oyalsocie ypublishing.o g/jou nal/ spa
Resea ch
Ci e his a icle: Downing CA, Weick G. 2020
Plasmonic modes in cylind ical nanopa icles
and dime s. P oc.R.Soc.A476: 20200530.
h ps://doi.o g/10.1098/ spa.2020.0530
Recei ed: 7 July 2020
Accep ed: 16 No embe 2020
Subjec A eas:
solid s a e physics, nano echnology, op ics
Keywo ds:
nanoplasmonics, nanopa icles, dime s,
quan um-size e ec s
Au ho o co espondence:
Guillaume Weick
e-mail: guillaume.w[email p o ec ed]
Plasmonic modes in cylind ical
nanopa icles and dime s
Cha les A. Downing1,2 and Guillaume Weick3
1Depa amen o de Física de la Ma e ia Condensada,
CSIC-Uni e sidad de Za agoza, 50009 Za agoza, Spain
2Depa men o Physics and As onomy, Uni e si y o Exe e , Exe e
EX4 4QL, UK
3Uni e si é de S asbou g, CNRS, Ins i u de Physique e Chimie des
Ma é iaux de S asbou g, UMR 7504, 67000 S asbou g, F ance
CAD, 0000-0002-0058-9746;GW,0000-0002-0617-5835
We p esen analy ical exp essions o he esonance
equencies o he plasmonic modes hos ed in
a cylind ical nanopa icle wi hin he quasi-s a ic
app oxima ion. Ou heo e ical model gi es us access
o bo h he longi udinally and ans e sally pola ized
dipola modes o a me allic cylinde wi h an a bi a y
aspec a io, which allows us o cap u e he physics
o bo h plasmonic nanodisks and nanowi es. We also
calcula e quan um mechanical co ec ions o hese
esonance equencies due o he spill-ou e ec ,
which is o ele ance o cylinde s wi h nanome ic
dimensions. We go on o conside he coupling o
localized su ace plasmons in a dime o cylind ical
nanopa icles, which leads o collec i e plasmonic
exci a ions. We ex end ou heo e ical o malism o
cons uc an analy ical model o he dime , desc ibing
he e olu ion wi h he in e -nanopa icle sepa a ion
o he esul an b igh and da k collec i e modes. We
commen on he eno maliza ion o he coupled mode
equencies due o he spill-ou e ec , and discuss
some me hods o expe imen al de ec ion.
1. In oduc ion
The op ical p ope ies o small me al clus e s ha e been
s udied h oughou he wen ie h cen u y [1], in a ield
which is now e e ed o as plasmonics [2]. Mode n
nanoplasmonics aims o con ine and con ol ligh a
he nanoscale, in an amalgama ion o pho onics and
elec onics [3]. I is en isaged ha applica ions will
a ise in a eas om da a s o age and mic oscopy o
2020 The Au ho s. Published by he Royal Socie y unde he e ms o he
C ea i e Commons A ibu ion License h p://c ea i ecommons.o g/licenses/
by/4.0/, which pe mi s un es ic ed use, p o ided he o iginal au ho and
sou ce a e c edi ed.
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ligh gene a ion and biopho onics [4–6]. In he las ew yea s, he sub ield o quan um plasmonics
has b anched away, whe eby quan um mechanical phenomena play a c ucial ole [7].
An in ensi ely s udied quasi-pa icle in plasmonics is he localized su ace plasmon (LSP),
a collec i e oscilla ion o conduc ion band elec ons, which a ises when a me allic nanopa icle
(NP) is i adia ed by ligh [2] o ho elec ons [8]. Explo ing how he esonance equency o
he plasmon changes depending on he geome y o i s hos ing NP is a undamen al ask o
he ield [9–14]. Recen ly, a numbe o g oundb eaking expe imen s [15–21] ha e p obed he
plasmonic esponse o me allic cylinde s, and in pa icula he limi ing cases o nanodisks and
nanowi es. Inspi ed by hese expe imen s, in his wo k, we de i e simple, analy ical exp essions
o he dipola plasmon esonances wi hin he quasi-s a ic limi ( alid when he dimensions o
he NP is smalle han he wa eleng h associa ed wi h he LSP esonance equency) in bo h he
longi udinal and ans e se pola iza ions ( ha is, along he cylind ical axis and pe pendicula o
i ). Ou model is based upon a calcula ion o he change in Ha ee ene gy o he NP due o he
collec i e displacemen o he alence elec ons. We assume ha he elec ons in he nanos uc u e
o m a body o app oxima ely uni o m densi y, which allows us o employ con inuum mechanics
and se up a simple equa ion o mo ion [22]. Impo an ly, ou analy ic heo y is alid o any
aspec a io o he cylinde , and as such is o ele ance o a wide ange o expe imen s. Ou wo k
he e o e complemen s p e ious heo e ical s udies o plasmonic cylinde s, which ha e ei he
employed he nanowi e app oxima ion [23–25], o ha e equi ed nume ics [26–31].
In ou model, he ine i able quan um co ec ions which a ise a he nanoscale a e add essed
by accoun ing o he so-called spill-ou e ec [32]. In his quan um size e ec , he esonance
equency is modi ied due o a p opo ion o elec ons spilling ou side o he small me allic
NP, hus lowe ing he a e age elec onic densi y inside he NP. This e ec a ises due o he
g ound-s a e many-body wa e unc ion, which de e mines he elec onic densi y, ha ing ails
which leak ou side o he sha p bounda y o he NP su ace, so ha a non-negligible numbe o
elec ons eside ou side o he clus e . The spill-ou e ec has been s udied his o ically in ela ion
o sphe ical NPs [22], and mo e ecen ly has been in es iga ed o plasmons in ul a-sha p g oo e
a ays [33].
Coulomb in e ac ions be ween LSPs housed in di e en NPs can gi e ise o collec i e
plasmons sp ead ou o e he combined nanos uc u e [34,35]. The s udy o collec i e plasmons
in NP a ays, including a chi ec u es buil om cylind ical NPs [36–40], has led o a weal h o
di e se physics, om plasmonic wa eguides [41,42] o ligh ha es e s [43,44] o analogues o
a opological insula o [45–48]. In his wo k, we a e conce ned wi h he simples example o a
coupled sys em, he NP dime [49–52], which cons i u es he building block o mo e complex
me as uc u es, and whe e insigh in o he na u e o coupled plasmons can be achie ed.
A se ies o expe imen s on nanoplasmonic dime s in he nea - ield coupling egime ha e
e ealed bo h b igh and da k plasmonic modes, whe e he dipole momen s a e o ien ed in-phase
o ou -o -phase, espec i ely [53–55]. In o de o accoun analy ically o such collec i e plasmonic
e ec s, we adap ou a o emen ioned heo y o he case o a dime o cylind ical me allic NPs.
We de i e simple exp essions o he b igh and da k mode esonance equencies o he sys em
as a unc ion o he in e pa icle sepa a ion, which allows o a clea desc ip ion o how he
plasmonic coupling scales wi h dis ance. Ou esul s supplemen heo ies o cylind ical dime s in
he li e a u e, which p edomina ely in ol e assump ions abou he aspec a io o he cylinde , o
equi e ime-consuming nume ical compu a ions [56–62]. We also commen on he spill-ou e ec
in he dime , and sugges some me hods o he expe imen al de ec ion o ou p edic ed e ec s.
This pape is o ganized as ollows. In §2, we calcula e he dipola esonances o a single
cylind ical NP and discuss hei espec i e decay a es. We ind he modi ica ions o he esonance
equencies due o he spill-ou e ec in §3. The heo y is ex ended o desc ibe collec i e e ec s in
a dime o cylind ical NPs in §4. Finally, we d aw some conclusions in §5.
2. Plasmonic modes in a single cylind ical nanopa icle
We conside a cylind ical NP o adius aand leng h L, con aining Ne alence elec ons wi h cha ge
−e<0andmassme(see he inse in igu e 1). We s a by neglec ing he elec onic spill-ou
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z
a
L
0
y
x
− L/ 2 + L/ 2
0.2
0.4
0.6
0.8
1.0
esonance equency/
w
p
0 2 4 6 8 10
L/a
w
0,
w
0,
disk
w
0,
wi e
0
0.2
0.4
0.6
0.8
2 4 6 8 10
L/a
w
0,
^
w
0,
^
disk
w
0,
^
wi e
1/ 2
(a)(b)
Figu e 1. Resonance equencies ω0,and ω0,⊥(solid ed lines), in uni s o he plasma equency ωp,asa unc iono he
aspec a io L/a o bo h he (a) longi udinal (c . (2.14)) and (b) ans e se (c . (2.20)) modes. Dashed g een lines: he disk limi
app oxima ions, om(2.15)and (2.21) o panels(a)and(b), espec i ely. Do edblue lines: he wi elimi app oxima ions, om
(2.16) and (2.22) o (a)and(b), espec i ely. Ho izon al dash-do ed line in (b): he asymp o ic esul ωp/√2, o L/a→∞.
Inse : ske ch o a cylind ical me allic nanopa icle o adius aand leng h L. (Online e sion in colou .)
e ec , and assume ha he densi y n( ) o alence elec ons is uni o m (wi h densi y n0) inside
he cylinde , and anishing ou side, i.e.
n( )=n0Θ(a− )ΘL
2+zΘL
2−z, (2.1)
whe e ( ,θ,z) a e he usual cylind ical coo dina es, and whe e Θ(x) is he Hea iside s ep unc ion.
Ou s a egy o ob ain he equencies o he plasmonic no mal modes along he longi udinal
(ˆ
z,α=) and ans e se (ˆ
,α=⊥) di ec ions1closely ollows he one p esen ed, o example,
in [22] o a sphe ical NP, which yields o he LSP esonance equency he well-known Mie
esul ωp/√3, wi h ωp he plasma equency.2We i s impose a igid shi uαo he elec on
dis ibu ion, which gi es ise o he displaced densi y n( −uα). Assuming ha uα=|uα|is small
wi h espec o he dimensions o he cylinde , we ha e n( −uα)≃n( )+δnα( ), wi h
δnα( )=−uα·∇n( ). (2.2)
We hen conside he esul ing change in he Ha ee ene gy (in cgs uni s)
δEα=e2
2d3 d3 δnα( )δnα( )
| − |, (2.3)
wi h espec o he equilib ium si ua ion. This quan i y gi es access o he es o ing o ce
Fα=− ∂
∂uα
(δEα)=−kαuα(2.4)
and o he esul ing sp ing cons an kα. The la e quan i y hen p o ides an exp ession o he
no mal mode equency
ω0,α=kα
Me
, (2.5)
whe e Me=Nemeco esponds o he o al elec onic mass.
Le us now conside he longi udinal (α=) and ans e se (α=⊥) pola iza ions each in u n,
which a ise om di e en elec onic dis ibu ion displacemen s uα.
1He e and in wha ollows, ha s designa e uni ec o s.
2No e ha a simila phenomenological app oach has been success ully applied by he au ho s o [63] o spin-dependen
dipole exci a ions, and excellen ag eemen was ob ained agains ime-dependen densi y unc ional heo y nume ical
calcula ions.
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(a) Longi udinal mode
We assume he longi udinal displacemen u=uˆ
z, such ha he change in densi y (2.2) is
δn( )=un0Θ(a− )δz−L
2−δz+L
2, (2.6)
whe e δ(x) is he Di ac del a unc ion. Equa ion (2.6) co esponds o a cha ge imbalance ha is
loca ed a he wo disks o adius aclosing he cylinde a z=±L/2 (c . he inse in igu e 1). In
o de o e alua e he modi ica ion o he Ha ee ene gy (2.3) due o he abo e densi y change,
we shall exploi he Laplace expansion o he New onian ke nel [64]
1
| − |=2
π
+∞

m=−∞∞
0
dkeim(θ−θ)cos k[z−z]Im(k <)Km(k >). (2.7)
He e, Im(x)andKm(x) a e modi ied Bessel unc ions o he i s and second kinds, espec i ely,
while <=min( , )and >=max( , ). Upon inse ing (2.6) and (2.7) in o (2.3), we a i e a
a se en-dimensional in eg al. A e ca ying ou he s aigh o wa d angula and Ca esian
in eg als, and using he ollowing esul o he double adial in eg al:
a
0
d a
0
d  I0(k <)K0(k >)=a2
2k2[1−2I1(ka)K1(ka)], (2.8)
we ind
δE=8π(en0u)2a3∞
0
dx
x2sin2L
2ax[1−2I1(x)K1(x)], (2.9)
which is ha monic in he displacemen u. E alua ing he i s e m in he abo e in eg al using
∞
0d sin2( )/ 2=π/2, and in eg a ing he second e m employing special unc ions, we ind
δE=4π(eun0)2a3πL
2a+4
3−gL
a. (2.10)
In he exp ession abo e, he unc ion g(x)isde inedas
g(x)=x
6x2+4K−4
x2−x2−4E−4
x2, (2.11)
whe e
K(x)=1
0
d
(1 − 2)(1 −x 2)and E(x)=1
0
d 1−x 2
1− 2(2.12)
a e he comple e ellip ic in eg als o he i s and second kinds, espec i ely. The mono onically
inc easing unc ion (2.11) has he ollowing asymp o ic expansions o small and la ge a gumen s:
g(x)≃4
3+(6ln2−1−2lnx)x2
4+O(x4), x1 (2.13a)
and
g(x)≃π
2x+1
2x−1
4x3+O(x−5), x1. (2.13b)
The esul (2.10), oge he wi h (2.4) and (2.5), hen yields he ollowing analy ic exp ession o
he esonance equency o he dipola longi udinal mode o he cylinde :
ω0,=ωp1+2a
πL4
3−gL
a. (2.14)
He e, he plasma equency o he conside ed me al is ωp=(4πn0e2/me)1/2, wi h he elec on
densi y n0=Ne/πa2L o he examined cylinde .
We plo in igu e 1a he longi udinal esonance equency (2.14) as a unc ion o he aspec a io
L/ao he cylinde as he solid ed line. As one can see om he igu e, ω0,is a mono onically
dec easing unc ion o he pa ame e L/a, wi h he limi ing alues limL/a→0{ω0,}=ωpand
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limL/a→∞{ω0,}=0, which coincide wi h he well-known asymp o ic esul s o a sphe oidal NP
[11]. Physically, he longi udinal mode so ens when he aspec a io o he cylinde inc eases,
since he a io o uncompensa ed cha ges o he compensa ed ones (by he ionic backg ound)
dec eases wi h inc easing L/a. We no e ha his end has been con i med expe imen ally [16].
We now conside he wo limi ing cases o (2.14), namely when he cylinde can be ea ed as
a nanodisk (L/a1) o a nanowi e (L/a1), and whe e insigh ul exp essions can be ob ained.
Le us i s examine he disk limi . Using he expansion (2.13a), (2.14) becomes
ωdisk
0,≃ωp1−L
4πa6ln2−1−2lnL
a,L
a1. (2.15)
Clea ly, his exp ession ends linea ly owa ds he plasma equency ωpin he ex eme pancake
limi (L→0); see he dashed g een line in igu e 1a. In he opposi e limi o a wi e, we ob ain
wi h (2.13b)
ωwi e
0,≃ωp8a
3πL,L
a1, (2.16)
which is plo ed as a blue do ed line in igu e 1a, showcasing he in e se squa e oo decay o
ze o equency.
(b) T ans e se mode
In o de o ha e access o he eigen equency o he ans e se dipola plasmonic mode, he e we
assume he a bi a y small displacemen u⊥=uˆ
x(see he inse in igu e 1), such ha he change
in elec on densi y (2.2) is
δn⊥( )=un0cos θδ( −a)ΘL
2+zΘL
2−z. (2.17)
Comple ing an analogous calcula ion as o ha o he p eceding case o he longi udinally
pola ized mode (c . §2a) leads o he ollowing equa ion o he change in he Ha ee ene gy (2.3):
δE⊥=8π(en0u)2a3∞
0
dx
x2sin2L
2axI1(x)K1(x). (2.18)
E alua ing he abo e in eg al hen yields
δE⊥=2π(eun0)2a3gL
a−4
3, (2.19)
whe e g(x) is de ined in (2.11). We hus ob ain an analy ic exp ession o he esonance equency
o he ans e se dipola plasmonic mode, using (2.4) and (2.5) wi h (2.19), as
ω0,⊥=ωpa
πLgL
a−4
3. (2.20)
We plo he ans e se esonance equency (2.20) in igu e 1bas he solid ed line, as a unc ion
o he aspec a io L/a. As is e iden om he igu e, ω0,⊥is a mono onically inc easing unc ion
o he pa ame e L/a, bounded by he wo limi s limL/a→0{ω0,⊥}=0 and limL/a→∞{ω0,⊥}=ωp/√2
( he la e limi is deno ed by he ho izon al dash-do ed line in he igu e). As is he case o
he longi udinal plasmonic mode, such asymp o ic limi s a e he same o a sphe oidal NP [11].
Con a y o he longi udinal mode shown in igu e 1a, he ans e se mode ge s ha de when he
aspec a io o he cylinde inc eases, since he a io o uncompensa ed cha ges ha si on he
longi udinal su ace o he cylinde o he compensa ed ones inc eases wi h inc easing L/a.
In igu e 1b, he limi ing cases o a nanowi e (L/a1, dashed g een line) and nanodisk (L/a
1, do ed blue line) a e also displayed, and ha e unc ional o ms which a ise di ec ly om (2.20)

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wi h he leading o de expansions (2.13). Explici ly, one inds
ωdisk
0,⊥≃ωpL
4πa6ln2−1−2lnL
a,L
a1 (2.21)
and
ωwi e
0,⊥≃ωp
√21−4a
3πL,L
a1. (2.22)
(c) Discussion: compa ison o sphe oids, sc eening e ec s, and damping a es
o he plasmonic esonances
In appendix A, we compa e ou analy ical esul s (2.14) and (2.20) o he LSP esonance
equencies o a cylind ical NP o he closed- o m exp essions o a sphe oidal pa icle wi h he
same aspec a io (e.g. [11]) and ind an excellen ag eemen . Such a co espondence be ween
bo h geome ies as been p e iously poin ed ou by Vene mo & Sih ola [65], who compa ed he
pola izabili y o a cylinde calcula ed by means o nume ical simula ions o ha o a sphe oid,
which is known analy ically [11]. The compa ison p esen ed in appendix A hus con i ms he
ele ance as well as he adequacy o ou app oach, which p o ides an analy ical unde s anding
o plasmonic modes o he cylinde geome y.
Thus a , ou app oach has neglec ed he possible dielec ic sc eening o he alence elec ons
by he delec ons (cha ac e ized by a dielec ic cons an d), which is o ele ance o noble
me al NPs, as well as he p esence o a dielec ic embedding medium (wi h cons an m). Fo
he sphe e geome y, he p esence o sc eening and he esul ing dielec ic misma ch no o iously
eno malizes [1] he Mie equency om ωp/√3 oωp/(d+2m)1/2. Wi hin ou heo e ical
app oach, i is s aigh o wa d o ealize ha when d≈m=, since he Ha ee ene gy (2.3)
is eno malized by a ac o −1, he esonance equencies in (2.14) and (2.20) ake on he same
exp essions, up o a eplacemen o ωpby ωp/√, leading o a edshi o he esonances. The
case d=mis much mo e in ol ed due o he complica ed o m o he Coulomb in e ac ion in
cylind ical coo dina es, e en wi hin he wi e limi [66], and is ou o he scope o he p esen wo k.
A inal commen is he e in o de abou he damping o he plasmonic exci a ions which we
ha e elucida ed hus a . Me allic nano-objec s a e subjec o adia i e and non- adia i e damping
mechanisms which b oaden he esonance o he collec i e exci a ion, such ha he o al decay
a e o he LSP modes a e gi en by γα=γ
α+γn (α=,⊥). Wi hin ou dipola app oxima ion,
he adia i e decay a es γ
αcan be eadily es ima ed om he elec omagne ic ield gene a ed in
he a - ield by a poin dipole [64] ca ying a cha ge −eNeand oscilla ing a he LSP esonance
equency ω0,α. E alua ing he o al powe adia ed by he dipole and he ene gy ini ially s o ed
in i , we ind
γ
α=ω2
pω2
0,α
6c3a2L, (2.23)
wi h c he speed o ligh in acuum. The adia i e damping a es γ
α hus depend on he cylinde
dimensions h ough he explici dependence a2Ldisplayed by he equa ion abo e, bu also
h ough he aspec - a io dependence o ω0,α( igu e 1), and inc eases wi h he dimensions o
he cylinde . Using he expansions (2.15), (2.16), (2.21) and (2.22), we ind o he longi udinal
mode γ ,disk
≃ω4
pa2L/6c3in he disk limi and γ ,wi e
≃4ω4
pa3/9πc3in he wi e limi , which,
in e es ingly, does no depend on L,sinceωwi e
0,goes o ze o o L/a1 (see (2.16)). Fo he
ans e se mode, we ind γ ,disk
⊥≃(6 ln 2 −1)ω4
paL2/24πc3and γ ,wi e
⊥≃ω4
pa2L/12c3.
The non- adia i e con ibu ion γn =γO+γL o he o al LSP linewid h, which is mode-
independen in a i s app oxima ion, can be di ided in o wo pa s. The i s pa co esponds
o he Ohmic, bulk-like con ibu ion γOwhich essen ially a ises om elec on–phonon and
elec on–elec on sca e ing. The expe imen s on single gold nano ods p o ec ed by a silica
shell o [16] epo a alue γO≈65 meV/¯
h. The second pa is he Landau damping decay
a e γL, a pu ely quan um-mechanical e ec [22,32] which comes om he con inemen o he
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elec onic eigens a es wi hin he NP, and which eads γL=A F/e , wi h Aa (ma e ial and
dielec ic en i onmen -dependen ) cons an o o de 1, F he Fe mi eloci y, and e an e ec i e
con inemen leng h. The expe imen s o [16] ha e shown ha e =(aL)1/2p o ides a good i
o he measu ed da a. In hese expe imen s on indi idual gold nano ods ha ing leng hs Lin
be ween 32 nm and 70 nm and adii ain he ange 4.3 nm o 11 nm, Landau damping was shown
o la gely domina e he size-dependen pa o he o al linewid h (which is in he 80–140 meV/¯
h
ange), while he maximal alue o he adia i e damping decay a e epo ed is only 15 meV/¯
h.
3. F equency eno maliza ion due o he spill-ou e ec
So a , ou app oach has been pu ely classical, and has neglec ed he spill ou o he elec onic
wa e unc ions ou side o he NP. This app oxima ion ollows om ou assumed ha d wall mean-
ield po en ial, esul ing in he app oxima e densi y o alence elec ons gi en by (2.1). Howe e ,
he quan um-mechanical spill-ou e ec is known o eno malize he LSP esonance equencies,
and is pa icula ly p ominen o NPs o only a ew nanome es in size [32]. We hus elax he
abo e ha d-wall app oxima ion, and assume ha he mean- ield po en ial (including bo h he
ionic posi i e backg ound and he elec on–elec on in e ac ions) seen by he alence elec ons o
he NP is gi en by
V( )=V0Θ( −a)Θ|z|−L
2, (3.1)
whe e V0=F+Wis he heigh o he po en ial, wi h Fand W he Fe mi ene gy and
he wo k unc ion o he NP, espec i ely. Such a hypo hesis has been es ed using densi y
unc ional ab ini io calcula ions using he local densi y app oxima ion in [67], and is a ai ly good
app oxima ion o he ealis ic mean- ield po en ial.
Due o he ini e heigh V0o he mean- ield po en ial (3.1), some pa o he alence elec ons
can spill ou o he cylind ical NP, e ec i ely inc easing i s leng h and adius acco ding o he
eplacemen s
L→˜
L=L+2,a→˜
a=a+⊥. (3.2)
He e, he small spill-ou leng hs Land ⊥ain he longi udinal (ˆ
z) and ans e se (ˆ
)
di ec ions, espec i ely, can be es ima ed om he a e age numbe o spill-ou elec ons Nand
N⊥in bo h o hese di ec ions acco ding o
=1
2
N
Ne
Land ⊥=1
2
N⊥
Ne
a. (3.3)
In he ollowing, we will es ima e Nand N⊥using semiclassical expansions, which will gi e
us access o he spill-ou leng hs and ⊥. We will hen inco po a e he p esc ip ion (3.2)
in o he mode equencies (2.14) and (2.20), which will hen p o ide us wi h an es ima e o he
eno malized esonance equencies.
(a) A e age numbe o spill-ou elec ons and spill-ou leng hs
A ze o empe a u e, he a e age numbe s o spill-ou elec ons in he longi udinal and ans e se
di ec ions a e gi en by
N=
occ

λ <a
|z|>L/2
d3 |ψλ( )|2and N⊥=
occ

λ >a
|z|<L/2
d3 |ψλ( )|2, (3.4)
espec i ely. He e, λlabels he bound s a es in he mean- ield po en ial (3.1) and he summa ions
un o e occupied s a es up o he Fe mi le el. The single-pa icle wa e unc ion ψλ( ) obeys he
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ime-independen Sch ödinge equa ion
−¯
h2
2me∇2+V( )ψλ( )=λψλ( ), (3.5)
wi h λ he co esponding eigenene gies. No e ha in (3.4), we dis ega d he negligible numbe
o spill-ou elec ons a ising a he co ne s o he cylind ical NP.
The choice o mean- ield po en ial (3.1) leads o a non-sepa able Sch ödinge equa ion (3.5).
Howe e , he eplacemen
V( )≃V0Θ( −a)+Θ|z|−L
2 (3.6)
is bo h an excellen app oxima ion o he o iginal V( ), wi h only he co ne s o he cylinde
de ia ing om he non-sepa able po en ial (3.1), and leads o an exac ly sol able p oblem.
Decomposing he sepa able po en ial (3.6) in o V( )=V ( )+Vz(z) wi h V ( )=V0Θ( −a)and
Vz(z)=V0Θ(|z|−L/2), he s a iona y Sch ödinge equa ion (3.5) hen eads
∂2
∂ 2+1
∂
∂ +1
2
∂2
∂θ2+∂2
∂z2+k2−2me
¯
h2V ( )+Vz(z)ψnm˜
n( )=0, (3.7)
whe e k=2me/¯
h2, and whe e mis he magne ic quan um numbe and n(˜
n) is he p incipal
quan um numbe due o he ans e se (longi udinal) mo ion. We sepa a e he a iables in (3.7)
using
ψnm˜
n( )=Fnm( ,θ)Z˜
n(z), (3.8)
and a e hus led o wo Sch ödinge equa ions in he educed eigen alues k and kz, espec i ely,
whe e k2=k2
+k2
z, whose solu ions a e gi en explici ly in appendix B. Using he esul s p esen ed
he e (see in pa icula (B10)), we a e hen able o e alua e he in eg als en e ing (3.4), which
a e app oxima ely gi en in he high-ene gy, semiclassical limi o k0a1andk0L1 (wi h k0=
(2meV0/¯
h2)1/2)by
 <a
|z|>L/2
d3 ψnm˜
n( )
2≃2
κzL
k2
z
k2
0
and  >a
|z|<L/2
d3 ψnm˜
n( )
2≃1
2κ a, (3.9)
whe e κ =(k2
0−k )1/2and κz=(k2
0−kz)1/2.
Upon subs i u ing he exp essions (3.9) in o (3.4), we hen eplace he summa ion o e he se
o quan um numbe s n,mand ˜
nby an in eg al o e wa e ec o k. We ake o he densi y o
s a es he leading-in-¯
hWeyl e m [68], which is app op ia e in he semiclassical limi kFa1and
kFL1 (wi h kF he Fe mi wa e ec o ). Fo ypical noble me als, such as Ag o Au, one has
kFa≃10 a[nm], so ha he semiclassical app oxima ion is sui able e en o nanome e-sized NPs
[69]. The a o emen ioned p esc ip ion leads o
N≃2V
(2π)3k<kF
d3k2
κzL
k2
z
k2
0
and N⊥≃2V
(2π)3k<kF
d3k1
2κ a, (3.10)
whe e he p e ac o o 2 accoun s o he spin degene acy and V=πa2Lis he olume o he
cylinde . Pe o ming he abo e in eg als in sphe ical coo dina es, we a i e a
N=k2
0a2
πkF/k0
0
dxx
4+1
−1
d 2
1−x2 2and N⊥=k2
0aL
4πkF/k0
0
dxx
2+1
−1
d 1
1−x2(1 − 2),
(3.11)
whe e k0>kF, and whe e he in eg als wi h espec o he dimensionless adial (x) and pola ( )
coo dina es a e ye o be pe o med. E alua ing he abo e in eg als (3.11), we ind he exp essions
N=(kFa)2
4πhF
V0and N⊥=k2
FaL
4πh⊥F
V0. (3.12)
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–1.0
−0.5
0
0.5
1.0
0 2 4 6 8 10
L/a
j
j
^
0.94
0.96
0.98
1.00
˜
w
0,
a
/
w
0,
a
0 0.025 0.050 0.075 0.100
1/k
F
a
0
0.1
0.2
0.3
0.4
0.25 0.50 0.75 1.00
k
F
^
k
F

F
/V
0

(a)(b)(c)
Figu e 2. (a) Spill-ou leng hs (3.15), scaled wi h he Fe mi wa e ec o kF, as a unc ion o he a io o he Fe mi ene gy o
mean- ield po en ial s eng h, F/V0.(b) Auxilia y unc ions jα(α=,⊥) om (3.17) as a unc ion o he aspec a io L/a.
(c) Reno malized esonance equencies ˜ω0,( iole lines) and ˜ω0,⊥(o ange lines) om (3.16) in uni s o he ba e equencies
ω0,αas a unc ion o he in e se size o he nanopa icle (do ed lines: F/V0=0.25; dashed lines: F/V0=0.50; solid lines:
F/V0=0.75), o L/a=1. (Online e sion in colou .)
He e, we ha e in oduced he auxilia y unc ions
h(x)=3
2−x1
x−1+2−3
2xa csin √xand h⊥(x)=1
√x+1−1
xa c anh √x.
(3.13)
Scaling he esul s (3.12) wi h he o al numbe o elec ons in he NP Ne=La2k3
F/3π, we ob ain
N
Ne=3
4kFLhF
V0and N⊥
Ne=3
4kFah⊥F
V0. (3.14)
Thus, he ac ion o spill-ou elec ons in bo h he longi udinal and ans e se di ec ions scales
wi h he in e se o he spa ial ex en o he cylinde (∝1/a,1/L), and so becomes inc easingly
impo an o pa icles wi h nanome ic dimensions.
Wi h he abo e esul s (3.14), we can now e alua e he spill-ou leng hs (3.3), which ead
kF=3
8hF
V0and kF⊥=3
8h⊥F
V0. (3.15)
Impo an ly, hese wo quan i ies do no depend on he NP dimensions Land a, and only on he
Fe mi ene gy F(o he Fe mi wa e ec o kF) and he dep h V0o he mean- ield po en ial (3.1).
The spill-ou leng hs (3.15) a e plo ed in igu e 2aas a unc ion o F/V0.Asonecansee om
he igu e, bo h o hese quan i ies smoo hly inc ease wi h he abo e-men ioned a io. Since kFis
ypically o he o de o 108cm−1 o alkaline o noble me als, and since F/V0is oughly o he
o de o 0.5 [32], he spill-ou leng hs (3.15) a e only o a ew en hs o an angs om. Howe e , as
we will see in he nex sec ion, such a iny sp ead o he elec onic wa e unc ions ou side o he
NP may ha e a non-negligible e ec on he LSP esonance equency.
(b) F equency edshi s due o he spill-ou e ec
We a e now in a posi ion o calcula e he eno malized esonance equency in he longi udinal
( ans e se) pola iza ion ˜ω0,(˜ω0,⊥) due o he spill-ou e ec . We accoun o he spill-ou o he
elec ons by ea ing he cylind ical NP wi h he e ec i e dimensions ˜
Land ˜
aas in (3.2). I ollows
om he di ec subs i u ion o hese e ec i e dimensions in o he esonance equencies (2.14)
and (2.20), which assumed ha d-wall con inemen o he alence elec ons, ha he eno malized
esonance equencies a e, o leading o de in he scaled spill-ou leng hs /Land ⊥/a(c . (3.15)),
gi en by
˜ω0,α≃ω0,α1−1+jαL
a
L−1−1
2jαL
a⊥
a,α=,⊥. (3.16)
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Appendix B. Sch ödinge equa ion wi h a cylind ical s ep po en ial
In his appendix, we p o ide de ails abou he bound-s a e solu ions o he Sch ödinge
equa ion (3.7), which enable us o e alua e semiclassically he a e age numbe o spill-ou
elec ons (3.4), in bo h he longi udinal and ans e se di ec ions.
Sepa a ing he a iables as in (3.8), he ans e se wa e unc ions Fnm( ,θ) a e subjec o he
ollowing Sch ödinge equa ion:
∂2
∂ 2+1
∂
∂ +1
2
∂2
∂θ2+k2
−2me
¯
h2V ( )Fnm( ,θ)=0. (B 1)
Wi h he ansa z Fnm( ,θ)=Rnm( )e
imθ/(2π)1/2, whe e he quan um numbe m∈Z, and wi h he
no a ion κ =(k2
0−k2
)1/2,whe ek0=(2meV0/¯
h2)1/2, one inds he ollowing bound s a e solu ions:
Rnm( )=Cnm ⎧
⎨
⎩
Jm(k ), ≤a,
Jm(k a)
Km(κ a)Km(κ ), >a,(B 2)
whe e Jm(x)andKm(x) a e he Bessel unc ions o he i s and second kinds, espec i ely. The
no maliza ion cons an in (B 2) is gi en by
Cnm =√2
aJm(k a)
Km(κ a)2
Km+1(κ a)Km−1(κ a)−Jm+1(k a)Jm−1(k a)−1/2
,(B3)
while he ans e se mo ion is subjec o ene gy quan iza ion ia he anscenden al equa ion
k Jm+1(k a)/Jm(k a)=κ Km+1(κ a)/Km(κ a), whose solu ions a e labelled wi h he quan um
numbe n.
The longi udinal wa e unc ions Z˜
n(z) en e ing (3.8) obey
d2
dz2Z˜
n(z)+k2
z−2me
¯
h2Vz(z)Z˜
n(z)=0, (B 4)
which is equi alen o he ex book quan um mechanics exe cise o a one-dimensional pa icle in
a squa e box [74]. The solu ions o (B 4) ha e ei he a symme ic (s) o an an isymme ic (a) pa i y,
which we speci y as Z˜
n(z)=Z˜
n,p(z), whe e he index p =(s, a). The e en bound s a e solu ions a e
gi en by
Z˜
n,s(z)=κz
1+κzL/2
⎧
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎩
cos kzL
2eκz(L/2+z),z≤−L
2,
cos (kzz),|z|<L
2,
cos kzL
2eκz(L/2−z),z≥L
2,
(B 5a)
whe e κz=(k2
0−k2
z)1/2>0. Simila ly, he odd solu ions ead
Z˜
n,a(z)=κz
1+κzL/2
⎧
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎩
−sin kzL
2eκz(L/2+z),z≤−L
2,
sin (kzz),|z|<L
2,
sin kzL
2eκz(L/2−z),z≥L
2.
(B 5b)
Bo h se s o eigen unc ions (B 5a)and(B5b) a e associa ed wi h an indi idual anscenden al
equa ion desc ibing he quan iza ion o ene gy due o he longi udinal con inemen , explici ly
an (kzL/2) =κz/kz(s modes) and an (kzL/2) =−kz/κz(a modes). The solu ions o hese equa ions
a e labelled wi h he hi d quan um numbe o he p oblem, ˜
n.

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Now ha he ull Sch ödinge equa ion (3.7) is sol ed, we p oceed wi h he e alua ion o he
in eg als en e ing (3.4), namely
Rin
nm =a
0
d |Rnm( )|2,Rou
nm =∞
a
d |Rnm( )|2(B 6)
and
Zin
˜
n,p =+L/2
−L/2
dz|Z˜
n,p(z)|2,Zou
˜
n,p =−L/2
−∞ ++∞
+L/2dz|Z˜
n,p(z)|2,(B7)
which desc ibe he p obabili y o inding he elec ons inside o ou side he cylind ical NP, in
ei he he ans e se o longi udinal di ec ions. Wi h (B 2), we ob ain o he ans e se in eg als
(B 6) he esul s
Rin
nm =Jm+1(k a)Jm−1(k a)−J2
m(k a)
Jm+1(k a)Jm−1(k a)−J2
m(k a)Km+1(κ a)Km−1(κ a)/K2
m(κ a)(B 8a)
and
Rou
nm =Km+1(κ a)Km−1(κ a)−K2
m(κ a)
Km+1(κ a)Km−1(κ a)−K2
m(κ a)Jm+1(k a)Jm−1(k a)/J2
m(k a).(B8b)
Simila ly, using (B 5), we ind o he longi udinal in eg als (B 7)
Zin
˜
n,p =1
1+κzL/2κzL
2+κ2
z
k2
0and Zou
˜
n,p =k2
z/k2
0
1+κzL/2.(B9)
In he high-ene gy semiclassical limi (k0a1, k0L1), which is well sui ed o he p oblem
a hand [69], we ind ha he exp essions (B 8) and (B 9) a e well app oxima ed by
Rin
nm ≃1, Rou
nm ≃1
2
1
κ a,Zin
˜
n,p ≃1andZou
˜
n,p ≃2
κzL
k2
z
k2
0
, (B 10)
which hen lead o (3.9).5
Appendix C. Toy model: wo coupled oscilla ing dipoles
In his appendix, we demons a e ha he esul s (4.8) and (4.13) o he esonance equencies
o he coupled modes in a dime o cylind ical NPs in he limi o la ge in e pa icle sepa a ion
dis ance (i.e. d/a1andd/L1) can be eco e ed om a oy model o wo coupled aniso opic
oscilla ing dipola momen s.
Le us conside wo ideal elec ic dipoles pi=−Nee i(i=1, 2), wi h i he associa ed
displacemen o he elec onic cloud wi h cha ge −Neeand mass Neme. The dime (wi h
in e pa icle dis ance d) is aligned along he z-di ec ion and each dipole oscilla es a he equency
ωdip
0,in he longi udinal (z) di ec ion and ωdip
0,⊥in he ans e se (x,y) di ec ions. The Lag angian
o he sys em desc ibed abo e eads
L=Neme
2
2

i=1˙
2
i−ωdip
0,⊥
2x2
i+y2
i−ωdip
0,
2z2
i−N2
ee2
d3 1· 2−3 1·ˆ
z 2·ˆ
z.(C1)
Using ha Nee2/me=ω2
pV/4π, wi h V he olume o he elec onic cloud, he Eule –Lag ange
equa ions o mo ion o he oy model (C 1) lead o he coupled mode esonance equencies
ωdip
τ,=ωdip
0,
2+2τω2
p
V
4πd3,ωdip
τ,⊥=ωdip
0,⊥
2+τω2
p
V
4πd3,τ=±.(C2)
5In his semiclassical limi , he leading o de exp essions (B 10) do no sa is y uni a i y, which equi es he inclusion o high-
o de e ms. Howe e , since he absen e ms a e o negligible impo ance o he ange o pa ame e s we conside in his
wo k we may omi hem. No ably, his semiclassical limi has been shown o be an excellen app oxima ion o a sphe ical
NP [69] and i has he signi ican ad an age o p o iding addi ional physical insigh .
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Wi h V=πa2L he olume o he cylinde conside ed in he main ex , he exp essions abo e
co espond o (4.8) and (4.13) wi h Ωgi en by (4.10) in he limi o d/a1andd/L1.
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