A icle h ps://doi.o g/10.1038/s41467-022-34393-4
Remo e nea -field spec oscopy o
ib a ional s ong coupling be ween o ganic
molecules and phononic nano esona o s
I ene Dolado
1,8
, Ca los Maciel-Escude o
1,2,8
, Eliza e a Nikulina
1
,
E genii Modin
1
,F ancescoCala alle
1
,ShuChen
1
,And eiBylinkin
1,3
,
F ancisco Ja ie Al a o-Mozaz
3
,JiahanLi
4
,JamesH.Edga
4
,
Fèlix Casano a
1,5
,Sau
l Vélez
6
,LuisE.Hueso
1,5
,RubenEs eban
2,3
,
Ja ie Aizpu ua
2,3
& Raine Hillenb and
5,7
Phonon pola i on (PhP) nano esona o s can d ama ically enhance he cou-
pling o molecula ib a ions and in a ed ligh , enabling ul asensi i e spec-
oscopies and s ong coupling wi h minu e amoun s o ma e . So a , his
coupling and he esul ing localized hyb id pola i on modes ha e been s udied
only by a -field spec oscopy, p e en ing access o modal nea -field pa e ns
and da k modes, which could u he ou undamen al unde s anding o
nanoscale ib a ional s ong coupling (VSC). He e we use in a ed nea -field
spec oscopy o s udy he coupling be ween he localized modes o PhP
nano esona o s made o h-BN and molecula ib a ions. Fo a mos di ec
p obing o he esona o -molecule coupling, we a oid he di ec nea -field
in e ac ion be ween ip and molecules by p obing he molecule- ee pa o
pa ially molecule-co e ed nano esona o s, which we e e o as emo e nea -
field p obing. We ob ain spa ially and spec ally esol ed maps o he hyb id
pola i on modes, as well as he co esponding coupling s eng hs, demon-
s a ing VSC on a single PhP nano esona o le el. Ou wo k pa es he way o
nea -field spec oscopy o VSC phenomena no accessible by con en ional
echniques.
S ong coupling be ween molecula ib a ions and in a ed pho ons
( ib a ionals ong coupling, VSC) leads o hyb id ligh -ma e s a es1–6.
They o e in iguing possibili ies o ul a-sensi i e ib a ional
spec oscopy7–10 and o modi ying chemical eac ions3,4. Typically,
VSC is achie ed wi h molecules embedded in o mic oca i ies, imply-
ing la ge mode olumes and la ge amoun s o molecules, which limi s
access o quan um phenomena ha may be accesible only o
nanoscale amoun s o molecules o a he le el o a ew molecules. In
his ega d, plasmonic in a ed esona o s a e a p omising ou e o
achie e VSC a he nanoscale7,8owing o hei d ama ically educed
mode olumes as compa ed o mic oca i ies. Al e na i ely, phonon
pola i ons (PhP) can be employed o VSC expe imen s, o e ing
s onge pola i on confinemen and la ge quali y ac o s9–11. Un o -
una ely, he a -field ex inc ion c oss-sec ion o indi idual PhP
Recei ed: 18 July 2022
Accep ed: 21 Oc obe 2022
Check o upda es
1
CIC nanoGUNE BRTA, 20018 Donos ia-San Sebas ián, Spain.
2
Ma e ials Physics Cen e , CSIC-UPV/EHU, 20018 Donos ia-San Sebas ián, Spain.
3
Donos ia
In e na ional Physics Cen e (DIPC), Donos ia-San Sebas ián, Spain.
4
Tim Taylo Depa men o Chemical Enginee ing, Kansas S a e Uni e si y, Manha an, KS
66506, USA.
5
IKERBASQUE, Basque Founda ion o Science, 48009 Bilbao, Spain.
6
IFIMAC-Condensed Ma e Physics Cen e , Ins i u o Nicolás Cab e a, and
Depa amen o de Física de la Ma e ia Condensada, Uni e sidad Au ónoma de Mad id, Mad id E-28049, Spain.
7
CIC nanoGUNE BRTA and Depa men o
Elec ici y and Elec onics, UPV/EHU, 20018 Donos ia-San Sebas ián, Spain.
8
These au ho s con ibu ed equally: I ene Dolado, Ca los Maciel-Escude o.
e-mail: .hi[email p o ec ed]
Na u e Communica ions | (2022) 13:6850 1
1234567890():,;
1234567890():,;
nano esona o s12–20 is ex emely small (due o hei small size com-
pa ed o he in a ed wa eleng h), challenging in a ed a -field spec-
oscopy. Fu he , sub adian da k modes—o e ing he ad an age o
longe li e imes—a e di ficul o p obe by a -field spec oscopy. These
p oblems can be ci cum en ed by nanoscale Fou ie T ans o m
In a ed (nano-FTIR) spec oscopy, which employs he s ong field
concen a ion a he apex o a me allic scanning p obe ip ( he nea -
field p obe)21,22 o enable nea -field spec oscopy and spa ial mapping
o bo h b igh and da k modes o indi idual phononic
nano esona o s17,18. Howe e , he nea -field mapping o VSC employ-
ing PhP nano esona o s has been elusi e so a .
Nano-FTIR spec oscopy has been employed o s udy he cou-
pling be ween molecula ib a ions and plasmonic esona o s, bu he
nea -field p obe i sel can also couple wi h he plasmonic esona o
and he molecules7,23,24, e en ually eaching s ong ip- esona o and
ip-molecule coupling. Al hough his coupling may be exploi ed o
on-demand con ol o VSC, i may challenge he p obing o he hyb id
pola i on modes ha a e exclusi ely o med by he molecule-
nano esona o coupling.
In e . 17, nano-FTIR spec oscopy was applied o s udy s ong
coupling be ween molecula ib a ions and p opaga ing PhPs on
ex ended, uns uc u ed h-BN laye s. In his expe imen , pola i on
in e e ome y had o be applied o ob ain he pola i on momen a and
hus he pola i on dispe sion. This measu emen p inciple and i s
associa ed da a analysis o e s he ad an age ha he ip-molecule and
ip-pola i on coupling does no a ec he pola i on dispe sion and
analysis o VSC. Howe e , such expe imen s do no allow o analyzing
he coupling be ween molecula ib a ions and localized PhP modes,
which will be o u mos impo ance o expe imen ally explo e, es and
e i y u u e VSC concep s employing indi idual PhP nano esona o s.
He e we demons a e ha hyb id pola i on modes o med by
ib a ional s ong coupling be ween a single PhP nano esona o and
molecula ib a ions can be s udied and imaged in eal space by nano-
FTIR spec oscopy. We minimize he influence o he ip by p obing
he molecule- ee pa o pa ially molecule-co e ed PhP nano -
esona o s wi h a non- esonan me allic ip, which we e e o as emo e
nea -field p obing. We also e i y ou expe imen al esul s ia com-
pa a i e nume ical simula ions, whe e he nea -field p obe is modeled
ei he by a poin -dipole sou ce ( ep esen ing a non-dis u bing nea -
field p obe) o by an oscilla ing me al ip ( ep esen ing a mo e ealis ic
and po en ially dis u bing nea -field p obe).
Resul s
Nano-FTIR spec oscopy o hal -co e ed phononic
nano esona o s
Ou expe imen is illus a ed in Fig. 1a. Hal o a hexagonal bo on
ni ide (h-BN) nano od is co e ed wi h CBP molecules (4,4′-bis(N-ca -
bazolyl)−1,1′-biphenyl; o ganic semiconduc o ) ha exhibi a ib a-
ional esonance a ω
CBP
= 1450 cm−1(Supplemen a y No e 1), as can be
ecognized om he nea -field spec um o a hin CBP laye shown by
he g een cu e in Fig. 1d, e. The nea -field p obe, he non- esonan
me allic ip25 o an a omic o ce mic oscope, is placed emo ely wi h
espec o he molecules a he opposi e ex emi y o he h-BN od. The
ip (oscilla ing no mal o he sample a equency Ω) concen a es an
illumina ing b oadband in a ed lase beam a i s apex o a nanoscale
nea -field spo , which exci es phonon pola i ons26–28 exhibi ing Fab y-
Pe o (FP) esonances9,18,19 in he h-BN nano od17.Figu e1bshowsa
simula ion o he second-o de FP mode exci ed by he nea fields o a
poin dipole (no e ha his mode is o en e e ed o as a da k mode,
since i canno be exci ed by a -field illumina ion due o i s ze o ne
elec ic dipole momen ; see Supplemen a y No e 2). The coupling
be ween he esona o mode and he laye o molecules— esul ing
om he s ong o e lap be ween he nea field o he esona o mode
and he molecules—is p obed by eco ding o he ip-sca e ed field
wi h an asymme ic Fou ie ans o m spec ome e , yielding bo h
nea -field ampli ude and phase spec a, s
3
(ω)andφ
3
(ω), espec i ely,
2.5
+-++ -
CBP
1400 14801440
0.9
0.8
1.0
1.2
1.1
1.3
L (Pm)
CBP
SiO
2
s
/s
3,Au
SiO
2
CBP
h-BN
Tip
E
sca
E
inc
500 nm
= 1437.4 cm
-1
E
CBP
h-BN
z
1.0
0.9
0.8
0.6
0.7
0.5
Ez/Ez,Au
L (Pm)
1400 14801440
CBP
3
0
0
(cm
-1
)
Z
(cm
-1
)
Z
Z
0.5
1.0
1.5
2.0
1
2
CBP
Z
CBP
Z
y
x
y
cd e
b
x
y
z
a
Fig. 1 | Tip-enhanced nea -field p obing o hal molecule-co e ed h-BN nano -
esona o s. a Illus a ion o he expe imen . bNume ical simula ion o he elec ic
field dis ibu ion a ω=1437.4cm
−1a ound an h-BN nano od o 1000 nm leng h,
250 nm wid h and 87 nm heigh , whose igh hal is co e ed by a 50 nm hick CBP
laye . The e ical a ow indica es a poin -dipole sou ce mimicking he ip.
cTopog aphy image o h-BN nano ods (leng h L, 87 nm heigh and 250 nm wid h)
ha a e hal co e ed by a 50 nm hick CBP laye . dExpe imen al nano-FTIR
ampli ude spec a eco ded a he posi ions ma ked in panel cby do s o he
espec i e colo . Fo be e isualiza ion, spec a o L= 1.0 and 1.3 μma escaledby
a ac o o 0.7. eSimula ed nano-FTIR ampli ude spec a, as explained in Fig. 3.Fo
he wid h and hickness o he h-BN ods we used he nominal expe imen al alues.
The leng hs Lwe e chosen such ha he second-o de FP mode unes ac oss he
molecula ib a ional esonance o he CBP molecules a ω= 1450 cm−1. We a i-
bu e he di e ences be ween Lin he expe imen and he simula ion o ab ica ion
unce ain ies such as apezoid-like od c oss sec ions.9,10 d,eG een cu es show
expe imen al and simula ed nano-FTIR ampli ude spec a o a 50 nm hick ba e
CBP laye on a 250 nm hick SiO
2
on Si subs a e. Red lines a e guides o he eye and
ma k peak posi ions. G ay-dashed lines indica e he CBP ib a ional esonance a
ω
CBP
= 1450 cm−1whose line wid h is 6.4 cm−1( e . 11). All spec a a e no malized o
ha ob ained on a Au e e ence su ace, and a e o se . We no e ha in he
expe imen al spec a (Fig. 1d) we obse e se e al smalle peaks, e.g., in he e-
quency ange be ween 1420 and 1440 cm−1. They can be a ibu ed o highe -o de
PhP modes10. In he simula ed spec a, whe e he nea -field p obe is desc ibed by a
poin -dipole sou ce (Fig. 1e), hese peaks a e only pa ially seen. A much be e
ep oduc ion o hese peaks is ob ained in nume ical simula ions, whe e he nea -
field p obe is desc ibed by a conical me allic ip ( o u he de ails and discussion
see Supplemen a y No e 4).
A icle h ps://doi.o g/10.1038/s41467-022-34393-4
Na u e Communica ions | (2022) 13:6850 2
whe e he index indica es ha he de ec o signal was demodula ed a
3Ω o supp ess backg ound signals (nano-FTIR spec oscopy; see
Me hods).
A opog aphy image o he se o h-BN nano ods o 250 nm wid h,
87 nm heigh and a ious leng hs LisshowninFig.1c, whe e one can
also see he hin, homogeneous CBP laye o 50 nm hickness co e ing
he igh hal o all esona o s. The expe imen al nano-FTIR ampli ude
spec a (Fig. 1d; phase spec a a e shown in Supplemen a y Fig. 3)
eco ded on he le od ex emi y (measu emen posi ion ma ked by
do s in Fig. 1c) clea ly show he esonance peaks o he fi s and
second-o de FP esonances (illus a ed by he schema ic abo e he
diag am in Fig. 1dand e ified by he expe imen al mode pa e n
shown in Fig. 2), which shi o highe equencies when he nano -
esona o leng h Lis educed om 1300 o 800 nm. Howe e , he peak
o he second o de mode does no c oss he CBP ib a ional eso-
nance a ω
CBP
=1450cm
−1. T acing he peak posi ions (ma ked by ed
lines) indeed e eals an i-c ossing beha io , indica ing ha he
nano esona o nea field couples wi h he molecula ib a ions o he
CBP laye . Mos impo an , he nano esona o -molecule coupling
(occu ing on he igh hal o he h-BN nano ods) can be well p obed
when he ip is placed on he le nano od ex emi y, ha is se e al
100 nm away om he molecules, whe e a di ec nea -field in e ac ion
be ween ip and molecules can be neglec ed (no e ha significan
nea -field in e ac ion be ween ip and sample occu s only o dis ances
smalle han he ip apex adius, he e abou 25 nm). The expe imen al
nea -field spec a can be well ep oduced by nume ical simula ions
(Fig. 1e), whe e he ip is modeled as a poin -dipole sou ce, which a e
desc ibed in mo e de ail in Fig. 3.
Spa io-spec al nea -field mapping o phononic nano esona o s
Remo e nea -field spec oscopy can be applied o nanoscale spa ial
mapping o he esona o -molecule coupling ia hype spec al
nanoimaging—e en o da k modes ha a e no accessible by a -field
spec oscopy. In he u u e, such a possibili y could be applied, o
example, o s udy ad anced esona o s uc u es whe e a a ie y o
di e en esona o modes may coexis and couple wi h he molecula
ib a ions. In Fig. 2we apply his capabili y o di ec expe imen al
iden ifica ion o he phononic esona o mode ha couples wi h he
molecula laye and o e i y ha he peak spli ing is caused by he
p esence o he molecules. To ha end, we eco ded nano-FTIR
ampli ude spec a along he p incipal axis o a ba e (Fig. 2a, b) and a
hal -co e ed (Fig. 2c, d) h-BN nano esona o . The leng hs Lo he
nano esona o s we e chosen such ha hei second-o de FP eso-
nance occu s a he molecula ib a ional esonance equency,
ω
CBP
=1450cm
−1. Fo each spec al peak we obse e s ong nea -field
oscilla ions along he p incipal nano esona o axis. The numbe o
oscilla ions inc eases s eadily wi h inc easing equency, e ealing a
se ies o longi udinal FP modes. The h ee nea -field maxima (in he
cen e and a he wo ex emi ies o he nano esona o ) a 1450 cm−1
clea ly e eal he second-o de (da k) FP mode. Mo e impo an , he
second-o de FP modal nea -field pa e n o he hal -co e ed nano -
esona o exhibi s a small spec al dip a he molecula ib a ional
esonance o CBP a 1450 cm−1(da k s ipe in Fig. 2d) all along he
p incipal esona o axis, which is absen in he hype spec al linescan
o he ba e nano esona o (Fig. 2b). Fo a be e quan i a i e com-
pa ison, we show in Fig. 2e he nano-FTIR ampli ude spec a eco ded
a he le ex emi y o he ba e and molecule-co e ed h-BN nano -
esona o s (posi ions ma ked by blue and ed do s in Fig. 2a, c,
espec i ely). We clea ly see ha he peak o he fi s -o de FP mode
( a away om he CBP esonance a ω
CBP
=1450cm
−1)isnea ly he
same o bo h nano esona o s, whe eas he peak o he second-o de
FP mode o he molecule-co e ed h-BN nano esona o exhibi s a
spec al dip occu ing a ω
CBP
, as ypical o he coupling be ween
molecula ib a ions and nano esona o modes7,9,10,23. The spa io-
spec al obse a ion p esen ed in Fig. 2demons a es ha he
nano esona o -molecule coupling can in p inciple be p obed a any
loca ion whe e he nano esona o mode can be ac i a ed by he nea -
field p obe. On he o he hand, Fig. 2d e eals ha he nea -field signal
on he molecule-co e ed pa o he nano esona o is s ongly
educed due o he inc eased ip-nano esona o dis ance and ha he
spec al line shape is modified due o he di ec nea -field in e ac ion
be ween ip and molecules, highligh ing he ad an ages o p obing he
molecule- ee pa o he esona o , ha is, p obing la ge nea -field
signals and educ ion o undesi able nea -field in e ac ion be ween ip
and molecules. The modifica ion o he line shape is mo e clea ly seen
in Fig. 2 , which compa es spec a eco ded on he molecule- ee
nano esona o pa (B and C) wi h spec um D ha was eco ded on
he molecule-co e ed pa . Specifically, we see ha he spec um D
exhibi s an asymme ic line shape and ha he dip is shi ed o sligh ly
highe equencies. This asymme ic line shape can be explained as
CBP
a
b
B
A
c
d
Heigh (nm)
0
140
0 0.5 1.0 1.5
min
max
1500
1400
1450
s
3
/s
3,Au
Posi ion (Pm)
0.5 1.0 1.5
SiO2SiO2CBP
CD
CBP
e
B
A
1390 14701430
B
C
D
1430 14701450
0
0.2
0.4
0
0.5
1.0
No m. s3
No m. s3
(cm
-1
)
Z
(cm
-1
)
Z
(cm
-1
)
Z
Posi ion (Pm)
Z
Z
0
Fig. 2 | Nano-FTIR line scans o ba e and hal molecule-co e ed h-BN nano -
esona o s. a Topog aphy image o a ba e h-BN nano od o 1.25μm leng h, 110 nm
heigh and 250 nm wid h. bNano-FTIR ampli ude spec a eco ded along he
ho izon al dashed black line in panel a.cTopog aphy image o an h-BN nano od o
1.1 μm leng h, 87 nm heigh and 250 nm wid h, which is hal co e ed wi h a 50 nm
hick CBP laye . dNano-FTIR ampli ude spec a eco ded along he ho izon al
dashed black line in panel c.eNano-FTIR ampli ude spec a o he ba e (blue) and
molecule-co e ed ( ed) h-BN nano od a posi ions ma ked by blue and ed do s in
panels aand c, espec i ely. Bo h spec a a e no malized o he peak maximum a
1408 cm−1. Nano-FTIR ampli ude spec a o he molecule-co e ed h-BN nano od a
posi ions ma ked by ed, o ange and black do s in panel c. The spec a a e no -
malized o he peak maximum a 1445 cm−1. The e ical g een dashed line ma ks
he equency o he molecula ib a ional esonance o CBP.
A icle h ps://doi.o g/10.1038/s41467-022-34393-4
Na u e Communica ions | (2022) 13:6850 3
supe posi ion o he symme ic nano-FTIR spec um o he molecule-
co e ed nano esona o (B) and he asymme ic nano-FTIR spec um o
a ba e CBP laye (such as he g een spec um in Fig. 1d) caused by he
di ec nea -field in e ac ion be ween ip and molecules. The la e
exhibi s a de i a i e-like spec al line shape, ha is, a peak and a dip o
he le and igh o he molecula esonance, espec i ely, which is a
ypical cha ac e is ic o nano-FTIR ampli ude spec a o molecula
laye s29.
Theo e ical desc ip ion o emo e nea -field p obing o VSC
To unde s and how he hyb id nano esona o -molecule modes man-
i es in he nea -field spec a, we fi s discuss in Fig. 3nume ical
simula ions whe e he ip is modeled as a poin -dipole sou ce17,30 ( ed
a ow in Fig. 3a, d) loca ed abo e he h-BN esona o . We ob ain
complex- alued nea -field spec um by e alua ing he e ical (z-)
componen o he elec ic field below he dipole (e alua ion posi ion
ma ked by a c oss in Fig. 3a, d) as a unc ion o equency, E
z
(ω).
Fu he simula ion de ails a e desc ibed in he Me hods sec ion.
Impo an ly, he dipole momen o he sou ce is kep cons an and
consequen ly is no modified by he fields o he nano esona o . This
simplified modeling o he ip allows o excluding a po en ial coupling
o he ip wi h he nano esona o and he molecules, and hus exclu-
si ely e eals he spec al nea -field signa u e o he coupling be ween
he nano esona o and he molecula ib a ions.
Figu e 3b shows he simula ed nea -field ampli ude spec um
|E
z
(ω)| (open symbols), o a h-BN nano od ha is hal co e ed wi h a
laye o pe mi i i y ε= 2.8 (co esponding o he pe mi i i y o CBP
wi hou he molecula esonance a 1450 cm−1). I s leng h was chosen
such ha he second-o de FP esonance is a ω=1450cm
−1,which
mani es s as a single peak in he nea -field ampli ude spec um.
Repea ing he simula ion when he h-BN nano od is hal co e ed by a
CBP laye (Fig. 3e), he nano esona o s’nea -field ampli ude peak
spli s in o wo peaks (open symbols), which is a di ec consequence o
he coupling be ween he nano esona o mode and molecula ib a-
ions. Fo quan i ying he coupling be ween he h-BN nano esona o
and he molecula ib a ions, we fi hesimula ednea -field spec a by
a model o wo coupled ha monic oscilla o s (Me hods), which
ep esen he nano esona o mode and he molecula ib a ion wi h
eigen equencies ω
i
and damping pa ame e s γ
i
( he index ideno es
ei he PhP o CBP). The wo oscilla o s a e coupled o each o he
h ough he coupling s eng h g. Fo a mos eliable analysis, we pe -
o med complex- alued fi ing o he simula ed nea -field spec a
EzωðÞ=∣EzωðÞ∣eiφωðÞ
, whe e |E
z
(ω)| and φ(ω) a e he ampli ude and
phase spec a (Fig. 3 ). Plo ing he fi s by ed solid lines in Fig. 3e, , we
find an excellen ag eemen wi h he simula ed spec a ( ed open
symbols), which confi ms he alidi y o he wo coupled ha monic
oscilla o model o fi hesimula ednea -field spec a.
Be o e analysing he expe imen al nano-FTIR spec a, we no e
ha he simula ed nea -field spec um o he h-BN nano esona o in
absence o molecula ib a ions yields a nea ly ci cula ajec o y in
hecomplexplane(Fig.3c). In e es ingly, when he nano esona o is
hal co e ed wi h CBP molecules, he opology o he ajec o y o he
complex- alued nea -field spec a changes. We find ha a small loop
appea s wi hin he main ci cula ajec o y (Fig. 3 ), which e eals he
coupling be ween he esona o mode and he molecula ib a ions,
i.e., ha no only a me e supe posi ion o wo modes is obse ed,
(Supplemen a y No e 5), simila o a ecen obse a ion in a -field
ellipsome y o exci onic coupling in classical mic o esona o s31.The
opology change o complex- alued ajec o ies may become an
in e es ing means o cha ac e izing coupling phenomena.
Quan i a i e analysis o expe imen al nano-FTIR spec a
Fo analysing he expe imen al nea -field spec a (Fig. 4a, b), we fi s
assume ha he ip is solely illumina ing he nano esona o (i.e., ha i
ac s like a dipole sou ce wi h a cons an dipole momen and o ha
eason does no need o be modeled as ano he coupled oscilla o ).
We hus apply he same model o wo coupled ha monic oscilla o s as
o he simula ed nea -field spec a o fi he expe imen al spec a. We
u he assume ha he PhP mode is no exci ed by a -field illumina-
ion because i is a da k mode. Figu e 4b shows he expe imen al nea -
field spec a o he hal -co e ed nano esona o s in he complex plane
(black lines), as well as he excellen fi ing esul s (o ange lines). The
co esponding ampli ude and phase spec a a e shown in Fig. 4aand
Supplemen a y Fig. 3, espec i ely. The small loops in he complex
plane clea ly e eal he coupling be ween he nano esona o mode
and he molecula ib a ions, as p edic ed by he simula ions in Fig. 3 .
We no e ha he size o he small loop sligh ly a ies in he expe i-
men al spec a o he di e en nano esona o s, which we a ibu e o
a ia ions o he esona o s´ quali y ac o due o ab ica ion unce -
ain ies. F om he fi s we de e mined o each expe imen al nea -field
spec um (i.e. o each nano esona o ) he coupling s eng h g, hePhP
eigen equency ω
PhP
and damping γ
PhP
, he molecula ib a ional
eigen equency ω
CBP
and damping γ
CBP
(see Me hods and Supple-
men a y Table 2). F om hese pa ame e s we calcula ed he eigen-
equencies o he new hyb id modes (see Me hods) o each
nano esona o , ω
±
, which a e shown in Fig. 4c (blue symbols) as a
unc ion o he nano esona o s´ eigen equencies ω
PhP
. We obse e a
clea an i-c ossing o he hyb id modes, which indica es s ong cou-
pling. To de e mine he coupling egime o each indi idual esona o ,
we ma k in Fig. 4e he ansi ion om weak o s ong coupling—
defined by he s anda d c i e ium g=|γ
PhP
+γ
CBP
|/4( e .32)—by black
dashed ho izon al lines. In e es ingly, we find ha gis abo e he black
dashed line o all indi idual esona o s, indica ing ha all o hem a e
s ongly coupled wi h he molecula ib a ions.
Impo an ly, applying he same fi ing p ocedu e o he simula ed
complex- alued nea -field spec a (whe e he ip is modeled as a poin -
dipole sou ce; see Me hods and Supplemen a y No e 3) yields hyb id
eigenmodes (Fig. 4d) and coupling s eng hs (Fig. 4 ) ha ma ch well
CBP
h-BN
Dipole
1440 14601450
1440 14601450
0.60
0.90
0.6
1.0
0.3
0
0.50
0.5 1
0.6 0.9
0
b
Ez /Ez,Au
e
a
Re[Ez /Ez,Au]
0.7 0.8
0.1
0.2
0.75
0.25
0.8
0.75
Im[Ez /Ez,Au ]
Re[Ez /Ez,Au]
c
d
Z
CBP
Z
Z
Z
CBP
(cm
-1
)
Z
(cm
-1
)
Z
H
=
2.8
h-BN
Dipole
Ez /Ez,Au
Im[Ez /Ez,Au ]
Fig. 3 | Theo e ical desc ip ion o nea -field p obing o VSC. a Simula ion geo-
me y, showing a poin -dipole sou ce abo e an h-BN nano od o 785 nm leng h,
87 nm heigh and 250 nm wid h, which is hal co e ed wi h a 50 nm hick laye o a
pe mi i i y ε=2.8.bAmpli ude o he z-componen o he elec ic field below he
dipole a he posi ion ma ked by a c oss in panel a(no malized o ha ob ained on
a Au e e ence su ace, ∣Ez=Ez,Au∣), as unc ion o equency ω.cz-componen o he
elec ic field below he dipole sou ce a he posi ion ma ked by a c oss in panel
a, plo ed in he complex plane. The a ow indica es inc easing equency ω.
dSimula ion geome y, showing a poin -dipole sou ce abo e he same h-BN
nano od as in panel abu hal co e ed wi h CBP molecules. eSame as panel b,bu
he h-BN nano od is hal co e ed by a 50 nm hick CBP laye . Same as panel c,bu
o a hal CBP-co e ed h-BN nano od. b,c,e, Open symbols show simula ion
esul s. Solid lines show fi s ob ained wi h b,ca single ha monic oscilla o model
and e, a coupled ha monic oscilla o model desc ibing he coupling be ween he
nano esona o modes and he molecula ib a ions.
A icle h ps://doi.o g/10.1038/s41467-022-34393-4
Na u e Communica ions | (2022) 13:6850 4
he expe imen al esul s. Since he ip is no in ol ed in he
nano esona o -molecule coupling in he simula ions, his quan i a i e
ag eemen ( o u he discussion see Supplemen a y No e 6) le s us
u he assume ha he coupling be ween he PhP mode and he
molecula ib a ions in ou expe imen can be well desc ibed wi h a
simple wo coupled ha monic oscilla o model, wi hou he need o
conside ing he ip as a hi d oscilla o . Ou analysis hus demons a es
he capabili y o emo e nea -field spec oscopy o p obe he hyb id
modes a ising om he coupling be ween a single PhP nano esona o
and nanoscale amoun s o o ganic molecules.
We no e ha he damping γ
PhP
ob ained om he simula ed nea -
field spec a is a ac o o h ee lowe han ha ob ained om he
expe imen al spec a (see Supplemen a y No e 3). We explain his
finding by PhP sca e ing and abso p ion a ab ica ion-induced i e-
gula i ies and damage o he h-BN nano od edges9,10, which is no
conside ed in he simula ion. Al hough he damping has only a mino
influence on he coupling s eng h gand on he de e mina ion o he
hyb id eigenmodes (see Eq. (10) in Me hods), i is a c ucial pa ame e
ha de e mines whe he a coupled sys em is in he weak o in he
s ong coupling egime. Ma king he ansi ion om weak o s ong
coupling in Fig. 4 by black ho izon al dashed lines (analogue o
Fig. 4e), we find ha i occu s a much lowe g o he simula ion as
compa ed o expe imen , which shows ha in simula ions we a e
deepe inside he s ong coupling egime han in he expe imen .
Influence o he oscilla ing ip and signal demodula ion
So a , we ha e no explici ly conside ed in he simula ions ha he ip
is a long me allic cone, ha he ip is oscilla ing, and ha he de ec o
signal is demodula ed a highe ha monics o he ip oscilla ion e-
quency. To elucida e he influence and impac o hese key ea u es on
he spec al mode posi ions and linewid hs, we pe o med nume ical
simula ions whe e he ip is modeled as a me al cone and he ip
oscilla ion and signal demodula ion a e conside ed. The esul s a e
shownin heSupplemen a yNo e6and7andsumma izedas ollows.
We fi s applied he coupled ha monic oscilla o model (same as used
in Fig. 3) ofi he simula ed nea -field spec a whe e he ip is modeled
as a me al cone bu wi hou conside ing he ip oscilla ion and signal
demodula ion. Compa ed o he fi ing pa ame e s ob ained om he
unpe u bed molecule-co e ed esona o spec a (ob ained om he
poin -dipole sou ce simula ions), we find ha he p esence o he
me allic ip yields negligible spec al shi s o he ba e h-BN nano -
esona o mode and he hyb id pola i on modes (<3 cm−1), a negligible
change o he sum o damping pa ame e s, γ
PhP
+γ
CBP
,andonlyasligh
educ ion o he coupling s eng h by a ac o o abou 1.26. In a sec-
ond simula ion, we addi ionally implemen ed ip oscilla ion and signal
demodula ion. Fi ing o he simula ed spec a yields nea ly he same
coupling s eng hs and mode posi ions as be o e. Howe e , o he
sum o he damping pa ame e s, γ
PhP
+γ
CBP
, we ob ain alues ha a e
educed by a ac o o abou 1.56 as compa ed o he esul s ob ained
om he simula ions whe e he ip is modelled as a poin -dipole
sou ce. Since he p esence o he ip, i s oscilla ion and signal demo-
dula ion can yield significan ly di e en alues o gand γ
PhP
+γ
CBP
,we
e-e alua e he coupling egime ob ained in he expe imen acco ding
o a modified condi ion, 1.26 g/[1.56(γ
PhP
+γ
CBP
)] > 0.25, whe e gand
γ
PhP
+γ
CBP
a e he pa ame e s (lis ed in Table 1) ob ained by fi ing he
expe imen al nea -field spec a. We find ha wo o he nano -
esona o s (leng h L=1.2µmandL=0.9µm) sa is y he s ong coupling
c i e ia, whe eas he o he nano esona o s a e a he onse o s ong
coupling. The ansi ion om weak o s ong coupling- acco ding o
his e-e alua ion—is ma ked in Fig. 4e by ed dashed ho izon al lines.
Since he modified condi ion o e alua ing he coupling egime is
mo e conse a i e han he o iginal one (g/[γ
PhP
+γ
CBP
]>0.25), he
Simula ions
Expe imen
c
Re[ 3]
1440
1450
1460
1440 1450 1460 1440 1450 1460
e
1440 1450 1460 1440 1450 1460
5
10
0
d
g(cm-1)
(cm
-1
)
V
Z
sc
sc
wc
+
Z
-
Z
PhP
Z
CBP
Z
PhP (cm-1)
Z
+
Z
-
Z
PhP
Z
CBP
Z
(cm
-1
)
Z
Z
CBP
Z
s
3
/s
3,Au
ab
CBP
1
2
0.9
0.8
1.0
1.1
L=
1.2
Pm
1420 1450 1480
0.2
0.3
0.1
0.25
0.15
-0.3
0.3
0
0.2
0.3
0.1
0
0.10-0.1
0.10
0-0.6 -0.3
000-0.1-0.1-0.1-0.2-0.2-0.2 0.10.10.1
wc
PhP (cm-1)
Z
PhP (cm-1)
Z
PhP (cm-1)
Z
Im[ 3]
V
Fig. 4 | Quan i a i e analysis o nano-FTIR spec a by coupled ha monic
oscilla o fi ing. a Black cu es show he expe imen al nano-FTIR ampli ude
spec a o h-BN nano ods o leng h L, which a e hal co e ed wi h CBP molecules
(same da a as in Fig. 1d). O ange cu es show fi s ob ained using he coupled
ha monic oscilla o model. Fo be e isualiza ion, he spec um o L=1.0μmis
scaled by a ac o o 0.7. bBlack cu es show he complex- alued expe imen al
nano-FTIR spec a, σ3ωðÞ=s3ωðÞeiφ3ωðÞ
, plo ed in he complex plane. O ange cu es
show fi s ob ained using he coupled ha monic oscilla o model. cEigen equencies
ω
±
o he hyb id modes, he nano esona o s´ ba e eigen equencies ω
PhP
(g ay
squa es), and ba e molecula ib a ional eigen equency ω
CBP
(g een squa es), all o
hem ob ained by fi ing o he complex- alued expe imen al nano-FTIR spec a.
dSame as panel c,bu ob ainedbyfi ing complex- alued simula ed nea -field
spec a o h-BN nano esona o s o leng h L=1.2 o0.8μm, which a e hal co e ed
by CBP molecules ( he co esponding ampli ude spec a a e shown in Fig. 1e; he
complex- alued spec a a e shown in Supplemen a y Fig. 3). The ip is modeled by a
poin -dipole sou ce as in Fig. 3.e, Coupling s eng h gob ained om he fi ing o
he expe imen al and simula ed nea -field spec a, espec i ely. Black and ed
dashed ho izon al lines indica e he ansi ion om weak (WC) o s ong (SC)
coupling defined by g=|γ
PhP
+γ
CBP
|/4 e .32 and g=|γ
PhP
+γ
CBP
| /3.23 (see
discussion below), espec i ely.
A icle h ps://doi.o g/10.1038/s41467-022-34393-4
Na u e Communica ions | (2022) 13:6850 5
ansi ion occu s in Fig. 4e a la ge g alues. We no e ha he ac o s
used o e-e alua e he coupling egime a e specific o ou s udy and
may change o di e en ip oscilla ion ampli udes o signal demo-
dula ion o de s.
Discussion
In summa y, we demons a ed ha in a ed nea -field spec oscopy
can be applied o nanoscale spa ial mapping o s ong coupling
be ween molecula ib a ions and esona ing da k PhP modes. To ha
end, we in oduced he concep o emo e nea -field p obing wi h a
non- esonan ip, whe e he ip and molecules a e sepa a ed such ha
he di ec nea -field in e ac ion be ween ip and molecules is a oided.
Such minimal-in asi e p obing o e s he ad an age ha he hyb id
nano esona o -molecule modes and coupling s eng hs can be de e -
mined wi hin he model o wo coupled ha monic oscilla o s wi hou
conside ing he ip as a hi d oscilla o , which significan ly simplifies
he coupling analysis and inc eases i s obus ness. On he o he hand,
nume ical simula ions o emo e nea -field spec oscopy including he
me al ip, i s oscilla ion and signal demodula ion e eal ha damping
pa ame e s may be unde es ima ed when ip oscilla ion and signal
demodula ion a e no aken in o accoun o fi ing he nea -field
spec a wi h a coupled ha monic oscilla o model.
Since nea -field spec oscopy is a ailable in he wide spec al
ange be ween isible and e ahe z equencies, we en ision emo e
nea -field p obing s udies o s ong coupling o di e en plasmonic
and phononic esona o s (which apa om h-BN could be made, o
example, om VO
5
o MoO
3
ha co e a ious in a ed equency
anges) wi h molecula ib a ions o exci ons in a ious esona o
geome ies. The possibili y o p obe da k modes and spa ially con ol
he selec i e exci a ion and p obing o coexis ing modes could pa e
he way o explo e s ong coupling configu a ions ha a e no acces-
sible by a -field spec oscopy.
Me hods
G ow h o monoiso opic 10B h-BN c ys als
The g ow h o 10B en iched h-BN c ys als was done by he me al flux
me hod33. Powde s o 40 g Fe, 25 g C , 1 g 10B we e mixed in an alumina
c ucible and placed in a single-zone u nace. A e a dwell ime o 24 h
a 1550 °C, he u nace was cooled a a a e o 4 °C/h o 1400 °C, o
p ecipi a e he h-BN c ys als on he me al su ace. This was subse-
quen ly ollowed by quick quenching o oom empe a u e.
Fab ica ion o h-BN nano esona o s
We fi s mechanically ex olia ed 10B en iched h-BN c ys als using blue
Ni o ape (Ni o Denko Co., SPV 224 P). Then he flakes we e ex olia ed
om he ape on o a anspa en polydime hyl-siloxane (PDMS)
s amp. Flakes o app op ia e hickness and size we e hen ans e ed
on o a Si/SiO
2
(250 nm) subs a e using he de e minis ic d y ans e
echnique34. Fo pa e ning he h-BN nano esona o s we used high-
esolu ion elec on beam li hog aphy using a double-laye poly(-
me hyl me hac yla e) (PMMA) esis (495 A4/950 A2). A e
de elopmen o he esis , a 3 nm- hick C laye was deposi ed on o he
sample by e-beam e apo a ion, ollowed by he mal e apo a ion o
40 nm o Al. A e li -o in ace one, chemical e ching o he h-BN flake
was pe o med using a SF6/A 1:1 plasma mix u e a 20 sccm flow,
100 mTo p essu e and 100 W powe o 60 s (RIE OXFORD PLAS-
MALAB 80 PLUS eac i e ion e che ). Fo emo ing he C -Al mask
om he h-BN s uc u es, he sample was imme sed in a ch omium
e chan (Sigma-Ald ich Co., 651826) o 20 min. A e insing in deio-
nized wa e , he sample was d ied using a N
2
gun.
The mal e apo a ion o CBP
4,4′-bis(N-ca bazolyl)-1,1′-biphenyl wi h sublimed quali y (99.9%)
(Sigma Ald ich) was he mally e apo a ed in an ul a-high acuum
e apo a o chambe (base p essu e <10−9mba ), a a a e o 0.1 nm s−1
using a Knudsen cell.
P epa a ion o a diamond mask o s uc u ed CBP deposi ion
A diamond mask was used o deposi ing he he mally e apo a ed
CBP molecules only on one hal o he h-BN nano esona o s. The mask
was cu om a polyc ys alline diamond film (500 nm hick) by ocused
Ga-ion beam (FIB) milling in a Helios 450S (FEI) dual beam sys em
(milling condi ions: 30 kV and 9.3 nA). Using an in si u pla inum
deposi ion, he diamond mask was a ached o he ungs en ip o an
Omnip obe mic omanipula o and physically placed on op o he
nano esona o s such ha i co e ed hal o each. The mask was fixed
on he sample su ace by in si u P -deposi ion. A e CBP deposi ion by
he mal e apo a ion (see abo e), he Omnip obe mic omanipula o
was app oached o he mask. The mask was fixed o he Omnip obe by
in si u P deposi ion. By FIB milling, he mask was de ached om he
sample and could be emo ed ia he Omnip obe mic omanipula o .
Nano-FTIR spec oscopy
We used a comme cial sca e ing- ype scanning nea -field op ical
mic oscope (s-SNOM) se up comp ising a nano-FTIR module (Neas-
pec/A ocube, Ge many), in which he oscilla ing (a a equency Ω≅
270 kHz) P /I -coa ed AFM ip (A ow-NCP -50, Nanowo ld, Nano-
Wo ld AG, Neuchâ el, Swi ze land) was illumina ed by p-pola ized mid-
IR b oadband adia ion gene a ed by a supe con inuum lase (a e age
powe o abou 0.5 mW; equency ange 1200–1700 cm−1). The
de ec o signal was demodula ed a a equency 3 Ω o e ec i e
backg ound supp ession. In e e og ams we e measu ed by eco ding
he demodula ed de ec o signal as a unc ion o he posi ion o he
e e ence mi o , d,a afixed ip posi ion. Fo apodiza ion o he
in e e og ams a Planck- ape window unc ion wi h ε=0.2was
applied. A e ze o-filling (4× padding) we Fou ie ans o med he
in e e og ams o ob ain complex- alued nea -field poin spec a,
E
s
(ω). Each spec um o Fig. 1d is an a e age o h ee spec a eco ded
in s eps o 40 nm along he y-axis. We no malized he ob ained poin
spec a o a e e ence spec um eco ded on gold, E
s,Au
(ω). The
spec al esolu ion was 6.25 cm−1.
Nume ical simula ions whe e he ip is modeled as a poin -
dipole sou ce
We pe o med he nume ical simula ions (Figs. 1e, 3,4d, ) using he
Radio F equency Module o COMSOL Mul iphysics so wa e. This
module sol es Maxwell’s equa ions in he equency domain based on
he Fini e Elemen Me hod (FEM). The h-BN nano esona o s we e
modeled as a ec angula s uc u e o w= 250 nm wid h (along he x-
di ec ion), d= 87 nm hickness (along he z-di ec ion) and a iable
leng h L(along he y-di ec ion). The esona o s a e on op o a 250 nm
hick laye wi h he pe mi i i y o SiO
2
, which is on op o a Si sub-
s a e. The CBP laye was modeled as a 50 nm hick laye co e ing hal
he leng h o he h-BN nano od. The ma e ial pe mi i i ies a e p o-
ided below. To ensu e nume ical con e gence o he simula ed nea -
field spec a, he comple e s uc u e (poin -dipole sou ce, h-BN
Table 1 | Pa ame e s ob ained by fi ing he expe imen al
complex- alued spec a shown in Fig. 4b ia hecoupled
ha monic oscilla o model
L(µm) ω
PhP
γ
PhP
ω
CBP
γ
CBP
gγ
PhP
+γ
CBP
1:26g
1:56ðγPhP +γCBP Þ
1.2 1441.2 9.4 1450.8 6.0 5.4 15.4 0.28
1.1 1447.8 8.9 1452.3 6.0 4.3 14.9 0.23
1.0 1451.0 8.8 1451.4 7.0 4.6 15.8 0.23
0.9 1454.4 10.2 1451.8 6.6 6.0 16.8 0.29
0.8 1462.4 10.7 1452.0 7.0 4.7 17.7 0.21
Allpa ame e s om he second o he se en h column a e exp essed in cm−1. E o s acco ding o
he s anda d de ia ion ob ained by fi ing he indi idual spec a a e p esen ed in Supplemen a y
Table 2.
A icle h ps://doi.o g/10.1038/s41467-022-34393-4
Na u e Communica ions | (2022) 13:6850 6
nano od and CBP laye ) was loca ed in a homogeneous ec angula
box (filled wi h ai ) o 8 × wwid h, 25 × ddep h and 4 × Lleng h. We use
pe ec ly ma ched laye s (PML) o he bounda ies o he simula ion
box and ee iangula elemen s o he nano od mesh and ee e -
ahed alelemen s o allo he s uc u es.
To ob ain he nume ical esul s shown in Figs. 1e, 3,4d, , we
modeled he ip as a poin -dipole sou ce o ien ed pe pendicula ly o
he subs a e (along he z-di ec ion). This model assumes ha he
elonga ed ip in he expe imen is o ien ed pe pendicula ly (z-di ec-
ion) o he h-BN nano od and is illumina ed by p-pola ized ligh . The
expe imen al signal de ec ed in he a field (E
s
) was app oxima ed by
he e ical componen o he elec ic nea field (E
z
) a an e alua ion
poin below he poin -dipole sou ce. In all he calcula ions, he poin -
dipole and he e alua ion poin s we e loca ed a coo dina es (x=0,
y=−L/2 + 50 nm,z= 350 nm) and (x=0,y=−L/2+50nm,z =65nm),
espec i ely. We se he o igin (x=0,y=0,z= 0) a he middle o he
op su ace o he nano od. The CBP laye co e ed he nano od o
y> 0. Fo compa ison wi h he expe imen al nea -field spec a, he
simula ed nea -field spec a we e no malized o E
z
ob ained when he
dipole sou ce is loca ed abo e a gold (i.e. e e ence) subs a e.
Pe mi i i y o h-BN
The pe mi i i y o he iso opically (10B) en iched h-BN (in-plane εhBN,?,
ou -o -plane εhBN,k) is desc ibed by a D ude-Lo en z model35
εhBN,j=ε1,j1+ ω2
LO,jω2
TO,j
ω2
TO,jω2iωΓj
!
,ð1Þ
whe e jindica es he in-plane (⊥) o ou -o -plane (||) componen . In Eq.
(1), ω
TO,j
and ω
LO,j
a e he TO and LO phonon equencies, Γjis he
damping cons an and ε1,jis he high- equency pe mi i i y. The
alues used o each cons an a e p esen ed in Table 2.
Pe mi i i y o CBP
We model he pe mi i i y o CBP as he sum o a non- esonan
backg ound plus ou oscilla o s which desc ibe he molecula ib a-
ions o CBP wi hin he equency ange o in e es . Thus, he pe -
mi i i y o CBP is gi en by11
εCBP =ε1,CBP +X
4
k=1
ω2
p,k
ω2
0,kω2iωΓCBP,k
,ð2Þ
whe e ε1,CBP =2:8 is he high- equency pe mi i i y and ω
p,k
,ω
0,k
and
Γ
CBP,k
a e he s eng h, na u al equency, and damping cons an o he
k- h oscilla o , espec i ely. The alues used o each cons an a e
p esen ed in Table 3.
Pe mi i i y o Si, SiO
2
and Au
Fo he nume ical calcula ions shown in he main ex , we conside ed
he h-BN nano od o be on op o a Si/SiO
2
subs a e. Fo he pe mi -
i i y o Si, we used a cons an alue ε
si
=12.TheSiO
2
pe mi i i y was
app oxima ed by he ollowing analy ical unc ion
εSiO2=a0+a1λ0+a2λ2
0+a3λ3
0+a4λ4
0+a5λ5
0+a6
λ0a7
2+a2
8
+ia9
λ0a7
2+a2
10
,
ð3Þ
wi h λ
0
being he ee space wa eleng h exp essed in mic ons,
a
0
=942.883,a
1
=−712.749 μm−1,a
2
=215.563μm−2,a
3
=−32.5483 μm−3,
a
4
= 2.45449 μm−4,a
5
=−0.0740283 μm−5,a
6
=−0.1404 μm2,
a
7
=8.078μm, a
8
= 0.1404 μm, a
9
=−0.1426 μm2and a
10
= 0.1426 μm.
The pe mi i i y o Au was aken om e . 36.
Coupled ha monic oscilla o model
To fi he da a shown in Figs. 3and 4,wedesc ibed hecoupling
be ween he Fab y-Pe o phonon pola i on esonance and he mole-
cula ib a ions by a classical model o wo coupled ha monic oscil-
la o s. The equa ions o mo ion ha desc ibe he coupled sys em a e
de e mined by he ollowing ela ions37–40
€
xPhP ðÞ+γPhP
_
xPhP ðÞ+ω2
PhPxPhP ðÞ2g_
xCBP ðÞ=FPhP ðÞ,ð4Þ
€
xCBP ðÞ+γCBP
_
xCBP ðÞ+ω2
CBPxCBP ðÞ+2g_
xPhP ðÞ=FCBP ðÞ,ð5Þ
whe e he do s deno e ime de i a i es, x
PhP
( ) ep esen s he PhP
mode wi h esonance equency ω
PhP
and damping cons an γ
PhP
,and
x
CBP
( ) ep esen s he molecula ib a ion o CBP wi h esonance e-
quency ω
CBP
and damping cons an γ
CBP
.gis he coupling s eng h
be ween he wo esona o s and F
PhP
( ), F
CBP
( ) ep esen he e ec i e
o ces ha d i e each esona o . The e ec i e o ces a e p opo ional
o he nea fields p o ided by he ip. In pa icula , we se F
CBP
( )=0as
in ou expe imen he nea fields o he ip do no ac di ec ly on he
CBP molecules.
Th ough a ime- o- equency Fou ie ans o m o Eqs. (4)and(5),
we find he ollowing s eady-s a e solu ions
^
xPhP ωðÞ=ω2
CBP ω2iγCBPω
ω2
PhP ω2iγPhPω
ω2
CBP ω2iγCBPω
2gωðÞ
2
^
FPhP ωðÞ,
ð6Þ
^
xCBPðωÞ=i2gω
ω2
PhP ω2iγPhPω
ω2
CBP ω2iγCBPω
2gωðÞ
2
^
FPhP ωðÞ,
ð7Þ
wi h ^
xPhP ωðÞ=F½xPhP ðÞ,^
xCBP ωðÞ=F½xCBP ðÞ,
^
FPhP ωðÞ=F½FPhP ðÞ
whe e Fis he ime- o- equency Fou ie ans o m.
Fi ing o ha monic oscilla o model o nea -field spec a
To fi he simula ed spec a (Figs. 3and 4d), we used Eq. (6)plusa
complex- alued o se e m (x
0
+y
0
i), which accoun s o he non-
pola i onic nea -field in e ac ion be ween he ip and he
Table 2 | Pa ame e s o calcula ing he in-plane and ou -o -
plane pe mi i i y componen s o h-BN acco ding o Eq. (1)
In-plane (ε
⊥
)Ou -o -plane(ε
||
)
ε
∞,j
32.8
ω
TO,j
1395 cm−1785 cm−1
ω
LO,j
1630 cm−1845 cm−1
Γ
j
2cm
−11cm
−1
Table 3 | Pa ame e s o calcula ing he pe mi i i y o CBP
acco ding o Eq. (2)
kω
0,k
[cm−1]ω
p,k
[cm−1]Γ
CBP,k
[cm−1]
1 1450.0 128 6.4
21478.6 47 4.4
3 1500.1 91 9.4
41507.4 99 6.1
A icle h ps://doi.o g/10.1038/s41467-022-34393-4
Na u e Communica ions | (2022) 13:6850 7
nano esona o :
^
xsim ωðÞ=^
xPhP ωðÞ+x0+y0i:ð8Þ
To fi he expe imen al spec a (Fig. 4a, b), we mul iply Eq. (6)bya
complex- alued ac o (eiϕ) ha accoun s o he d i o he in e -
e ome e be ween he measu emen o he nano esona o spec a
and he e e ence measu emen on a fla gold su ace:
^
xexp ωðÞ=^
xPhP ωðÞeiϕ+x0+y0i:ð9Þ
In he fi ing p ocedu e, we used as ee pa ame e s ω
PhP
,γ
PhP
,g,
^
FPhP ωðÞ,x
0
,y
0
and ϕ. The pa ame e s ω
CBP
and γ
CBP
a e fi ing pa a-
me e s limi ed o he anges (1450, 1452.5) cm−1and (6,7) cm−1,
espec i ely. F om he fi pa ame e s we ob ain he eigen equencies
ω
±
o he coupled sys em using he ollowing ela ion
ω±=1
2ωPhP +ωCBP
±1
2Re ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
4∣g∣2+Δωi
2Δγ
2
s
2
43
5,ð10Þ
wi h Δω=ω
PhP
-ω
CBP
and Δγ=γ
PhP
-γ
CBP
.Equa ion(10) can be ob ained by
sol ing o he eigen alues o Eqs. (4)and(5), oge he wi h he secula
app oxima ions ω2
PhP ω2
±
=ðωPhP +ω±ÞðωPhP ω±Þ≈2ω±ðωPhP
ω±Þand ω2
CBP ω2
±
=ðωCBP +ω±ÞðωCBP ω±Þ≈2ω±ðωCBP ω±Þ.
Da a a ailabili y
Da a ha suppo he esul s o his wo k a e a ailable upon eason-
able eques om he co esponding au ho .
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Acknowledgemen s
This wo k was suppo ed by he MCIN/AEI/10.13039/501100011033
unde he Ma ía de Maez u Uni s o Excellence P og am (CEX2020-
001038-M) and he P ojec s RTI2018-094830-B-100, PID2021-
123949OB-I00, PID2019-107432GB-I00 and PID2021-122511OB-I00, as
well as by he G aphene Flagship (G apheneCo e3, No. 881603). J.L. and
J.H.E. a e g a e ul o suppo om he O fice o Na al Resea ch (Awa d
No. N00014-20-1-2474), o he BN c ys al g ow h. S.V. acknowledges
financial suppo by he Comunidad de Mad id h ough he A accionde
Talen o p og am (g an no. 2020-T1/IND-20041). C.M.-E., R.E., and J.A.
ecei ed unding om g an no. IT 1526-22 om he Basque Go e nmen
o consolida ed g oups o he Basque Uni e si y.
Au ho con ibu ions
R.H., I.D., and C.M.-E concei ed he s udy. h-BN c ys als we e g own by
J.L., supe ised by J.H.E. I.D. ab ica ed he h-BN nano esona o s and
pe o med he expe imen s and da a analysis. C.M.-E. pe o med he
calcula ions and da a analysis. E.N. and E.M. ab ica ed he diamond
mask. F. Cala alle deposi ed he molecula laye , supe ised by F.
Casano a and L.E.H. S.C. and F.J.A.-M. con ibu ed o simula ions. A.B.
and R.E. pa icipa ed in he da a analysis. S.V. supe ised he ab ica ion
o samples. R.H., I.D., and C.M.-E. w o e he manusc ip wi h inpu om
all au ho s. All au ho s con ibu ed o scien ific discussion and manu-
sc ip e isions. R.H. and J.A. supe ised he wo k.
Compe ing in e es s
R.H. is co- ounde o Neaspec GmbH, a company p oducing sca e ing-
ype scanning nea -field op ical mic oscope sys ems, such as he one
used in his s udy. The emaining au ho s decla e no compe ing
in e es s.
Addi ional in o ma ion
Supplemen a y in o ma ion The online e sion con ains
supplemen a y ma e ial a ailable a
h ps://doi.o g/10.1038/s41467-022-34393-4.
Co espondence and eques s o ma e ials should be add essed o
Raine Hillenb and.
Pee e iew in o ma ion Na u e Communica ions hanks he o he
anonymous e iewe (s) o hei con ibu ion o he pee e iew o
his wo k.
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