SPECIAL ISSUE - RESEARCH ARTICLE
Theo e ical ea men o single-molecule scanning Raman
picoscopy in s ongly inhomogeneous nea ields
Yao Zhang
1,2
| Zhen-Chao Dong
2
| Ja ie Aizpu ua
1
1
Ma e ials Physics Cen e , CSIC-
UPV/EHU and Donos ia In e na ional
Physics Cen e (DIPC), Donos ia-San
Sebas ián, 200018, Spain
2
He ei Na ional Labo a o y o Physical
Sciences a he Mic oscale and Syne ge ic
Inno a ion Cen e o Quan um
In o ma ion and Quan um Physics,
Uni e si y o Science and Technology o
China, He ei, 230026, China
Co espondence
Ja ie Aizpu ua, Ma e ials Physics Cen e ,
CSIC-UPV/EHU and Donos ia
In e na ional Physics Cen e (DIPC),
Donos ia-San Sebas ián 200018, Spain.
Email: [email p o ec ed]
Funding in o ma ion
Eusko Jau la i za, G an /Awa d Numbe s:
IT1164-19, KK-2019/00101; Minis e io de
Ciencia e Inno ación, G an /Awa d
Numbe : PID2019-107432GB-I00;
Na ional Na u al Science Founda ion o
China, G an /Awa d Numbe : 21804125
Abs ac
Tip-enhanced Raman spec oscopy (TERS) o a single molecule is commonly
desc ibed by conside ing he change in he pola izabili y o he molecule wi h
espec o a no mal coo dina e induced by homogeneous illumina ion. How-
e e , he local ields induced by nanoscale and a omic-scale ea u es a he su -
ace o me allic clus e s and nanogaps show s ong inhomogenei ies in hei
spa ial dis ibu ion, which induces b eaking o Raman selec ion ules. In his
con ex , he spa ial ex ension o he molecula elec onic s a es subjec ed o
s ongly a ying local ields challenges he alidi y o he poin –dipole app oxi-
ma ion as an adequa e desc ip ion o TERS in such con igu a ions. He e, we
in oduce a gene al ea men o simula e single-molecule TERS spec a and
hei ene gy- il e ed ib a ional inge p in s maps, in which he pola iza ion
p ope ies o he single molecule and ha o he op ical enhancing nan-
o esona o can be calcula ed sepa a ely and hen con enien ly combined o
ob ain he o al Raman c oss sec ion o he molecule unde he s ongly inhomo-
geneous ield. We apply he gene al me hod o s udy ip-enhanced scanning
Raman picoscopy o a 4,4
0
-bipy idine and biphenyl molecules in he p oximi y
o a sil e icosahed al clus e wi h a ew a oms a he ip apex mimicking an
enhancing picoca i y. The pola iza ion o he molecules is calcula ed wi hin
densi y unc ional heo y (DFT), and he op ical esponse o he ip is calcula ed
wi hin a classical a omis ic disc e e–dipole app oxima ion. The Raman spec a
a e ound o be ex emely sensi i e o he spa ial dis ibu ion o he local ields
and o he o ien a ion o he molecule. Ou calcula ions show ha he spa ial
mapping o molecula ib a ional inge p in s, as p obed by a ip wi h a omic
p o usions, is capable o e eal in amolecula ea u es o a single molecule in
eal space and hus es ablish a obus basis o scanning Raman picoscopy.
KEYWORDS
nanoan ena, picoca i y, plasmonics, scanning Raman picoscopy, su ace-enhanced Raman
spec oscopy, ip-enhanced Raman spec oscopy
Addi ional suppo ing in o ma ion ela ed o he de ailed de i a ion o he molecula pola izabili y exp essed in e ms o he a omic con ibu ions and
he ib a ional modes o he biphenyl molecule wi h hei co esponding elec ic ield dis ibu ions may be ound online in he Suppo ing
In o ma ion sec ion a he end o his a icle.
Recei ed: 30 Ap il 2020 Re ised: 28 Augus 2020 Accep ed: 2 Sep embe 2020
DOI: 10.1002/j s.5991
This is an open access a icle unde he e ms o he C ea i e Commons A ibu ion-NonComme cial License, which pe mi s use, dis ibu ion and ep oduc ion in any
medium, p o ided he o iginal wo k is p ope ly ci ed and is no used o comme cial pu poses.
© 2020 The Au ho s. Jou nal o Raman Spec oscopy published by John Wiley & Sons L d
296 J Raman Spec osc. 2021;52:296–309.wileyonlinelib a y.com/jou nal/j s
1|INTRODUCTION
Since he concep o ip-enhanced Raman spec oscopy
(TERS) was i s p oposed in 1985
[1]
and expe imen ally
demons a ed in 2000,
[2-5]
his echnique has become a
powe ul ool o ob ain chemical iden i ica ion o molecu-
la species in he nanoscale, showing bo h high spec al
sensi i i y and exquisi e spa ial esolu ion when com-
bined wi h scanning unneling mic oscopy (STM) o
a omic o ced mic oscopy (AFM),
[6-13]
pa icula ly when
ope a ed a ul ahigh- acuum and low- empe a u e con-
di ions.
[14-16]
These unique p ope ies a e o igina ed om
he nanome ic localiza ion and enhancemen o local
elec omagne ic ields a he apex o a me allic p obing
ip, which o en o ms a ca i y be ween he me allic ip
and he subs a e, leading o he exci a ion o localized
su ace plasmons (LSPs).
[17]
A a ge molecule o molecu-
la laye loca ed benea h he ip o inside he ca i y expe-
iences he enhancemen o he local ield bo h om he
incoming adia ion, as well as om he ou going adia-
ion; hus, he enhancemen o he Raman signal is ound
o scale wi h he ou h powe o he local ield enhance-
men .
[18-20]
This elec omagne ic ield enhancemen can
be in ensi ied o dec eased due o o he e ec s connec ed
wi h chemical pola iza ions o he molecule-subs a e
sys em
[21]
o op omechanical eedback mechanisms,
[22-25]
which ha e been ecen ly explo ed as addi ional enhanc-
ing mechanism o molecula Raman sca e ing in
plasmonic ca i ies.
The Raman signal is ex emely sensi i e o he spa ial
dis ibu ion o local ields induced by he incoming ligh
ha p obes he molecules; he e o e, i u ns o be o pa -
amoun impo ance o de elop an accu a e quan i a i e
es ima ion o he op ical nea - ield esponse in speci ic
TERS con igu a ions. A a ie y o heo e ical me hods
including classical and quan um app oaches ha e been
adop ed o es ima e he local ield and he co esponding
Raman signal enhancemen .
[26-28]
Bo h classical and
quan um me hods o en adop a smoo h and con inuous
desc ip ion o me allic in e aces and molecula objec s
o ob ain he dis ibu ion o local ields based on he solu-
ion o Maxwell's equa ions o o he Sch ödinge equa-
ion, espec i ely. Among he classical me hods o ob ain
local ields, he ini e-elemen me hod (FEM),
[29,30]
he
ini e-di e ence in ime-domain (FDTD) me hod,
[31,32]
and he bounda y elemen me hod (BEM),
[33-36]
among
o he s, ha e been commonly applied o me allic
nanoan enas and ips. In all o hese me hods, he local
ield p ope ies a e usually de e mined by he smoo h
bounda ies o a nanoscale mo phology o he nanos uc-
u e ( ip o ip-subs a e ca i y) wi h a mo e o less
sophis ica ed desc ip ion o he oughness. E en in quan-
um desc ip ions o he op ical esponse o me allic
ca i ies, a jellium model
[37-41]
o he elec on gas, which
elies on a smoo h desc ip ion o he elec onic densi y a
he me al- acuum in e ace, is o en conside ed as a alid
app oach o ob ain he op ical pola iza ion o me allic
nano esona o s wi hin he use o he ime-dependen
densi y unc ional heo y (TDDFT). Beyond hese con in-
uous app oaches, a omis ic models ha e e he op ical
esponse o ma e o i s a omic cons i uen s p o ide a
oadmap o e eal he ole o subnanome ic ea u es in
me allic ca i ies and ips ha in oduce signi ican di e -
ences in he local ield dis ibu ions a ound he
nano esona o s ( ips and/o gaps) and hus on i s ac ion
on o he molecules deposi ed nea by.
[42-45]
Recen s udies ha e shown he abili y o TERS o
achie e submolecula esolu ion in he Raman signal o
di e en ib a ional inge p in s o a single molecule a
low empe a u e and ul ahigh acuum
[16]
and e en
iden i y wo di e en adjacen molecules.
[46]
The spa ial
esolu ion o single-molecule TERS has now eached
he Ångs öm le el, d i en by he de elopmen o he
scanning Raman picoscopy (SRP) echnique,
[47,48]
which
elies on he exquisi e con ol o he localiza ion o nea
ields a a ew a oms on he apex o he me allic scan-
ning ips, he so-called pico esona o o picoca i y,
capable o esol e he ib a ions o a single chemical
bond. Rema kably, his ex eme le el o ield localiza-
ion does no only occu a he apex o a ip in scan-
ning p obe mic oscopy con igu a ions bu is also
p esen in s anda d su ace-enhanced Raman spec os-
copy (SERS) o molecules loca ed in na ow gaps.
[49]
A omic-scale p o usions in such plasmonic gaps a e
also capable o p oducing picoca i ies ha localize he
induced nea ields o he a omic scale and hus p obe
single bonds o molecules and induce b eaking o
Raman selec ion ules.
[25,50,51]
In o de o add ess his ex eme localiza ion o ligh ,
i is necessa y o implemen a me hodology ha con-
side s he a omis ic na u e o bo h he me allic
nanoclus e and he single molecule in o de o accoun
o he s ong inhomogenei y o he nea - ield dis ibu-
ion in he desc ip ion o he Raman signal. Recen ly,
Ba b y e al. p oposed a ull quan um a omis ic calcula-
ion o he plasmonic esponse o a me allic ca i y based
on TDDFT,
[28]
which showed he possibili y o localiza-
ion o plasmonic local ields below 1 nm
3
a a omic-scale
e ices and edges, simila o hose exis ing in scanning
ips. In e es ingly, i has been shown ha hese a omic-
scale ea u es o he plasmonic esponse can be eason-
ably well ep oduced wi h he use o a “classical”
a omis ic app oach ha ollows he quan um mechanical
elec on densi y p o ile o he su ace a oms,
[52]
hus ali-
da ing he op ical nea - ield esponse ob ained o me al-
lic clus e s and nanopa icles by o he classical s udies
ZHANG ET AL.297
such as hose based on he disc e e–dipole app oxima ion
(DDA).
[43-45]
A single molecule posi ioned in a plasmonic ield
localized a he a omis ic scale shows a beha io o he
pola iza ion o ally di e en om ha o a molecule
loca ed in an homogeneous ield. I he spa ial dis ibu-
ion o he local ield is compa able o e en smalle han
he size o he p obed molecule, he di e en a oms
wi hin he molecule will be subjec ed o di e en local
ield ampli udes and di ec ions, esul ing in he b eak-
down o Raman-ac i e c i e ia and selec ion ules based
on molecula symme ies.
[53]
Di e en s a egies ha e
been adop ed o desc ibe he in luence o local ields on
he pola iza ion o a molecule, which include e alua ing
he in eg als be ween he molecula wa e unc ions and
he local ield ampli udes o desc ibe he Raman in e ac-
ion. In his con ex , we can ci e o ins ance he
Gaussian- ype ield ep esen a ion
[54]
o he locally in e-
g a ed Raman pola izabili y densi y me hod.
[45]
Howe e ,
hese me hods o en equi e a comple e calcula ion
whene e he molecula sys em o i s ela i e con igu a-
ion wi hin he ca i y is modi ied, jeopa dizing he possi-
bili y o ob aining massi e Raman maps o molecula
sys ems wi h gene ali y.
He e, we in oduce a simpli ied gene al ea men o
single-molecule Raman sca e ing in he p esence o a
gene al nano esona o ha localizes ligh , including he
ex eme case o localiza ion induced by a pico esona o .
Wi hin his ea men , he op ical esponse o he nano-
s uc u e is i s calcula ed by s anda d classical elec o-
magne ic simula ion me hods (e.g., FEM, BEM, o
TDTD), and he pola iza ion o he single molecule is
calcula ed wi h he use o quan um chemis y packages
(e.g., Gaussian, VASP, o QChem). We i s calcula e he
G eens unc ion and local elec ic ield o he nan-
o esona o o choice, conside ing he pa icula inciden
illumina ion used in he Raman sca e ing p ocess. The
molecula pola izabili y de i a i es a e calcula ed om
he a omic con ibu ions o he molecule in o de o
compose he Raman dipole momen , and om he o al
Raman dipole, one can ob ain he Raman sca e ing
c oss sec ion. The Raman spec a a e hen ob ained by
summing up he con ibu ion om all he ib a ional
modes. As an example o he p ocedu e, he e we adop
a ully a omis ic model o desc ibe bo h he op ical
esponse o a me allic p obing ip and ha o he Raman
dipole momen o he molecule. This app oach allows
o exposing he e ec o a omic-scale ea u es and in e -
ac ions h ough he ou pu o he Raman signal, e eal-
ing ex eme spec al and spa ial dependencies o he
ib a ional inge p in s on hese ea u es. The ip is
desc ibed as an Ag
1415
nanoclus e wi h he use o
an a omis ic DDA me hod o ep oduce he s ongly
inhomogeneous ields a ound he a oms o he ip apex.
Two di e en single molecules (4,40-bipy idine [44BPY]
and biphenyl) a e conside ed, and hei chemical com-
posi ion is desc ibed a a quan um a omis ic le el wi hin
DFT. The molecules a e loca ed unde he s ongly inho-
mogeneous local ields o he sil e apex wi h speci ic
ealisi ic o ien a ions. The Raman signal o he single
molecules can be e ie ed om he ull a omis ic calcu-
la ion ha conside s he inhomogenous illumina ion
induced by he local ields o he sil e apex. We analyze
he Raman spec a o di e en molecula o ien a ions
and simula e he eal-space ene gy- il e ed Raman maps
o single molecules and molecula dime s as ob ained by
heo e ically scanning he ip o e he molecules. This
ea men o Raman sca e ing o single molecules nea
a picoca i y can be gene alized o any molecule and any
nanos uc u e shape as i elies on he sepa a e calcula-
ion o hei op ical p ope ies independen ly. Such a
gene al p ocedu e can enable sys ema ic and massi e
calcula ion o expe imen al Raman images o a a ie y
o molecules as ob ained wi h SRP and hus allows o
in e p e a ion o single-molecule Raman sca e ing a
in amolecula le el.
2|THEORETICAL TREATMENT
We in oduce he e he gene al heo e ical ea men o
add ess he Raman sca e ing c oss sec ion, dσRaman
k=dΩ,
unde inhomogeneous illumina ion in a sys em com-
posed by a gene ic nano esona o and a molecule. The
low cha o he o e all simula ion p ocedu e is shown
in Figu e 1. In sho , he op ical esponse o he nan-
o esona o can be calcula ed by any me hod o elec o-
magne ism, om which we can ob ain he wo-poin
G een's unc ion G
$ð , 0Þo he sys em ela ing posi ions
and 0as well as he local elec ic ield Eloc ðÞinduced
by he ex e nal illumina ion a posi ion . The op ical
p ope ies o he molecule can be ob ained om a a ie y
o me hods o quan um chemis y, om which he de i -
a i es o he a omis ic molecula pola izabili y, ∂α
mol
/∂ξ
n
,
co esponding o each a omic coo dina e ξ
n
o a om
ncan be ob ained. These wo sepa a ed calcula ions can
hen be combined o p o ide he a omis ic Raman
dipole con ibu ion o each a om n o he o al Raman
dipole momen pRaman
k. The emission o he Raman dipole
momen co esponding o a gi en k h ib a ional mode
o he molecule, sca e ed ou by he nano esona o o
he a ield, can be desc ibed h ough he co esponding
G een's unc ion, G
$
kð ∞, nÞ. The o al Raman spec um
can be ob ained by adding up he con ibu ion om all
he ib a ional peaks. The de ails o his model will be
explained in he ollowing sec ions.
298 ZHANG ET AL.
2.1 |A omis ic classical
elec odynamical calcula ion o he p obe
local nea ield
We i s ocus on he op ical esponse o a me allic nano-
pa icle as a p obing nano esona o ha p oduces he
local dis ibu ion o nea ields. We conside a classical
a omis ic app oach o calcula e he nano esona o 's
esponse, bu any o he me hod ha p ope ly accoun s
o he a omic-scale ea u es a he su ace o he nan-
o esona o is equally alid. In his a omis ic desc ip ion,
he plasmonic esponse o he nano esona o can be
ela ed o he pola iza ion o each cons i uen a om,
including he dipole–dipole in e ac ions be ween he di -
e en a omic dipoles. This me hod is he so-called
DDA.
[55]
The a omis ic pola izabili y o each i h a om, α
i
,
depends on he pa icula ma e ial, and he elec o-
magen ic in e ac ion be ween wo a oms is es ima ed
om hei poin –dipole in e ac ion o simplici y. The
pola ized a omic dipole momen a each i h a om o he
nano esona o , p
i
, induced by he elec ic ield ac ing on
he a om, E
i
, can be exp essed as pi=αiEi. The elec ic
ield ha each a om expe iences no only con ains he
inciden elec ic ield Einc
ibu also includes he ield p o-
duced by he o he a omic dipole momen s as
Ei=Einc
i−X
j≠i
Aijpj=α−1
ipi,ð1Þ
whe e Aij =−3^
ij
^
ij −I
= 3
ij deno es he dipole–dipole
in e ac ion be ween a oms iand jand
ij
is he dis ance
be ween hose a oms, wi h he di ec ion o he dis ance
deno ed by he uni ec o ^
ij . We de ine he diagonal
e m Aii =α−1
i, and he abo e equa ion can be w i en as
a linea se o equa ion o each i h a om o he o al
numbe o a oms o he nano esona o , N,as
X
N
j=1
Aij pj=Einc
i:ð2Þ
Equa ion (2) can be sol ed nume ically o ob ain he
sel -consis en a omis ic pola iza ions, p
i
, and hence, he
abso p ion c oss sec ion, σ
abs
, o he nano esona o can
be exp essed in e ms o hese induced a omis ic dipole
momen s as
σabs =4πω
cIm X
N
i=1
piEinc
()
=Einc
2,ð3Þ
whe e Im{} is he imagina y pa o {}, Nis he o al num-
be o a oms in he nano esona o , ωis he equency o
he inciden ield, and cis he speed o ligh .
The local elec ic ield, E
loc
( ), induced a a gene ic
posi ion can be es ima ed om he sum-up o he a om-
is ic dipole con ibu ions as
Eloc ðÞ=X
N
i=1
3pi − i
ðÞ½ − i
ðÞ
− i
jj
5−
pi
− i
jj
3
!
,ð4Þ
whe e
i
is he posi ion o each a om in he nan-
o esona o . The alue o E
loc
is one o he key ing edien s
FIGURE 1 Flow cha o he
p ocedu e o simula ion o single-molecule
Raman sca e ing. See he ex o an
explana ion o he magni udes calcula ed
wi hin each s ep o he app oach
ZHANG ET AL.299
o calcula e he Raman c oss sec ion o a molecula in-
ge p in loca ed in he icini y o he nano esona o , as
obse ed in he low cha o he p ocedu e in Figu e 1.
2.2 |Calcula ion o he molecula
a omis ic Raman pola izabili ies
wi hin DFT
In o de o accoun o he inhomogenei y o he local
nea ields ac ing on he di e en a oms o a molecule,
we will exp ess he o al Raman dipole o a molecule in
e ms o he a omis ic Raman pola izabili ies. We i s
conside he induced dipole momen o a single molecule,
p
ind
, pola ized by an inciden elec ic ield, which can be
w i en as
pind =αmolEinc =αmol
0Einc +X
k
∂αmol
∂Qk
0
QkEinc +oQ
2
k
,
ð5Þ
whe e α
mol
is he pola izabili y o he single molecule,
E
inc
is he inciden elec ic ield, and Q
k
is he no mal
coo dina e o he k h ib a ional mode. The second e m
o Equa ion (5) co esponds o he Raman sca e ing p o-
cess, om which we can de ine he Raman dipole
momen , pRaman
k, and he Raman pola izabili y, αRaman
k,as
pRaman
k=∂αmol
∂Qk
0
QkEinc αRaman
kEinc:ð6Þ
The de ailed desc ip ion o he de i a ion o he
molecula pola izabili y exp essed in e ms o he a omic
con ibu ions can be ound in he Supplemen a y In o -
ma ion. In b ie , he Raman pola izabili y, usually
exp essed in e ms o no mal coo dina es Q
k
, can be
e e ed o he a omic coo dina es ξ
(n)
o each a om no
he molecule, by applying a coo dina e ans o ma ion.
Hence, one can exp ess he Raman pola izabili y as
αRaman
k=∂αmol
∂Qk
0Qk=ffiffiffiffiffiffiffiffiffiffiffiffiffi
ℏ
2μkωk
X
M
n=1
ϕnðÞ
k
∂αmol
∂ξn
ðÞ
0
X
M
n=1
αn
ðÞ
k
ð7Þ
wi h M he numbe o he a oms in he molecule, μ
k
and
ω
k
he educed mass and wa enumbe o he k h ib a-
ional mode, ϕðnÞ
k he no malized displacemen o he n h
a om co esponding o he k h ib a ional mode, and ξ
(n)
he coo dina e o he n h a om. F om Equa ion (7), we
de ine αðnÞ
kas he a omis ic Raman pola izabili y o he
n h a om co esponding o he k h ib a ional mode.
Unde his ans o ma ion, we can unde s and he
Raman pola izabili y o a molecule as a esul o he con-
ibu ions o he Raman pola iza ion o each a om, αðnÞ
k.
The a omis ic Raman dipole momen o he n h a om in
he molecule o he k h ib a ional mode, p
n,k
, induced
by an inciden elec ic ield E
inc
, can hus be w i en as
pRaman
n,k=αnðÞ
kEinc:ð8Þ
In o de o ob ain he de i a i e o he pola izabili y
co esponding o each a omic coo dina e, as well as he
educed mass μ
k
, ib a ional wa enumbe ω
k
, and no mal-
ized displacemen ϕnðÞ
k o be used in Equa ion (7), we can
adop any easonable scheme wi hin molecula quan um
chemis y calcula ions. In pa icula , we use he e he
DFT package implemen ed in he Gaussian09 so wa e
[56]
wi h he hyb id unc ional B3LYP and he 6-31G(d) basis
se o calcula e all he ele an ib a ional pa ame e s
and hence ob ain he a omis ic ull Raman pola izabili y
enso o each a om o he molecules unde s udy.
2.3 |Calcula ion o inhomogeneous
ield-enhanced Raman sca e ing
In his sec ion, we ou line he me hodological app oach ha
allows o calcula ing he Raman c oss sec ion o a molecule
subjec ed o a s ongly inhomogeneous ield. I he single
molecule is loca ed in an inhomogeneous local elec ic ield,
he a omis ic Raman dipole momen o he n h a om o he
molecule o he k h ib a ional mode induced by he local
elec ic ield Elocð nÞcan be de ined as
pRaman
n,k=αnðÞ
kElocð nÞ:ð9Þ
The local ield induced by he nano esona o on each
a om o he molecule, Elocð nÞ, can be ob ained om he
solu ion o Equa ion (4), which, wi hin he a omis ic
desc ip ion p esen ed in his sec ion, can be w i en as
Elocð nÞ=X
N
i=1
G
$ n, i
ðÞpi=X
N
i=1
G
$
niX
N
j=1
A−1
ij Einc
jðωÞ,ð10Þ
whe e G
$ n, i
ðÞG
$
ni is he G een's unc ion o he
dipole momen p
i
and ωis he equency o inciden
ligh . The alue o Elocð nÞcould be de i ed om any
o he me hod o sol e Maxwell's equa ion, as a as he
a omis ic ea u es on he su aces o he nano esona o
a e p ope ly desc ibed in he bounda y condi ions.
[52]
300 ZHANG ET AL.
The ac ion o he local ield on each a om nallows o
de e mining he a omis ic Raman pola izabili ies
olllowing Equa ion (9). Hence, he ield p oduced by he
con ibu ion om all he a omis ic Raman dipoles o he
molecule a an a bi a y posi ion can be exp essed as
Emol ðÞ=X
M
n=1
G
$
k , n
ðÞpRaman
n,k,ð11Þ
whe e G
$
k , n
ðÞis he G een's unc ion o he a omis ic
Raman dipole momen pRaman
n,kwi h adia ion equency
(ω−ω
k
). In pa icula , he a - ield adia ion o he pola -
ized molecule, Emol
a ∞
ðÞ, can be also exp essed as a sum
o he con ibu ions om each a om in he molecule:
Emol
a ∞
ðÞ=X
M
n=1
G
$
k ∞, n
ðÞpRaman
n,k,ð12Þ
which allows o de e mining he Raman sca e ing c oss
sec ion as
dσRaman
k
dΩ/X
M
n=1
G
$
k ∞, n
ðÞ
pRaman
n,k
2
:ð13Þ
In his exp ession, he nanoan enna e ec induced
by he nanoclus e is wo old. On he one hand, he
a - ield G een's unc ion in Equa ion (13) p o ides an
enhancemen o he adia ion o he molecula dipole.
On he o he hand, he a omis ic Raman dipole
momen is also enhanced by he local ield acco ding o
Equa ion (9). Bo h con ibu ions p o ide he well-
known jE
loc
j
4
dependence on he local ield o he inal
Raman in ensi y. In he ollowing sec ion, we will show
he esul s o his me hod when applied o calcula e he
Raman sca e ing o a molecule enhanced by a sil e
clus e as a p obing ip.
3|SCANNING RAMAN
PICOSCOPYOFASINGLE
MOLECULE
We show now he applica ion o he me hod desc ibed
abo e o calcula e he Raman spec um o an exempla y
molecule in he s ongly inhomogeneous ield o a
picoca i y p oduced by he ew a oms in a me allic clus-
e . The Raman signal o such a sys em allows o pe -
o ming SRP, as shown below. We i s p o ide he
de ails o he op ical esponse o bo h picoscale esona o
and molecule o la e add ess i s Raman sca e ing spec-
um and picoscopy.
3.1 |Plasmonic esponse o a picoca i y:
A sil e nanoclus e
We i s conside he ypical con igu a ion o a picoscale
esona o composed o a ew p o uding a oms in an Ag
clus e . This ype o con igu a ion is simila o he one
o en ound in scanning p obe mic oscopy ips, o
ins ance. As poin ed ou in he me hodological sec ion,
any me hodology o sol e Maxwell's equa ions in such an
a omic-scale sys em could be used o p o ide he solu ion
o he a - ield and nea - ield esponse o he clus e .
He e, we choose he classical DDA o pu he emphasis
on he a omis ic aspec s o he ield localiza ion. The clus-
e con igu a ion is buil up om a egula icosahed on
wi h 1415 sil e a oms, whe e each a om is ega ded as a
sphe e wi h an a omic adius = 1.598 Å, and a c ys al
la ice cons an (a= 4.0897 Å). The co esponding a omic
pola izabili y, shown in he inse o Figu e 2b, is deduced
om he Clausius–Mosso i ela ionship wi h he expe i-
men al op ical pa ame e s o sil e .
[57]
Figu e 2b shows a
ypical abso p ion spec um o his nanoclus e , as calcu-
la ed om DDA calcula ions ollowing he p ocedu e
desc ibed in he me hodological sec ion. A esonan peak
can be obse ed a 3.47 eV when he elec ic ield o an
inciden planewa e is pola ized along he z-axis (see axis
in Figu e 2a). The co esponding local elec ic ield dis i-
bu ions a e shown in Figu e 2c–h o di e en ene gies
and e alua ion planes. I he equency o he inciden
elec ic ield is a om he esonan peak (o - esonan
condi ion, o ins ance a ω= 2.0 eV), he local elec ic
ield in he yz-plane (clus e side c oss sec ion as shown
in Figu e 2c) exhibi s a gene al dipola ea u e wi h
clea ly isible ield enhancemen close o he apex o he
nanoclus e , as “ho -spo s.”The spa ial con inemen o
hese ho -spo s is a he a omic scale, clea ly con ined o
he las a om o pai o a oms in he apex o he clus e .
This e ec has been iden i ied in he li e a u e as a ligh -
ning od e ec a he a omic scale, p oduced by he “cu -
a u e”o he a omic p o usions,
[52]
an e ec alida ed
by a omis ic ab-ini io TDDFT calcula ions o such clus-
e 's esponse.
[28,58]
This ul a-con ined local ield can be
obse ed mo e clea ly om he ield dis ibu ion in he
xy-plane (c oss sec ion below he nanoclus e apex) a di -
e en dis ances om he apex. Fo he closes e alua ion
dis ance (d= 0 nm deno es a xy-plane angen o he su -
ace o he apex a om), he la e al ex ension o he ho -
spo is smalle han 1 nm (see Figu e 2d), whe eas o a
xy-plane mo e dis an om he apex a om (d= 0.5 nm),
he ield dis ibu ion is mo e sp ead (see Figu e 2e).
Unde esonan incidence (ω= 3.47 eV), he a omic-scale
ho -spo s a e much mo e localized, as illus a ed in
Figu e 2 –h. E en o a xy-plane unde he a omis ic ip
as a as 0.5 nm away (Figu e 2h), he local elec ic ield
ZHANG ET AL.301
is s ill con ined in a subnanome e egion. The inhomo-
genei y and s ong localiza ion o he local ields as hose
ound he e a e hus excellen candida es o se a omic-
scale ligh sou ces enabling SRP,
[16,46,48,59-61]
as well as
o he a omically esol ed molecula spec oscopies, such
as luo escence.
[62]
3.2 |Raman esponse o molecules:
4,4'-BPY and biphenyl
As shown in he me hodological sec ion, he ea men o
he Raman pola izabili y o he molecules is a key ing e-
dien in he calcula ion o he Raman sca e ing c oss sec-
ion. We conside he e he 4,4'-BPY and biphenyl
molecules o show he applica ion o he me hodology.
Thei chemical s uc u es a e shown in Figu e 3a, whe e
bo h molecules exhibi a simila double- ing geome y,
excep o he eplacemen o he 4- and 40-si es by N
a oms, in he 4,4'-BPY molecule. Fo homogeneous illu-
mina ion, he Raman pola izabili y o a pa icula ib a-
ional mode can be e ie ed om he de i a i e o he
molecula pola izabili y, as di ec ly ob ained om DFT
calcula ions. The esul s o he Raman spec a o bo h
molecules show e y simila ea u es, as shown in
Figu e 3b, because o he simila molecula s uc u e.
Th ee main peaks a e dominan , and labeled as
12
,Ω
and
8a
modes, ollowing he Va sányi's symbols.
[63,64]
Figu e 3c shows he schema ics o he ib a ional mo ion
co esponding o hese h ee ib a ional modes o he
4,4'-BPY molecule.
We show now he main di e ences in he Raman
pola iza ion o he h ee ele an modes when he mole-
cules a e inhomogenously illumina ed. To ha end, we
i s ob ain he a omis ic Raman dipole o each a om in he
molecule, induced by he pa icula dis ibu ion o he illu-
mina ion, ollowing Equa ion (9), and hence, calcula e he
co esponding elec ic ield p oduced by he Raman pola -
ized molecule, as gi en by Equa ion (4), bu wi h he a om-
is ic dipole p
i
eplaced by he a omis ic Raman dipole o
each na om, pRaman
n,k, o each k h mode. We i s apply
his p ocedu e o he case o homogeneous illumina ion.
The induced ield in ensi ies jE
mol
j om he pola ized
molecule, as ob ained om Equa ion (11), a e calcula ed
o he h ee ele an modes o an inciden elec ic ield
pola ized along he z-di ec ion, exhibi ing he ela ion-
ship be ween he a omic ib a ions and elec ic ield dis-
ibu ions o each o hem. The ields a e calcula ed on
he wo py idine ing planes o he 44BPY molecule
a he han on he xy-plane. The ield dis ibu ions in
Figu e 3d show ha mos ly he ib a ing a oms wi hin
each mode con ibu e o he pola ized Raman dipole, as
FIGURE 2 (a) A omis ic s uc u e o
aAg
1415
nanoclus e wi h a egula
icosahed on s uc u e. The o ien a ion is
e e ed o he axis displayed on he
bo om-le co ne . (b) Abso p ion
spec um o he nanoclus e in (a) o an
inciden planewa e wi h elec ic ield
pola ized along he z-axis, E
z
. The inse
shows he eal and imagina y pa o he
a omic pola izabili y o he Ag a oms
o ming he nanosclus e . (c–e) Induced
local elec ic ield dis ibu ion in he
p oximi y o he nanoclus e o an
inciden op ical planewa e wi h o -
esonance ene gy (ω= 2.00 eV) in he
(c) yz-plane, in he (d) xy-plane a a
dis ance o d= 0 nm, and in he (e) xy-
plane a a dis ance d= 0.5 nm om he
apex o he nanoclus e . ( )–(h) show he
same local ield dis ibu ions as in (c)–(e),
o inciden ligh in esonance wi h he
nanoclus e dipola mode (ω= 3.47 eV)
302 ZHANG ET AL.
obse ed in he co esponding induced Raman elec ic
ield. This in oduces a way o dis inguish di e en ib a-
ional modes by speci ying he co esponding local
esponses om each a om. Fu he mo e, his e ec o he
local induced dipole could be mo e e icien ly p oduced
by a local elec ic ield sou ce and hus would enable
single-molecule imaging in eal space. The ib a ional
modes o he biphenyl molecule and he co esponding
elec ic ield dis ibu ions can be ound in he Supple-
men a y In o ma ion. Howe e , i he inciden elec ic
ield is no a plane wa e bu an ex emely localized ield,
he molecula esponse is di e en , as shown in
Figu e 3e. In his case, he ield dis ibu ion o he inci-
den local ield is conside ed o adop an ad-hoc Gaussian
unc ion in he h ee dimensions, o 0.2 nm ull wid h a
hal maximum, and loca ed 0.5 nm away om he cen e
o he 44BPY molecule la e ally (along he y-axis). Com-
pa ed o he homogeneous illumina ion, his ex eme
case o localized illumina ion makes each a om o expe i-
ence a di e en ield in ensi y, as obse ed in Figu e 3e,
esul ing in a e y di e en o al molecula pola iza ion.
This e ec will de e mine he inal ac i a ion o no o a
FIGURE 3 (a) Schema ics o he
chemical s uc u es o 4,4
0
-bipy idine
(44BPY) and biphenyl molecules om
side and op iews. (b) Calcula ed
Raman spec a o 44BPY and biphenyl
molecules wi hin densi y unc ional
heo y (DFT) unde homogeneous
illumina ion. The h ee main
ib a ional peaks a e labeled as
12
,Ω,
and
8a
modes, espec i ely.
(c) Vib a ional mo ion o he h ee
dominan ib a ional modes (
12
,Ω,
and
8a
) o he 44BPY molecule,
associa ed wi h ing mo ions. The ed
and blue a ows highligh he
cha ac e is ic mo ion o each mode.
(d) No malized elec ic ield
dis ibu ion wi hin he wo py idine
ing planes o he h ee di e en
ib a ional modes o he 44BPY
molecule pola ized by a plane wa e
pola ized along he z-di ec ion, as
indica ed by whi e a ows. The dashed
lines indica e ha he wo py idine ing
planes a e no coplana . (e) Elec ic
ield dis ibu ions o he same modes
as in (d), in esponse o a localized ield
cha ac e ized by a h ee-dimensional
Gaussian p o ile wi h cen al
pola iza ion and ull wid h a hal
maximum o 0.2 nm, cen e ed 0.5 nm
away om he cen e o he molecule
along he z-axis. The maximum
magni ude o his Gaussian-like local
ield a he o iginal poin is se o be E
0
ZHANG ET AL.303
ib a ional mode in he Raman spec um, as we will see in
he nex sec ion. The Gaussian-like local ield used he e
has been adop ed in o de o show he main ea u es o he
ac i a ion o a oms ela ed o each pa icula mode in an
academic way. In he nex sec ion, we will de ail his e ec
in he Raman spec um, as iden i ied in he ealis ic case
o a molecule loca ed in he local ield o a picoca i y gen-
e a ed by he a oms o a clus e , mimicking a ip.
3.3 |TERS o a single molecule in a
picoca i y
3.3.1 |Inhomogeneous ield-enhanced
Raman spec oscopy
Simila o he e ec p oduced by an ex emely localized
gaussian sou ce o ligh shown in he p e ious sec ion,
he apex o a me allic nanoclus e mimicking he a omic
p o usions a he end o a ip can equally p oduce an
ex emely con ined local elec ic ield dis ibu ion o
p obe a molecule. This s ongly inhomogeneous ield will
make each a om in he molecule o expe ience a di e en
elec ic ield in ensi y, ha is, a di e en alue o Elocð nÞ
o each o i s a oms (Equa ion 9), as isually shown on
he le panels o Figu e 4a–d. The a omis ic molecula
Raman dipole momen pRaman
n,kinduced by his inhomoge-
neous nea ield can be calcula ed ollowing he p oce-
du e desc ibed in Sec ion 2.3 and in he Suppo ing
In o ma ion. These a omis ic Raman dipoles ac as
induced ligh sou ces whose elec ic ield a he i h a om
o he nanoclus e can be exp essed in e ms o he
G een's unc ion as
ERaman
i,k=X
M
n=1
G
$
k i, n
ðÞpRaman
n,k
=X
M
n=1
G
$
in,kαnðÞ
kX
N
i0=1
G
$
ni0,kX
N
j0=1
A−1
i0j0Einc
j0:
ð14Þ
This elec ic ield hus u he pola izes he a oms in
he nanoclus e , esul ing in an enhanced Raman emis-
sion, which can be desc ibed ollowing Equa ion (2) as
X
N
j=1
Aij pðkÞ
j=ERaman
i,k:ð15Þ
And hence, he nanoclus e -enhanced Raman sca e -
ing c oss sec ion o he molecule can be ob ained:
FIGURE 4 (a) Le : schema ic o a 4,4
0
-bipy idine (44BPY) molecule loca ed on op o a Ag
1415
nanoclus e a a dis ance o 0.5 nm
unde inciden elec omagne ic ield wi h equency ω= 2.00 eV. Righ : calcula ed Raman spec a o single 44BPY molecule wi h di e en
o ien a ions: up igh s anding (g een spec um), lying (b own spec um), and la lying (pu ple spec um). The Raman spec um o a 44BPY
molecule in ee space is also shown in black o e e ence. (b) Schema ic (le ) and calcula ed Raman spec a ( igh ) o a 44BPY molecule a
a dis ance o 0.5 nm om he apex o he nanoclus e a a equency ω= 3.47 eV o he same o ien a ions as in (a). (c) and (d) Same as in
(a) and (b) o a biphenyl molecule
304 ZHANG ET AL.
