength ofthe on-site potential is varied.

Let us consider the response of a quasi-one-dimensionalnanostructured system to a time-varying external electro-magnetic field with frequency �. We assume an atomic chainof N atoms and length L that is modeled by a lattice with Nsites with the intersite distance a. We assume the total num-ber of noninteracting electrons Nel. This assumption is sup-ported by the direct measurement of atomic chains,3 whichdemonstrates that the eigenstates of the electrons are in goodagreement with the model of noninteracting electrons in aninfinite well potential. The Hamiltonian for an electron witha background on-site potential Vos�x� can be written as

H = −�2

2m�

�2

�x2 + Vos�x� , �1�

where m� is the effective electron mass. The backgroundpotential is either taken to be uniform or varies from site tosite. The corresponding eigenproblem is solved by the nu-merical diagonalization.

Within the RPA approximation, the dielectric function isgiven by10

��q,�� = 1 − V�q��0�q,�� , �2�

where �0 is the retarded density-density correlation functionfor pair-bubble interactions. For a system with eigenenergiesEi and eigenfunctions �i, the density-density correlationfunction in quasi-one-dimension is

�0�q,�� =1

L�ij

f i − f j

Ei − Ej − �� − i��Mij�2, �3�

with matrix elements Mij = �i�eiqx�j� between the eigenstates iand j. Here, f i=1 / �exp�Ei /kBT�+1 is the Fermi distributionfunction, and � is a small level broadening constant. TheFourier transform of the Coulomb potential in quasi-one-dimension is given by V�q�=4e2 ln�qa�,10 where the latticespacing a is used as a cutoff for the small q divergence. Inthe following calculations we focus on the long-wavelengthlimit q→0.

From the dielectric response, the loss function L�q ,��can be obtained via

L�q,�� = Im �−1�q,�� . �4�

Collective �plasmon� excitation appears as a large spike inthe loss function, which occurs when the real part of � van-ishes and the imaginary part of � is sufficiently small. Besidethe plasmon resonance, the system’s response may exhibitsingle-particle excitations at different frequencies �i.

Let us now focus on the dependence of the characteristicresonance frequencies �i on the size of the system, specifi-cally looking for a possible transition from quantum to clas-sical scaling. The excitation energies can be determined bythe zeros in the dielectric function and depend on the differ-

ence between the eigenenergies of occupied and unoccupiedstates. Let us give an analysis of the two limiting cases: forsmall system sizes and large system sizes, correspondingly.

For an infinite square-well potential of the length L, thesingle-particle eigenenergies are El= l22ta2L−2, where l=1,2 , . . . is an integer quantum number, and t= �2

2m�a2 is thecharacteristic energy scale in the system. For relatively smallsystem’s sizes L, the optical response of the system is ex-pected to be dominated by single electron transitions. In thiscase, the multiple poles in the loss function L�q ,�� are alsoexpected to scale as L−2. In contrast, the excitation spectrumfor a quasi-one-dimensional system in the limit of L→ andin the long-wavelength limit q→0 is known to be dominatedby a collective response at the plasma frequency,10

�p qa�0�ln�qa��1/2 + O�q2� , �5�

with �0=�2ne2 /m�a2, where n is the electron density. Keep-ing the number of electrons in the system fixed, the electrondensity scales as n�L−1, and the plasmon energy is expectedto scale as �p�L−1/2. This scaling behavior is also valid forthree-dimensional nanostructures, assuming scaling of onlyone dimension of a nanostructure, while keeping the othertwo dimensions fixed.

To summarize, we predicted two different scaling regimesfor frequencies of the most pronounced excitations in thesystem �i, with the changes of the system’s size L, while thenumber of electrons Nel is fixed. The first scaling regime is�i�L−2, in this regime dominant excitations have single-particle nature. We called this regime quantum because thescaling resembles scaling of the eigenenergies of a particle ina box. In the second scaling regime, the single-particle polesmerge into one plasmon resonance �p, which scales as �p�L−1/2. We denote this regime as classical.

In Fig. 1, the poles of the dielectric loss function are plot-ted as a function of the system’s size for a fixed number ofelectrons Nel=50, considering chain lengths with 4 to 64atomic sites. Each circle represents a peak in the loss func-tion, with the diameter of the circle proportional to itsstrength. The solid line is the weighted average over all thepoles, given by

�̄ =�i

�i � L�q,�i�

�iL�q,�i�

. �6�

As the length of the chain is increased, the finite-size energyspacings decrease, and the poles eventually merge into asingle plasmon feature.8 In the inset, the weighted loss func-tion is fitted by the quantum ��L−2� and classical ��L−1/2�scaling laws. From the figure, it is evident that there is asmooth transition from the quantum regime for small atomicchains to the classical regime for longer chains. The intersec-tion of the two fits yields a critical length scale Lcr, whichdepends on the electron density. For the parameters chosenfor Fig. 1, Lcr14a. Additionally, we check that the plasmonresonance given by Eq. �5� is recovered in the limit of longchains.

Now, let us investigate how the critical size Lcr dependson the electron density in the system. In Fig. 1, the number

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of electrons has been kept fixed, so the electron density n andthe dominant plasmon frequency �p both decrease with in-creasing system’s size. It may be difficult to reproduce thissituation with differently doped semiconductor nanorods11

and impossible with real atoms. In order to model realisticatom chains with different number of atoms,3 we keep theelectron density n constant and change the number of sites N.

In Figs. 2�a� and 2�b�, the loss function poles are plottedas a function of the system’s size for densities of 1 and 2electron�s� per site. For a fixed electron density, the single-particle excitation frequencies are expected to scale in thequantum regime �small L� as

�i = �ENel+i − ENel�/� � ��Nel + i�2 − Nel

2 L−2 � L−1, �7�

where Nel�nL and then approach a constant value �p in thebulk �L→�. The solid line in Fig. 2�a� represents the aver-age frequency given by Eq. �6�, and the dotted line repre-sents the plasmon resonance, which occurs at 6.1t for a den-sity of one electron per site.

For this case, the critical length is determined by thelength for which the dominant pole converges to within 1%of its L→ value. Using this criterion, the critical length isLcr17a for a filling of 1 electron per site. In Fig. 2�b�, theplasmon resonance is 8.3t and the critical length is Lcr24a for a filling of 2 electrons per site. For size L=20a,illustrated by Fig. 2�b�, one has twice electrons in the systemthan in Fig. 1�a�, but surprisingly, there is no establishedcollective response. This example illustrates the crucial dif-ference in the response of quasi-one-dimensional and three-dimensional nanostructures. For quasi-one-dimensional sys-tems, the critical length Lcr increases with the electrondensity n, whereas it is expected to decrease with increasingdensity in three dimensions. Below we present our analysisof this anomalous behavior of the critical length Lcr.

The characteristic critical distance for collisionless plasmais the distance when the averaging of the oscillating field fora moving particle happens.12 For a degenerate electron gas, itis the distance at which an electron travels at Fermi velocityvF during one period of the collective field oscillation Lcr2vF /�p. At the same time, the Landau damping, which isclosely connected with the spatial dispersion of plasma, oc-curs also for a critical wave vector kcr=2 /Lcr�p /vF. Onecan also formulate the criterion in a different way that thecollective excitations, i.e., plasmons, do not damp for thephase velocities in plasma vph=�p /kcr much larger than thetypical single-particle velocity vF, vph�vF.12 Since �p��nand the Fermi velocity scales differently with electron den-sity n in quasi-one- and three dimensions, the critical lengthdepends on the density of electrons as Lcr�n1/2 in quasi-one-dimension �Lcr�n−1/6 in three-dimensional case�, consistentwith the results shown in Figs. 2�a� and 2�b�.

We suggest to check our theoretical predictions experi-mentally by measuring the optical response of atom chainsmade of monovalent �such as potassium or sodium� and mul-tivalent atoms �for example, aluminum�. Atom chains shouldbe placed on nonmetallic surface. Based on our theoreticalanalysis, the formation of collective �plasmon� responseshould be observed for shorter chains made of monovalentatoms and for longer chains made of multivalent atoms. Onthe contrary, in the case of three-dimensional nanostructures,such as spherical clusters, the formation of plasmon collec-tive mode will be observed for smaller radius of clustersmade of multivalent atoms.

Let us now consider inhomogeneous quasi-one-dimensional systems made of different species of atoms. Al-ternatively, one can consider a chain of coupled artificialatoms made of semiconductor quantum dots with a con-trolled number of electrons. To model such systems, we as-sume an on-site potential that takes on alternating valuesbetween consecutive atomic sites. Starting with a small sys-tem to understand the underlying delocalization effects, the

10 20 30 40 50 6010

0

101

102

L

ω[t]

20 40 6010

0

101

102

L

ω[t]

Σω⋅L(ω)/ΣL(ω)ω

0/L2

ωN

/√ L

FIG. 1. �Color online� Low energy poles in L�q ,�� as a func-tion of the system’s size L �in units of the lattice spacing a� for afixed number of electrons Nel=50. The solid line is a weightedaverage of L�q ,�� in the frequency space. �=0.05t and q=0.05a−1. In the inset, we show the transition from quantum L−2 toclassical L−1/2 scaling.

10 20 304

5

6

7

8

9

10

11

L

ω[t]

(a)

ϖω

p−bulk

10 20 30 404

6

8

10

12

14

16

18

L(b)

ϖω

p−bulk

FIG. 2. �Color online� Low energy poles in L�q ,�� for constantelectron density n. The dotted line is the plasmon resonance L→. �=0.05t. �a� n=1 electron per site, �p=6.1t. �b� n=2 elec-trons per site, �p=8.7t.

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wave functions of the low-energy eigenstates in a doublewell are plotted in Fig. 3�a�. These illustrate the formation ofbonding and antibonding combinations, which in the thermo-dynamic limit of a multiwell chain merge into degenerateenergy bands. For clarity, the eigenfunctions in this figure areoffset by their energy gap with the ground state, �i−�0. Theeigenenergies of the lowest four states, i.e., bound stateswithin the well, are doubly degenerate and well separatedfrom higher states, reminiscent of the spectrum of an atom.As the eigenenergies approach the potential barrier height,the spacing between energies decreases and a transition oc-curs from localized to delocalized states, as observed in�4–�6.

For sufficiently large alternating on-site potentials andlow electron densities, the dominant low-energy excitationsof larger chains can be approximated by consecutive quasi-infinite wells, connected via perturbative tunneling matrixelements, . To illustrate this point, consider first the case oftwo wells. Using leading-order perturbation theory, theeigenstates of this system are �i�� =�i�� , where �i are theeigenenergies of isolated infinite wells. This is a good ap-proximation as long as � ��i−� j�. Then, for even number ofelectrons Nel in the system, the dominant transitions occurbetween the energy levels within the individual wells �iand �i+1. However, for odd number of electrons Nel, transi-tions are also allowed between bonding and antibondingstates �i�� and �i�� .

In Fig. 3�b�, the corresponding poles of the dielectric loss

function are shown for increasing number of electrons Nel inthe system. For odd numbers of electrons at low densities,the lowest excitation corresponds to the transition between�0 and �1 for one electron and between states �2 and �3 forthree electrons. In contrast, the higher-energy excitationsclose to �=15t correspond to transitions between energybands and are the lowest available excitation for commensu-rate fillings. With increasing number of electrons, a singlelow-energy pole starts to dominate the excitation spectrum.This excitation is a precursor of the collective plasmonmode, and it scales as �Nel, as indicated by the solid line.

The effects discussed for the two-well case are amplifiedby increasing the number of wells. For example, the neardegeneracy of the low-energy states increases with the num-ber of wells, such that for M identical wells, there will be Mstates with nearly degenerate energies. For commensuratefillings, the excitation spectrum is then dominated by transi-tions between the lowest two energy bands. As more elec-trons are added, intraband transitions, which occur on a scaleof , become more dominant.

In Fig. 4, the poles in the loss function of an 11-site chainare compared for �a� a uniform on-site potential and �b� analternating potential. Each circle represents a peak in the lossfunction, and the solid lines connect the poles with the maxi-mum loss. In the uniform case �Fig. 4�a�, there are severalenergy bands that increase linearly with increasing numberof electrons, as the difference in consecutive eigenenergiesEi+1−Ei� �i+1�2− i2 depends linearly on the quantum num-ber i. As more electrons are added and lower states are filled,transitions occur between eigenstates with higher energies.

In Fig. 4�b�, the low-energy poles in the dielectric lossfunction are shown for the case of an alternating on-site po-tential, leading to five wells of depth 20t. At low numbers ofelectrons, the peak absorption energy increases dramatically

1 2 3 4 5−10

0

10

20

30

40

Atomic Site

Wav

efun

ctio

n

(a)

ψ0

ψ1

ψ2

ψ3

ψ4

ψ5

ψ6

0 2 4 6 80

5

10

15

20

25

30

# of electrons

ω[t]

(b)

FIG. 3. �Color online� �a� Electronic wave functions of adouble-well structure with alternating on-site potential, Vos=30t,offset by �i−�0. The on-site potential is shown by the shaded re-gions. �b� Low-energy poles of the dielectric loss function for thispotential. The size of the circles represents the corresponding peakstrength.

0 5 10 15 20 25 300

5

10

15

20

# of electrons

ω[t]

Vos

=0t

(a)

0 5 10 15 20 25 300

5

10

15

20

# of electrons

ω[t]

Vos

=20t

(b)

FIG. 4. �Color online� Poles in the loss function of a chain withN=11 sites, �=0.05t, q=0.05a−1. �a� Uniform on-site potentialVos=0t. �b� Alternating on-site potential Vos=0 or 20t. The thin lineconnects the loss function peak with maximum intensity.

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whenever the number of electrons is a multiple of the num-ber of wells, making an analog of a “closed shell.” For lessthan five electrons, the loss function is dominated by transi-tions between the lowest energy states associated with eachwell. Between five and ten electrons, the dominant transi-tions are between the second energy states associated witheach well. For five electrons, the lowest set of eigenstates�shell� is occupied and the dominant transition is between thefirst and second energy of each well. The pattern repeatsagain for ten electrons, which corresponds to two electronsper well. Above ten electrons, the highest occupied energylevel is comparable to the well height so that the electronsare no longer strongly localized in the wells and the excita-tion energies approach the energies for the case with theuniform potential. This example illustrates the existence of“magic numbers” in arrays of alternated local potentials,which are expected to be most pronounced at the small num-ber of electrons limit.

In conclusion, we calculated the dielectric-response func-tion for quasi-one-dimensional nanostructures using linear-response theory and RPA in the long-wavelength limit q→0. We have discovered a smooth transition from the quan-tum L−2 to classical L−1/2 scaling as a function of the systemsize L for the fixed number of electrons Nel in the system. We

have determined the critical Lcr length at which the collectiveresponse is established for different electron densities n. Wefound the Lcr�n1/2 in quasi-one-dimensional nanostructures,opposite to Lcr�n−1/6 for three-dimensional nanostructures.We proposed an experiment to verify our theoretical predic-tions. We also studied inhomogeneous nanostructures, mod-eled by an alternating on-site potential. It was found thatelectronic bands are formed via tunneling between adjacentpotential wells. Depending on the number of electrons, intra-band or interband transitions dominate the dielectric lossfunction, leading to a characteristic sequence of “magic” fill-ings. Verification of these model predictions by more sophis-ticated ab initio calculations and by luminescence experi-ments, e.g., in �Alx ,Ga1-x�As layered heterostructures, isanticipated.

We are grateful to A.F.J. Levi for useful discussions andacknowledge the Center of Integrated Nanotechnology atLos Alamos and Sandia National Laboratories for its support.This work was carried out under the auspices of the NationalNuclear Security Administration of the U.S. Department ofEnergy at Los Alamos National Laboratory under ContractNo. DE-AC52-06NA25396.

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