Atomistic simulations of electronic structure and coherent transport in 1D & 2D carbon-basednanoarchitectures

Uni e sidad del Pa´ıs Vasco / Euskal He iko Unibe si a ea
Doc o al Thesis
A omis ic simula ions o elec onic s uc u e and
cohe en anspo in 1D & 2D ca bon-based
nanoa chi ec u es
Au ho :
Xabie Diaz de Ce io Palacio
Supe iso :
A an Ga cia-Lekue
2025
(cc) 2025 Xabie Diaz de Ce io Palacio (cc by 4.0)
This hesis has been ca ied ou a Donos ia In e na ional Physics Cen e (DIPC).
Abs ac
The minia u iza ion o silicon-based ield e ec ansis o s (FETs) o e he las 60 yea s
has led o ema kable imp o emen s in he pe o mance o elec onic de ices. Howe e ,
hese basic elec onic componen s ha e al eady eached sizes o a ew nanome e s, whe e
hei cha ac e is ics s a o be a ec ed by quan um mechanical phenomena. One-a om-
hick ma e ials, and specially hose based in g aphene, eme ge as p omising al e na i es
o silicon. In pa icula , i has been p edic ed ha a omically p ecise nanome e -size
po es can in oduce impo an modi ica ions in he elec onic p ope ies o g aphene, o
example opening a gap in he band s uc u e o in oducing quan um in e e ence e ec s
in he elec on anspo . In his hesis, we heo e ically s udy he elec onic p ope ies o
po ous 1D and 2D ca bon-based nanoa chi ec u es ab ica ed wi h a omic p ecision using
he on-su ace syn hesis echnique. We employ densi y unc ional heo y and igh -binding
models in combina ion wi h non-equilib ium G een’s unc ions o simula e he elec onic
s uc u e and cohe en quan um elec on anspo in di e en size scales, anging om
he a omic o ealis ic de ice le els. We shed ligh on a new syn he ic ou e o ab i-
ca e po ous g aphene nano ibbons (GNRs) and we in es iga e how hose nanopo es a ec
he elec onic s uc u e o he sys em. Besides, we explo e new s a egies o une he
elec onic aniso opy and quan um elec on anspo in nanopo ous g aphene (NPG),
a la e al a ay o coupled GNRs. We show ha enginee ing he molecula b idges con-
nec ing adjacen GNRs can be used o enhance he elec onic anis opy and con ol he
elec onic Talbo in e e ence e ec , a phase cohe en phenomena go e ning la ge-scale
elec on anspo in NPG. Apa om in insic modi ica ions o he a omic s uc u e, we
explo e he s acking o NPG on g aphene. We analyze he dependence o he elec onic
p ope ies and he Talbo e ec on he wis angle be ween he laye s and p o ide inge -
p in s o he expe imen al e i ica ion o ou p edic ions. Finally, we add ess he o igin
o he decep i e elec onic densi y con inemen in spec oscopic measu emen s pe o med
wi h scanning unneling mic oscopy on GNRs and NPG. All in all, he simula ions e-
po ed in his hesis p o ide suppo o he unde s anding and analysis o expe imen al
measu emen s. Mo eo e , ou p edic ions open new pa hways owa ds he design o elec-
onic de ices based on he nanoscale con ol o elec onic cu en s and phase-cohe en
anspo phenomena in ca bon-based nanoa chi ec u es.

Resumen (summa y in Spanish)
Du an e los ´ul imos 60 a˜nos, la minia u izaci´on de ansis o es de e ec o de campo (FET)
basados en silicio ha p oducido ex ao dina ios a ances en el endimien o de disposi i os
elec ´onicos. Sin emba go, es os undamen ales componen es elec ´onicos ya han alcan-
zado ama˜nos de unos pocos nan´ome os, donde sus ca ac e ´ıs icas comienzan a es a
in luenciadas po en´omenos mec´anico-cu´an icos. En es e sen ido, los ma e iales de un
solo ´a omo de g oso , especialmen e aquellos basados en g a eno, su gen como al e na i-
as p ome edo as al silicio. En pa icula , se ha p edicho que la c eaci´on con p ecisi´on
a ´omica de po os de ama˜no nanom´e ico puede in oduci modi icaciones impo an es
en las p opiedades elec ´onicas del g a eno, como la ape u a de un “gap” de ene g´ıa
en su es uc u a de bandas o la in oducci´on de e ec os de in e e encia cu´an ica en el
anspo e elec ´onico. En es a esis, ealizamos un es udio e´o ico sob e las p opiedades
elec ´onicas de nanoa qui ec u as po osas 1D y 2D basadas en ca bono, las cuales son ab-
icadas con p ecisi´on a ´omica median e la ´ecnica de s´ın esis en supe icie. En pa icula ,
empleamos la eo ´ıa del uncional de densidad y modelos de enlace ue e en combinaci´on
con unciones de G een ue a de equilib io pa a simula la es uc u a elec ´onica y el
anspo e cu´an ico de elec ones en escalas de ama˜no que a ´ıan en e la escala a ´omica
y la de disposi i os ealis as. A ojamos luz sob e una nue a u a sin ´e ica pa a ab ica
nanocin as de g a eno (GNRs) po osas y analizamos c´omo los nanopo os a ec an a la es-
uc u a elec ´onica del sis ema. Adem´as, explo amos nue as es a egias pa a op imiza la
aniso op´ıa y anspo e elec ´onicos en g a eno nanopo oso (NPG), una ma iz la e al de
GNRs acopladas. Demos amos que la ingenie ´ıa de los puen es molecula es que conec an
GNRs adyacen es puede se u ilizada pa a aumen a la aniso op´ıa elec ´onica y con o-
la el e ec o de in e e encia Talbo elec ´onico, un en´omeno de cohe encia de ase que
gobie na el anspo e de elec ones a g an escala en NPG. Apa e de las modi icaciones
in ´ınsecas de la es uc u a a ´omica, explo amos el apilamien o de NPG sob e g a eno.
Analizamos el e ec o que el ´angulo de o aci´on ela i o en e NPG y g a eno iene sob e
las p opiedades elec ´onicas y el e ec o Talbo y p opo cionamos ca ac e ´ıs icas dis in i-
as del sis ema pa a la e i icaci´on expe imen al de nues as p edicciones. Finalmen e,
abo damos el o igen de la con inaci´on enga˜nosa de densidad elec ´onica en mediciones es-
pec osc´opicas ealizadas con mic oscop´ıa de e ec o ´unel en GNRs y NPG. En conjun o,
las simulaciones epo adas en es a esis p opo cionan sopo e pa a el en endimien o y
an´alisis de mediciones expe imen ales. Adem´as, nues as p edicciones ab en nue as ´ıas
hacia el dise˜no de disposi i os elec ´onicos basados en el con ol de la co ien e elec ´onica
y en´omenos de anspo e cohe en e en nanoa qui ec u as de ca bono.
Lis o publica ions
This hesis is based on he ollowing published con ibu ions:
I. C. Mo eno, X. Diaz de Ce io, M. Teno io, F. Gao, M. Vilas-Va ela, A. Sa asola, D.
Pe˜na, A. Ga cia-Lekue, and A. Muga za. “On-su ace syn hesis o po ous g aphene
nano ibbons media ed by phenyl mig a ion”. Communica ions Chemis y 7, 219
(2024).
II. C. Mo eno, X. Diaz de Ce io, M. Vilas-Va ela, M. Teno io, A. Sa asola, M. B and-
byge, D. Pe˜na, A. Ga cia-Lekue, and A. Muga za. “Molecula b idge enginee ing
o uning quan um elec onic anspo and aniso opy in nanopo ous g aphene”.
Jou nal o he Ame ican Chemical Socie y 145, 8988-8995 (2023).
III. X. Diaz de Ce io, A. Bach Lo en zen, M. B andbyge, A. Ga cia-Lekue. “Twis ing
nanopo ous g aphene on g aphene: elec onic decoupling and chi al cu en s”. Nano
Le e s (2025) (accep ed).
IV. I. Pique o-Zulaica*, E. Co al-Rasc´on*, X. Diaz de Ce io*, A. Riss, B. Yang, A.
Ga cia- Lekue, M. A. Khe -Elden, Z. M. Abd El-Fa ah, S. Nobusue, T. Kojima,
K. Seu e , H. Sakaguchi, W. Auw¨a e , and J. V. Ba h. “Decep i e o bi al con-
inemen a edges and po es o ca bon-based 1D and 2D nanoa chi ec u es”. Na-
u e Communica ions 15, 1062 (2024). (*These au ho s con ibu ed equally o his
wo k).
O he publica ions no co e ed in his hesis:
•D.M.A. Mackenzie, M. Galbia i, X. D. de Ce io, I.Y. Sahaliano , T.M. Radchenko,
J. Sun, D. Pe˜na, L. Gammelgaa d, B.S. Jessen, J.D. Thomsen, P. Bøggild, A.
Ga cia-Lekue, L. Camilli and J.M. Ca idad. “Un a eling he elec onic p ope ies
o g aphene wi h subs i u ional oxygen”. 2D Ma e ials 8, 045035 (2021).
6Chap e 1. In oduc ion
(b)(a)
Figu e 1.3: (a) On-su ace syn hesis ou e ( op) and STM images o NPG (bo om).
The bo om igh STM image was aken using a CO- ip. Figu e aken om Re . [111].
Rep in ed wi h pe mission om AAAS. (b) Mul i-scale DFT+TB elec on anspo sim-
ula ion showing he Talbo in e e ence e ec in NPG. Figu e ep oduced om Re . [112]
wi h pe mission om he Royal Socie y o Chemis y. Adap ed wi h pe mission om
Re . [113]. Copy igh 2019 Ame ican Chemical Socie y.
achie e a omically p ecise g aphene nanos uc u es [81]. OSS p ocedu es s a om small
building-block molecules ha a e assembled as “lego”-pieces in o complex a ge geome-
ies. These molecula p ecu so s a e assembled ia he mally ac i a ed eac ions ca -
alyzed by chemically ac i e me allic su aces (e.g. Au(111)) (see Fig. 1.2a). The elegance
o his bo om-up ab ica ion me hod elies on he ac ha i allows o sys ema ically
ep oduce a omic le el modi ica ions o he p oduc geome y h ough chemically con-
olled adjus me s o he p ecu so monome . As a consecuence, since he ab ica ion o
he i s a omically p ecise GNR 15 yea s ago [82], a my iad o no el ca bon-based nanoa -
chi ec u es exhibi ing all kinds o sizes and geome ies has been designed and ab ica ed
using his me hod. Fo example, GNRs wi h a mchai (AGNR) [82], zigzag (ZGNR) [83]
and chi al (ch-GNR) [84] edge o ien a ions ha e been ealized, whose wid h, edge s uc-
u e and chemical composi ion can be u he modi ied (see Fig. 1.2b) [85–89]. Equally,
a wide ange o 0D nanog aphenes [90–93] and 2D molecula [94,95] assemblies ha e
been ob ained. Mos impo an ly, his has led o a ema kable a ie y o ailo ed elec-
onic p ope ies, including unable semiconduc ing bandgaps [85,86,96], localized and
en angled spins [97–107], and opological bounda y s a es [99,100,102,107–110], highly
a ac i e ea u es o nanoelec onics, sensing and quan um in o ma ion echnologies.
Pa icula ly a ac i e is he abili y o enginee po es a he nanoscale in g aphene [70,
94,114]. In 2018, a ully co alen a omically p ecise NPG was syn hesized o he i s
ime by la e ally using AGNRs o al e na ing 7-a om and 13-a oms wid h (7-13-AGNR),

7
se ing a signi ican miles one in he OSS communi y (see Fig. 1.3a) [111]. This 2D
s uc u e can be iewed as a la e al pe odic a ay o semiconduc ing GNRs ha a e con-
nec ed by in e ibbon molecula b idges. In e es ingly, because he in e ibbon elec onic
coupling is weak compa ed o he kine ic ene gy along he ibbon g ow h-di ec ion, he
esul ing NPG exhibi s ema kable elec onic aniso opy. Mo eo e , heo e ical s udies
ha e e ealed ha such a weak in e ibbon coupling is esponsible o elec onic cu en s
in NPG exhibi ing he elec onic analog o he Talbo in e e ence e ec [113], an op i-
cal phenomena a ising om he in e e ence be ween weakly coupled op ical wa eguides
(see Fig. 1.3b). This ex ends he obse a ion o elec on analogs o op ical phenomena
o semiconduc ing nanos uc u ed g aphene. Fu he mo e, designing new s a egies o
op imize hese e ec s and handle nanoscale cu en s in NPG a ises as a p omising a enue
o esea ch.
An addi ional s iking ea u e o NPG is he eme gence o so-called po e s a es, elec-
onic s a es ha a e s ongly con ined o he nanopo es [111]. These elec onic s a es
ha e been desc ibed as 1D analogs o he image po en ial s a es (IPSs) o g aphene [115],
eme ging on he side o GNR edges a he han abo e he g aphi ic su ace. Due o
hei po en ial sesi i i y o o eign molecula agen s, po e s a es ha e boos ed he pe -
spec i e o using NPG as a molecula nanosenso . Howe e , hese s a es lie a e y high
ene gies abo e he Fe mi su ace, which may hinde hei applicabili y [115,116]. Thus,
unde s anding hei p ope ies and con olling hei ene ge ics a e impo an issues o be
add essed.
Howe e , be o e g aphene nanos uc u es can be used in p ac ical applica ions, an
a omic-le el cha ac e iza ion o hei elec onic p ope ies is equi ed. F om he expe i-
men al side, scanning p obe me hods (SPMs), such as he a omic o ce mic oscope (AFM)
[117] and he scanning unneling mic oscope (STM) [118], ha e demons a ed excep ional
pe o mance in ackling his ask [119]. By measu ing he cu en unneling be ween a
me allic ip and he sample, he STM can p o ide bo h local a omic and spec oscopic
cha ac e iza ion o he sample wi h ema kable sensi i i y o a omic-scale ea u es, making
he elec onic s uc u e and i s eal-space p ope ies accessible [119,120]. By unc ional-
izing he ip wi h a CO molecule, STM and AFM can p o ide high esolu ion images o
he a omic s uc u e, as shown in Fig. 1.3a o NPG, in which e en indi idual C-C bonds
a e iden i ied [121,122].
Along wi h expe imen al ad ances, heo y and compu a ion has de eloped as an es-
sen ial pa ne in he cha ac e iza ion o g aphene-based nanoa chi ec u es. As such,
simula ions and heo e ical s udies play a key ole in unde s anding and p edic ing he
chemical-physical p ocesses o he no el unc ional uni s, e.g. GNRs o NPG, a he
nanoscale. In pa icula , Densi y Func ional Theo y (DFT) has been one o he mos
widely employed me hod o s udy he a omic and elec onic s uc u es o ma e ials, as i
o e s accu a e ab ini io p edic ions in exchange o accep able compu a ional cos . How-
e e , using DFT may become un easible when s udying he p ope ies o la ge-scale sys-
ems consis ing o hund eds o housands o e en millions o a oms. The challenge hen
elies on building simple models whose accu acy is compa able o ha p o ided by ab
ini io me hods. In e es ingly, igh -binding (TB) models can be employed as a i s ap-
p oxima ion o he p oblem. Despi e hei accu acy being o en signi ican ly lowe han
DFT, he TB app oxima ion allows o sys ema ic in es iga ions o la ge sys ems ha
8Chap e 1. In oduc ion
would ake ages o ca y ou using DFT [123,124]. Besides, he ully quan um mechani-
cal non-equilib ium G een’s unc ion (NEGF) app oach can be used o calcula e elec on
anspo p ope ies. The unde lying elec onic s uc u e can be calcula ed using DFT o
simpli ied models such as TB. Rema kably, a mul i-scale app oach combining DFT, TB
and NEGFs has ecen ly allowed elec on anspo simula ions wi h ab ini io accu acy in
g aphene-based de ices o ealis ic size, o example made om NPG (see Fig. 1.3b) [112,
113,125]. All in all, he abo e-desc ibed a omis ic simula ion ools ha e been success ul
o unde s anding expe imen s and p edic ing no el elec onic p ope ies in a wide a ie y
o ca bon-based sys ems.
Ou line o he hesis
In his hesis, we combine DFT, TB and NEGFs simula ions in o de o a ionalize ex-
pe imen al measu emen s and p edic no el phase-cohe en elec onic and anspo phe-
nomena in 1D and 2D ca bon-based nanoa chi ec u es, om he a omic le el o a ealis ic
de ice-scale. In pa icula , he hesis is o ganized as ollows:
In Chap e 2, we desc ibe he heo e ical me hods applied in his hesis. In pa icula ,
we explain he basic heo ems and concep s o DFT, as well as ele an implemen a ion
de ails. We also in oduce he main ea u es o di e en kinds o TB models. Addi ionally,
he non-equilib ium G een’s unc ion (NEGF) me hod and i s applica ion o quan um
elec on anspo calcula ions is p esen ed.
Chap e 3 is ocused on he s udy o wo syn hesized po ous ca bon-based nanoa -
chi ec u es: a po ous GNR and a NPG s uc u e wi h ailo ed in e ibbon molecula
b idges. We sc u inize he phenyl mig a ion eac ion enabling he o ma ion o nanopo es
in GNRs and examine hei e ec on he elec onic p ope ies. Besides, using a combined
DFT+TB mul i-scale app oach, we add ess he ole o enginee ed molecula b idges in he
elec onic aniso opy o NPG. In pa icula , we explo e how he chemical and s uc u al
con igu a ion o he b idges can be used o ine- une he elec onic Talbo in e e e ence
pa e n in NPG.
In Chap e 4, we p opose an al e na i e app oach o modi y he elec onic p ope ies
o NPG by conside ing bilaye s composed o NPG s acked on g aphene wi h di e en
in e laye wis angles. Based on a simple TB model, we show ha he in e laye coupling
is de e mined by he wis angle. We u he demons a e ha he elec onic anspo and
he Talbo in e e ence pa e n in NPG a e highly sensi i e o he in e laye a angemen .
Nex , in Chap e 5 we disen angle he o igin o elec onic densi y con inemen a
he edges and nanopo es o ca bon-based nanoa chi ec u es, a common e ec ha has
led o dubious in e p e a ions in he ield o OSS. Combining DFT calcula ions wi h
a Fou ie analysis o he elec onic wa e unc ions, we a ionalize scanning unneling
spec oscopy (STS) measu emen s in gul - ype GNRs and NPG, showing ha elec onic
densi y con inemen is likely o a ise due o expe imen al limi a ions o he STM a he
han om in insic wa e unc ion p ope ies.
Finally, in Chap e 6, he main conclusions o he hesis a e summa ized and he
u u e pe spec i es a e discussed.
CHAPTER 2
Me hods
In he p esen chap e we in oduce he heo e ical me hods applied in his hesis. These
include densi y unc ional heo y (DFT), he igh -binding (TB) me hod and he non-
equilib ium G een’s unc ion (NEGF) o malism o quan um elec on anspo simula-
ions.
2.1 Densi y Func ional Theo y
DFT is an ab ini io a ia ional heo y ha p o ides an e icien app oach o s udy many-
body sys ems. I s elegance and powe elies on he ac ha he o al ene gy o a sys em
composed o many elec ons and ions can be exp essed only in e ms o he elec onic
densi y. Minimizing he o al ene gy wi h espec o he elec on densi y p o ides he
g ound s a e ene gy and densi y, which de e mine all he p ope ies o he sys em [126].
Al hough he basic concep s o he heo y we e in oduced in he 1960s, i was hanks
o he de elopmen s implemen ed in he ollowing decades ha DFT became a p ac ical
me hod. Since he 1990s, boos ed by an excep ional imp o emen o he compu a ional
capabili ies, i has become one o he mos widely employed me hods o s udying he
elec onic s uc u e o a oms, molecules and ma e ials [127].
Likewise, DFT is one he main me hods applied o calcula e he elec onic s uc u e
o ca bon-based nanoa chi ec u es, and i plays a cen al ole in his hesis. The e o e,
a e in oducing he many-body p oblem o elec onic s uc u e, his sec ion is de o ed
o explaining he basic heo ems and concep s o DFT, as well as he app oxima ions and
me hods adop ed o i s p ac ical applica ion.
2.1.1 The many-body p oblem
S udying he elec onic s uc u e o molecules o c ys als, implies sol ing he Sch ¨odinge
equa ion o sys ems composed o a la ge numbe o ions and elec ons, wha is commonly
known as he many-body p oblem. Fo such a sys em composed o M ions and N elec ons,
10 Chap e 2. Me hods
he many-body Hamil onian ope a o is de ined as
H=−ℏ2
2me
N
X
i=1 ∇2
i−1
2MI
M
X
I=1 ∇2
I+
N
X
i=1
M
X
I=1
ZIe2
4πϵ0| i−RI|
+
N
X
i=1
N
X
j>i
e2
4πϵ0| i− j|+
M
X
I=1
M
X
I>J
ZIZJe2
4πϵ0|RI−RJ|,
(2.1)
whe e ions, wi h mass MI, cha ge ZIeand posi ion RI, a e deno ed by uppe case sub-
sc ip s, while elec ons, wi h mass me, cha ge eand posi ion i, a e deno ed by lowe case
subsc ip s. Thus, he sys em consis s o a o al o 3(M+N) spa ial deg ees o eedom:
3M co esponding o he ions and 3N o he elec ons.
Fo many pu poses whe e he in e ac ions be ween elec ons and ionic ib a ions can
be neglec ed, his ex emely complex Hamil onian is commonly simpli ied by applying he
Bo n-Oppenheime app oxima ion [128]. The la e s a es ha , as ions a e much hea ie
han elec ons, ionic posi ions can be conside ed o be ixed and beha e as pa ame e s.
The ionic kine ic ene gy (second e m in Eq. (2.1)) can be acco dingly neglec ed and he
ion-ion in e ac ion e m ( i h e m in Eq. (2.1)) can be aken as a e e ence ene gy. By
adop ing Ha ee a omic uni s ℏ=me=e= 4πϵ0= 1, he undamen al Hamil onian o
he elec ons is hen w i en as ollows:
ˆ
Helec =ˆ
T+ˆ
Vex +ˆ
Vin ,(2.2)
whe e he elec onic kine ic ene gy ope a o is
ˆ
T=−1
2
N
X
i=1 ∇2
i,(2.3)
he ex e nal po en ial ˆ
Vex is he ionic po en ial ac ing on elec ons,
ˆ
Vex =−
N
X
i=1
M
X
I=1
ZI
| i−RI|,(2.4)
and ˆ
Vin is he elec on-elec on Coulomb in e ac ion,
ˆ
Vin =
N
X
i=1
N
X
j>i
1
| i− j|.(2.5)
The Bo n-Oppenheime app oxima ion p o ides a signi ican educ ion in he numbe
o deg ees o eedom o he many-body sys em. Ye , i s ill yields a complex many-elec on
Sch ¨odinge equa ion wi h no s aigh o wa d solu ion. The ea ly app oaches by Ha ee
(1928) [129] and Fock (1930) [130] achie ed a ema kable simpli ica ion o his p oblem
by ea ing elec ons as independen pa icles. Applying he a ia ional me hod hey
ob ained sel -consis en single-pa icle equa ions desc ibing he mo ion o elec ons in an
e ec i e po en ial due o he ions and he es o elec ons. Al hough hei app oaches laid
2.1. Densi y Func ional Theo y 11
he ounda ions o e ec i e single-pa icle me hods, hey ailed o add ess he elec onic
co ela ions ha a ise om elec on-elec on Coulomb in e ac ions (Eq. (2.5)). Accu-
a ely desc ibing such elec onic co ela ions is he main challenge o elec onic s uc u e
me hods, which has mo i a ed he sea ch o e icien al e na i e me hods o sol e he
many-elec on p oblem [126].
2.1.2 Hohenbe g-Kohn heo ems
Ins ead o a emp ing o calcula e he many-body wa e unc ion o he sys em, he p ob-
lem can be ema kably simpli ied by exp essing i only in e ms o he elec onic densi y.
Thomas [131], Fe mi [132] and Di ac [133] made he i s a emp s o app oxima e he
o al ene gy o a many-body sys em as an explici unc ional o he densi y. Howe e ,
i was no un il 1964 when Hohenbe g and Kohn in oduced he heo ems ha demon-
s a ed ha he o al ene gy o he sys em could indeed be de ined as a unc ional o he
elec onic densi y [134]:
•Theo em I: The ex e nal po en ial, Vex ( ), ( he ion-elec on in e ac ion po en ial,
in ou case) is uniquely de e mined, excep o a cons an , by he g ound s a e
elec onic densi y, n0( ).
•Theo em II: A uni e sal unc ional o he ene gy E[n( )] can be de ined in e ms
o he elec onic densi y n( ), o any ex e nal po en ial. The global minimum o
E[n( )] will be he exac g ound s a e ene gy o he sys em, and he densi y n( )
ha minimizes ha ene gy will be he exac g ound s a e elec onic densi y, n0( ).
I ollows ha he ull Hamil onian is ully de e mined, excep o a cons an shi , by
he g ound s a e elec onic densi y n0( ). This implies ha he many-body wa e unc ions
o all s a es and all he p ope ies o he sys em a e ully de e mined by n0( ). Mo eo e ,
he g ound s a e densi y n0( ) can be exac ly achie ed om he ene gy unc ional E[n( )],
which can be w i en as:
E[n] = T[n] + Eex [n] + Ein [n],(2.6)
whe e T[n] is he kine ic ene gy and
Eex [n] = ZVex ( )n( )d3 , (2.7)
is he ex e nal ene gy. Ein [n] is he Coulomb ene gy o in e ac ing, co ela ed elec ons,
and is composed o he classical Ha ee e m and a non-classical e m desc ibing exchange
and co ela ion:
Ein [n] = EHa ee[n] + Encl[n],(2.8)
whe e
EHa ee[n] = 1
2Zn( ′)n( )
| − ′|d3 d3 ′.(2.9)
As shown by Eq. (2.6), he key idea o exp essing he o al ene gy as a unc ional
o he elec onic densi y p o ides a ema kable simpli ica ion o he many-body p oblem

12 Chap e 2. Me hods
(Eq. (2.2)), as he complexi y o he la e is educed o he h ee deg ees o eedom o
n( ) [126].
The Hohembe g-Kohn heo ems a e he basis o DFT. Impo an ly, unlike p e ious
app oaches o he many-body p oblem, DFT is an exac heo y o co ela ed many-body
sys ems, as no app oxima ions ha e been made apa om he Bo n-Oppenheime ap-
p oxima ion. Ne e heless, despi e he beau i ul simplici y o Hohenbe g and Kohn’s
p oposal, T[n] and Encl[n] a e s ill unknown many-body unc ionals. Fo ins ance, he e
is no known exp ession o he kine ic ene gy in e ms o he densi y, and he only way o
compu ing he exac kine ic ene gy Tis using he many-body wa e unc ion co espond-
ing o he in e ac ing sys em. The challenge hen elies in inding a densi y unc ional
o m o he o al ene gy ha allows o a p ac ical applica ion o he heo y.
2.1.3 The Kohn-Sham ansa z
In 1965, Khon and Sham p esen ed an app oach o ob aining an explici exp ession o
he o al ene gy unc ional [135]. This consis s in eplacing he in e ac ing many-body
sys em by an auxilia y independen -elec on sys em, assuming ha he in e ac ing g ound
s a e densi y o he o me is equal o he independen -elec on densi y o he la e . This
assump ion is called “non-in e ac ing-V- ep esen abili y”, and leads o a desc ip ion in
e ms o independen -elec on equa ions.
In pa icula , he elec onic densi y n( ) o an auxilia y independen -elec on sys em
is gi en by
n( ) =
N
X
i=1 |ψKS
i( )|2,(2.10)
whe e ψKS
i( ) a e he independen -elec on wa e unc ions, o Kohn-Sham wa e unc ions.
Fo p ac ical easons, he e we neglec he elec onic spin deg ee o eedom. The Kohn-
Sham ene gy is hen w i en as ollows:
EKS =Ts[n] + Eex [n] + EHa ee[n] + Exc[n],(2.11)
whe e he kine ic ene gy,
Ts[n] = −1
2PN
i=1 Rd3 ψKS*
i( )∇2ψKS
i( )
=1
2PN
i=1 Rd3 |∇ψKS
i( )|2,
(2.12)
is simpli ied o he sum o he kine ic ene gy o each elec on, and Exc[n] is he exchange
and co ela ion ene gy. Conside ing ha he Kohn-Sham ene gy ep esen s he o al
ene gy o he sys em and equa ing bo h Eqs. (2.11) and (2.6), Exc[n] is exp essed as
Exc =T[n]−Ts[n] + Ein [n]−EHa ee[n].(2.13)
This exp ession desc ibes he physical meaning o Exc[n], which accoun s bo h o he
non-classical e ec s o he in e ac ing Coulomb ene gy (see Eq. (2.8)) and he kine ic
e ms beyond he independen -elec on desc ip ion o he kine ic ene gy.
2.1. Densi y Func ional Theo y 13
In e ac ing many-body sys em Auxilia y independen -elec on sys em
Elec ons
In e ac ions/co ela ions
Ex e nal po en ial
E ec i e Kohn-Sham po en ial:
Independen Kohn-Sham elec ons
Equi alen
Same elec onic densi y
Figu e 2.1: Illus a ion o he Kohn-Sham ansa z, whe e he many-body in e ac ing
sys em is eplaced by an auxilia y independen -elec on sys em wi h he same g ound
s a e densi y.
One can hen apply a a ia ional minimiza ion o Eq. (2.11) wi h espec o ψKS*
i( )
employing he Lag ange mul iplie me hod, which leads o he so-called Kohn-Sham equa-
ions:
h−1
2∇2+Vex ( ) + VHa ee( ) + Vxc( )iψKS
i( ) = εKS
iψKS
i( ),(2.14)
whe e
VHa ee( ) = Zn( ′)
| − ′|d3 ′,(2.15)
Vxc( ) = δExc[n]
δn( ),(2.16)
and εKS
ia e he Kohn-Sham eigen alues.
Eq. (2.14) is a Sch ¨odinge -like equa ion o independen elec ons mo ing in an
e ec i e po en ial
VKS( ) = Vex ( ) + VHa ee( ) + Vxc( ),(2.17)
which depends sel -consis en ly on he elec onic densi y (see Fig. 2.1). The Kohn-Sham
equa ions mus be sol ed i e a i ely o ind he g ound s a e solu ion o he single-elec on
wa e unc ions and he elec onic densi y, as desc ibed in Fig. 2.2. In addi ion, he
equa ions a e gene al o any exp ession o Exc[n], and i Vxc was known, sol ing he
equa ions would p o ide he exac solu ion o he in e ac ing sys em. The cen al p oblem
o DFT is ha he e is no an exac known exp ession o Exc[n], which has mo i a ed
decades o esea ch ocused on inding sui able app oxima ions o exchange-co ela ion
e ec s [126,136].
2.1.4 Exchange and co ela ion unc ionals
A e y con enien poin o he Kohn-Sham ansa z is ha independen -pa icle kine ic
ene gies and he long- ange Ha ee po en ial en e as sepa a e e ms in Eq. 2.14. All
14 Chap e 2. Me hods
Ini ial Guess
n( )
Calcula e he e ec i e po en ial
VKS( )=Vex ( )+VHa ee( )+Vxc( )
Sol e he Kohn-Sham equa ions
[−1
2∇2+VKS]ψKS
i=εKS
iψKS
i
Compu e he elec onic densi y and he o al ene gy
n( )=∑
i
|ψKS
i( )|2→E[n( )]
Con e ged?
Compu e ou pu quan i ies
n( ),E[n( )]→
YES
NO
Figu e 2.2: Sel -consis en i e a i e cycle o sol ing Kohn-Sham equa ions. Upon mak-
ing an ini ial guess o he elec onic densi y he e ec i e Kohn-Sham po en ial is ob ained.
By sol ing he Kohn-Sham equa ion he single-pa icle wa e unc ions a e ob ained, and
he co esponding new elec onic densi y and he o al ene gy a e compu ed. I hese a e
he same as hose ob ained in he p e ious i e a i e s ep, con e gence is achie ed and
ou pu quan i ies a e calcula ed. O he wise, he new elec onic densi y is employed o
build a new e ec i e po en ial and pe o m ano he s ep.
he es o elec onic e ec s included in Exc[n] a e hen ypically expec ed o be sho -
anged and can be app oxima ed ia local o nea ly local unc ionals. Exc[n] can hen be
exp essed as:
Exc[n] = Zn( )ϵxc([n], ),(2.18)
whe e ϵxc([n], ) is he exchange-co ela ion ene gy pe pa icle a posi ion and depends
on he densi y in some neighbo hood o such posi ion. Ve y good esul s ha e been
ob ained employing ema kably simple app oxima ions o ϵxc([n], ). Nex , we e iew
some o he mos commonly used ones.
Local densi y app oxima ion
The simples way o desc ibe Exc is he local densi y app oxima ion (LDA), which akes
he exchange and co ela ion ene gy pe pa icle in a uni o m elec on gas, ϵuni
xc [n], and
2.1. Densi y Func ional Theo y 15
subs i u es in ha exp ession he a ying elec onic densi y o ou sys em n( ). In pa -
icula , he exchange ene gy pe pa icle in he LDA is gi en by
ϵuni
x[n] = −3
43
πn( )1
3.(2.19)
Al hough he co ela ion e m ϵuni
c[n] canno be de e mined analy ically, app oxima e
pa ame ized exp essions ha e been p oposed which a e i ed o nume ical esul s ob-
ained o example using quan um Mon e Ca lo me hods [137]. The LDA exchange-
co ela ion unc ional ELDA
xc [n] is hen calcula ed by adding he exchange and co ela ion
e ms [126]:
ELDA
xc [n] = Zd3 n( )hεuni
x[n( )] + εuni
c[n( )]i
=Zd3 n( )εuni
xc [n( )].
(2.20)
Gene alized g adien app oxima ion
As a nex s ep o LDA, he gene alized g adien app oxima ion (GGA) sugges s εxc[n] o
be no only a unc ion o n( ), bu also o i s g adien . Consequen ly, EGGA
xc [n] can be
exp essed as
EGGA
xc [n] = Zd3 n( )εxc[n( ),|∇n( )|].(2.21)
Nume ous o ms o εxc ha e been p oposed. P obably he simples GGA unc ional is
he one by Pe dew, Bu ke and E nze ho (GGA-PBE) [138], which we employ in mos o
he calcula ions epo ed in his hesis. Al hough GGA p o ides signi ican imp o emen s
o e LDA [126], bo h app oxima ions ha e been success ully es ed in sys ems whe e n( )
a ies slowly. Howe e , hey can be inadequa e in sys ems wi h apidly a ying elec onic
densi ies. This is illus a ed by he ac ha di e en GGA unc ionals can di e ge no ably
in hose si ua ions, while hey yield almos iden ical esul s o he wise [126]. Besides, LDA
and GGA app oxima ions ail o desc ibe long- ange an de Waals o ces, which can be
impo an o ins ance in calcula ions o molecules adso bed on subs a es. Acco dingly,
addi ional ex ensions o he abo e-desc ibed app oxima ions ha e been p oposed.
App oxima ions including dispe sion o ces
Elec on dispe sion o ces, also known as an de Waals o ces, a e long ange a ac ing
in e ac ions p oduced by ins an aneous a ia ions o he elec onic densi y in one egion
o he sys em. These luc ua ions induce a co esponding edis ibu ion o he densi y in
ano he egion, gi ing ise o an a ac i e o ce decaying wi h a leading e m −1
6. In
con as , he locali y o he LDA and GGA p o ides an exponen ial decay o he binding
cu es, consequen ly ailing o add ess a wide a ie y o sys ems whe e non-local dispe sion
o ces play a c ucial ole [139].
In o de o accu a ely desc ibe hese sys ems, se e al co ec ions o LDA and GGA
ha e been de eloped. These can be mainly classi ied in wo g oups. The i s one consis s
22 Chap e 2. Me hods
whe e g0
α(E) = [ESα−Hα]−1is he G een’s unc ion o he isola ed elec ode α. In
p ac ice, SE ma ices unca e ou p oblem o ini e dimensions, while hey accoun o
he unpe u bed, semi-in ini e pa o he elec ode. Physically, hey can be iewed as
e ec i e Hamil onians a ising om he coupling o he de ice egion wi h he elec odes.
They induce a eno maliza ion and b oadening o he de ice egion eigen alues, which
e lec s he ac ha de ice egion eigens a es a e pe u bed and hei li e ime is educed.
Such a ini e li e ime indica es he a e a which elec ons a e injec ed in o o d ained
om he de ice by he elec odes. The p obabili y o an incoming elec on p opaga ing
in o he sca e ing egion and being d ained by an elec ode again de e mines he ans-
po cha ac e is ics o he de ice. Impo an ly, he Landaue -Bu ike o malism and he
NEGFs me hod can be ex ended o an a bi a y numbe o elec odes [125,156,158].
2.3.2 Impo an quan i ies om he G een’s unc ion
Some impo an quan i ies ha can be w i en in e ms o he de ice G een’s unc ion a e
he ansmission unc ion, spec al unc ion, densi y o s a es and he bond- ansmissions
[125,156,157].
T anmission unc ion
The ansmission unc ion p o ides he p obabili y o an elec on being ansmi ed be-
ween elec ode αand β. I can be calcula ed om he G een’s unc ion as ollows:
Tαβ(E) = T [GDΓαG†
DΓβ],(2.38)
whe e
Γα=i[Σα−Σ†
α].(2.39)
Γαis he b oadening o he isola ed de ice egion eigen alues due o coupling o elec ode
α, and ep esen s he a e a which elec ons a e injec ed in o o d ained om he de ice.
Acco ding o he Landaue -B¨u ike o malism, he cu en be ween he wo elec odes
is gi en in e ms o he co esponding ansmission unc ion as:
Iαβ =2e
hZdETαβ(E)[ α(E;µα, τα)− β(E;µβ, τβ)],(2.40)
whe e α/β a e he Fe mi dis ibu ions o each elec ode, wi h chemical po en ial µα/β and
empe a u e τα/β.
Spec al unc ion and densi y o s a es
The spec al unc ion co esponding o elec ode α eads:
Aα(E) = GDΓαG†
D.(2.41)
F om he spec al unc ion, he spec al con ibu ion o he densi y o s a es coming om
each elec ode can hen be w i en as:
ADOSα(E) = 1
2πRe T [Aα(E)S].(2.42)

2.3. Quan um elec on anspo wi h non-equilib ium G een’s unc ion 23
Besides, he de ice densi y o s a es can be exp essed as:
DOS(E) = −1
πIm T [GD(E)S].(2.43)
O en he de ice densi y o s a es can be exp essed as he sum o spec al densi y o s a es
o each elec ode such ha DOS(ϵ) = ΣαADOSα(ϵ). Howe e , i a bound s a e exis s in
he de ice egion ha does no couple o he elec odes, his will no be cap u ed by he
sum o e he spec al densi y o s a es o all elec odes, and he p e ious equali y will no
hold.
Bond ansmissions
Bond ansmissions p o ide a use ul way o isualizing eal-space cu en densi ies coming
om a pa icula elec ode [113,159,160]. They ep esen he local cu en lowing
be ween wo a oms. In pa icula , he o bi al ansmission, i.e. he ansmission be ween
wo o bi als iand jo igina ing om an elec ode α, is gi en by:
Tij(E) = i[(HDji −SDji)Aαij(E)−(HDij −ESDij)Aαji(E)].(2.44)
The bond ansmission be ween wo a oms µand νis hen gi en as a sum o e he
ansmission be ween all o bi als in each a om:
Tµν(E) = X
i∈µX
j∈νTij(E).(2.45)
I ollows om Eq. (2.40) ha bond ansmissions can be ex apola ed o non-ze o bias
condi ions by in eg a ing o bi al ansmissions in he pe inen bias window. The co e-
sponding quan i ies a e called o bi al cu en s
Jij(E) = 2e
hZdETij(E)[ α(E;µα, Tα)− β(E;µβ, Tβ)],(2.46)
and bond cu en s
Jµν(E) = X
i∈µX
j∈ν
Jij(E).(2.47)
The e a e di e en ways o plo ing bond ansmissions [112,123,159,160]. Following
he me hod employed in p e ious wo ks s udying NPG s uc u es [112,113,161,162],
in his hesis we choose o ep esen hem as 2D colo maps in which bond ansmissions
o igina ing om each si e a e summed and in e pola ed on o a 2D g id. Al hough in o -
ma ion abou he di ec ionali y o cu en s is los using his me hod, i p o ides a de ailed
isualiza ion o elec on low in la ge de ices o ealis ic size [125].
2.3.3 De ails o TB+NEGF simula ions
In his hesis, we pe o m quan um cohe en elec on anspo simula ions by combining
TB Hamil onians and NEGFs as implemen ed in he TRANSIESTA u ili y TBTRANS
[157]. Sys em geome ies and Hamil onians a e se -up using he SISL py hon so wa e
24 Chap e 2. Me hods
CAP
Sel -ene gy
i
∆
CAP
Sel -ene gy
i
∆
T anspo di ec ion
Figu e 2.4: Example o a NPG de ice equipped wi h op and bo om elec ode SEs
(yellow a ea), and le and igh CAPs ( ed a ea). CAPs a oid backsca e ing o he
le and igh de ice edges. iand indica e he posi ions whe e CAPs s a and end,
espec i ely.
[163]. In ou calcula ions, he calcula ed de ice G een’s unc ion is exp essed in he
ollowing gene al o m:
GD(E, k) = "SD(k)(E+iη)−HD(E, k)−δH−X
α
Σα(E, k)#−1
.(2.48)
whe e kis he Bloch wa e- ec o in ans e se pe iodic di ec ions pe pendicula o he
anspo di ec ion.
The e a e a ious ways o desc ibe he TB de ice Hamil onian HD. On he one hand,
p uned Hamil onians can be ob ained, only e aining a educed se o he ull DFT ma ix
elemen s. This is he s a egy adop ed in Chap e 3. On he o he hand, one can de ine
a TB Hamil onian whose ma ix elemen s depend on pa ame e s ha ha e o be i ed
o ab ini io calcula ions o expe imen s, as we do in Chap e 4.
The e m Σα(E, k) can be used o in oduce he e ec o elec odes (see Fig. 2.4) and
o he pe u bing componen s in he sys em. Following a ecen ly de eloped mul i-scale
me hod, such pe u ba ions can be ea ed wi h DFT and hen be embedded ia SE
objec s in o la ge-scale de ices desc ibed by TB [112,113]. Fo example, his s a egy
allows o include he e ec o a STM ip in con ac wi h some a oms in he de ice om
DFT calcula ions, as illus a ed in Fig. 2.5. Al e na i ely, such DFT-based pe u ba ions
2.3. Quan um elec on anspo wi h non-equilib ium G een’s unc ion 25
Figu e 2.5: Illus a i e ep esen a ion o he mul i-scale me hod employed o embedding
DFT-calcula ed componen s in o a la ge-scale de ice desc ibed by TB. Figu e ep oduced
om Re . [112] wi h pe mission om he Royal Socie y o Chemis y.
can be eplaced by pa ame ized SE objec s de ined by hand, so ha hey app oxima ely
ep oduce he e ec o an ab ini io SE a a lowe compu a ional cos [125,161]. The
la e is he app oach adop ed in his hesis o simula e he p esence o a STM ip in he
anspo simula ions.
The δHobjec can be used op ionally o accoun o he e ec o ex e nal pe u ba ions
o he Hamil onian, such as cha ge-ca ie doping o magne ic ields [123,125]. He e, we
use i o in oduce complex adso bing po en ials (CAPs) in egions a ound he ini e
de ice edges, as shown in Fig. 2.4. The posi ion dependen CAP e ms a e gi en by:
W( ) = ℏ2
2m2π
∆  ( ),(2.49)
whe e
( ) = 4
c2∆
−2 i+ 2+∆
− 2−2.(2.50)
He e, c= 2.62, iand a e he posi ions whe e he CAP egion s a s and ends, e-
spec i ely, and ∆ = − iis he CAP egion wid h [164]. In he selec ed egion, he
CAP in oduces on-si e imagina y e ms −iW( ) whose magni ude g ows p og essi ely
as app oaches , i.e. as he dis ance o he edge dec eases, e en ually di e ging on
he de ice edge ( = ). This in oduces an on-si e decay a e ha adso bs elec ons
in he a oms close o he edge. The e iciency o CAPs imp o es wi h hei wid h ∆ ,
and i has been shown ha ew nanome e s-wide egions a e able o comple ely d ain
elec ons and a oid backsca e ing o he de ice edges [123,125]. CAPs can he e o e
be employed as a compu a ionally less cos ly al e na i e o SEs in o de o mimic open
bounda y condi ions in ans e se di ec ions, and hey indeed play a c ucial ole in many
o he simula ions epo ed in his hesis.
CHAPTER 3
Po ous 1D and 2D nanoa chi ec u es: disen angling
phenyl mig a ion and elec onic aniso opy
The NPG ob ained by using AGNRs o al e na ing 7-a om and 13-a om wid h (7-13-
AGNRs) exhibi s a highly aniso opic elec onic s uc u e. On he one hand, VBs and CBs
ha e a ma ked longi udinal (L) cha ac e , hei dispe sion being s ong along he g ow h
di ec ion o he ibbons and weak along he pe pendicula di ec ion. Mo eo e , due o
he weak in e ibbon coupling, he o he wise degene a e on ie bands coming om he
wo using ibbons (see uni cell o NPG in Fig. 3.1a) hyb idize and exhibi a small ene gy
spli ing [111]. This gi es ise o he elec onic Talbo in e e ence e ec , which can be
unde s ood as a s anding wa e pa e n ha a ises om he supe posi ion o equal-ene gy
wa e unc ions coming om wo longi udinal bands wi h a small momen um di e ence
(see Fig. 3.1b) [113]. On he o he hand, al hough no explo ed in his wo k, NPG also
hos s elec onic bands o ans e sal (T) cha ac e a highe ene gies, whose dispe sion
is small along he g ow h di ec ion o he ibbons and s onge along he pe pendicula
di ec ion. In addi ion, he po ous s uc u e leads o he eme gence o so-called po e s a es
(P) a high ene gies. These esemble he supe a om molecula o bi als (SAMOs) ound
in ulle enes [165] and ha e gene a ed signi ican in e es o hei po en ial applica ion
in molecula sensing (see Fig. 3.1a) [111].
In e es ingly, he ema kable lexibili y p o ided by OSS has allowed o op imize he
p omising elec onic p ope ies o NPG h ough a omic-scale a ia ions o i s geome y
[166–168]. Fo example, la e al supe la ices o elec os a ically ga ed GNRs ha e been
ab ica ed upon subs i u ion o indi idual N a oms in hei in e ibbon molecula b idges
[168]. F om a heo e ical pe spec i e, simila s a egies o molecula b idge enginee -
ing ha e also been conside ed in o de o une he Talbo e ec . In pa icula , chemi-
cal coupling modi ica ions o chemical unc ionaliza ion can cause des uc i e quan um
in e e ence in he b idges, ab up ly quenching in e ibbon elec onic ansmission and
enhancing he aniso opy o elec onic cu en s [161,162]. Thus, de eloping a syn he ic
s a egy owa ds expe imen al molecula b idge enginee ing as well as de ising al e na i e
con ol knobs o ine- uning quan um in e e ence e ec s a e key o ha nessing cohe en
anspo phenomena in NPG de ices.
Op imizing he ene ge ics o po e s a es by modi ying he size and shape o nanopo es
is ano he impo an equi emen in o de o ully exploi he mul i unc ional capabili ies
o NPG and o he po ous nanoa chi ec u es. In his ega d, he design o new syn-
he ic mechanisms wi hin he OSS oolbox may con ibu e o keep c ea ing inno a i e

28 Chap e 3. Po ous 1D and 2D nanoa chi ec u es
(a) (b)
La e al
usion X
Z
Figu e 3.1: (a) Elec onic band s uc u e o single a 7-13-AGNR (le ) and NPG ( igh ).
Longi udinal (L), ans e sal (T) and po e (P) s a es a e highligh ed in yellow, pu ple
and g een, espec i ely. The a omic s uc u es o he 7-13-AGNR and NPG a e schema -
ically depic ed below he band s uc u es. Figu e aken om Re . [111]. Rep in ed wi h
pe mission om AAAS. (b) In e e ence pa e n eme ging om he supe posi ion o wa e
unc ions coming om equal-ene gy s a es in he wo longi udinal CBs. Adap ed wi h
pe mission om Re . [113]. Copy igh 2019 Ame ican Chemical Socie y.
geome ies. In pa icula , su ace-assis ed he mally-ac i a ed phenyl mig a ion has been
ecen ly epo ed o induce in e nal ans o ma ions in molecules [169,170]. Achie ing
selec i e con ol on his ype o eac ions is a necessa y condi ion o each he a ge
p oduc s sys ema ically and eliably.
In his chap e , we in oduce wo o iginal, expe imen ally syn hesized ca bon-based
nanoa chi ec u es ab ica ed om he same pa en GNR ia di e en syn he ic ou es:
1) a po ous GNR ob ained h ough a phenyl mig a ion eac ion a he ibbon edge; and
2) a NPG s uc u e wi h enginee ed molecula b idges syn hesized by he la e al c oss-
coupling o pa en ibbons. A e in oducing hei syn hesis p ocedu e and suppo ed by
expe imen al measu emen s, we heo e ically un eil he pa icula eac ion pa h leading
o phenyl mig a ion, as well as he e ec o po e o ma ion on he elec onic s uc u e
o he GNRs. In he case o NPG, we explo e he in luence ha i s di e en s uc u al
pa ame e s ha e on he elec onic aniso opy and quan um cohe en anspo phenomena.
3.1 On-su ace syn hesis o po ous GNRs and NPG
The syn he ic ou e ollowed o ob ain po ous GNRs and NPG is summa ized in Fig. 3.2,
along wi h STM images o he p oduc s a e each eac ion s ep. Bo h esul ing s uc-
3.1. On-su ace syn hesis o po ous GNRs and NPG 29
u es sha e he same p ecu so monome , called 2’-di([1,1’-biphenyl]-4-yl)-10,10’-dib omo-
9,9’bian h acene (DBP-DBBA), which is e y simila o he DP-DBBA monome em-
ployed o syn hesize he NPG s uc u e in Re .[111]. The only di e ence be ween hem
is he addi ion o single phenylene uni s a he le and igh sides o he DBP-DBBA,
which a e key elemen s o he syn hesis o he po ous GNRs and NPG in oduced in
his chap e . The OSS o bo h s uc u es sha e he same ini ial syn hesis s eps desc ibed
below, all he way un il he o ma ion o so-called Ph-7-13-AGNRs. The p ocedu e is pe -
o med in ul a-high acuum condi ions, and s a s by deposi ing p ecu so DBP-DBBA
molecules on a Au(111) su ace held a oom empe a u e (Fig. 3.2a). Subsequen sub-
s a e annealing a T1= 200◦C igge s he so-called Ullmann coupling eac ion be ween
monome s, leading o he o ma ion o 1D polyme ic chains (Fig. 3.2b). By inc easing he
annealing empe a u e o T2= 400◦C he cyclodehyd ogena ion eac ion in he polyme s
is ac i a ed, which induces hei plana iza ion and he o ma ion o Ph-7-13-AGNRs.
The la e consis o 7-13-AGNRs o med o al e na ing 7-C-a om and 13-C-a om-wide
sec ions [111], wi h addi ional phenyl ings ha appea single-bonded o he edges o
13-C-a om-wide sec ions (Fig. 3.2c). Impo an ly, Ph-7-13-AGNRs e ain he p ochi al
con igu a ion o hei pa en monome (i.e. hey a e chi al when con ined in 2D), and can
appea in Sand Ro ien a ions.
A his poin , expe imen s e eal ha u he annealing he subs a e can igge wo
di e en scena ios depending on he su ounding en i onmen o Ph-7-13-AGNRs.
1) I Ph-7-13-AGNRs a e aligned and ee o di use la e ally, dehyd ogena i e c oss
coupling is he mally induced a T3= 450◦C and adjacen ibbons use o o m
co alen 2D NPG s uc u es (Fig. 3.2e). In con as o NPGs ob ained om 7-
13-AGNRs [111], h ee di e en bonding con igu a ions a e ob ained depending on
he chi ali y o he ibbons and he si e in he phenyl side-g oup pa icipa ing in
he co alen bond (o ho,pa a o me a, see Fig. 3.2c): homochi al ibbons c ea e
pa a-pa a (pp) o me a-me a (mm) con igu a ions, while he e ochi al ibbons bind
ia pa a-me a (pm) coupling. Bonds be ween o ho posi ions a e no ound, as hey
a e s e ically hinde ed by he hyd ogens a he adjacen phenyl uni and he ac
ha i would in ol e he o ma ion o mo e han one bond. The obse a ion o la ge
de ec - ee NPG lakes demons a es he success ul o ma ion o NPG wi h dis inc
in e ibbon b idges.
2) When he la e al coupling be ween GNRs is inhibi ed, a new eac ion pa h opens a
T3= 450◦C. In his scena io, he ou e mos phenylene ing o he ibbon mig a es
om he C2(C3) si e o he neighbo ing equi alen posi ion C3(C2) a he edge
o he 13-C-a om-wide sec ion (see Fig. 3.3a and Fig. 3.4a). In some cases, his
mig a ion is ollowed by a dehyd ogena i e c oss coupling wi h an adjacen phenyl
side g oup, leading o he o ma ion o GNRs unc ionalized wi h [18]-annulene po es
([18]-annulene GNRs) (Fig. 3.2d).
In he nex wo sec ions we p esen combined expe imen al and heo e ical s udies o
each o hese nanos uc u es. An special emphasis is pu on he ole o ou DFT-based
simula ions in co obo a ing he expe imen al STM cha ac e iza ion and p edic ing he
elec onic p ope ies o hese ma e ials.
30 Chap e 3. Po ous 1D and 2D nanoa chi ec u es
DBP-DBBA monome
Polyme
Ph-7-13-AGNR
T1 = 200 ºC Ullmann Coupling
T2 = 400 ºC Cyclodehyd ogena ion
NPG
T3 = 450 ºC T3 = 450 ºCDehyd ogena i e C oss CouplingPhenyl mig a ion
[18]-annulene GNR
(a)
(b)
(c)
(d) (e)
Figu e 3.2: S ep-by-s ep syn he ic ou e ollowed o ob ain [18]-annulene GNRs and
NPG, along wi h STM images o he p oduc s a each s ep. (a) DBP-DBBA p ecu -
so monome . Sand Rindica e he wo possible enan iome s eme ging when con ining
he monome o 2D. (b) Polyme chain o med ia Ullmann coupling o DBP-DBBA
molecules. (c) Ph-7-13-AGNR. Illus a ion ( igh ): O ho,me a and pa a labels in he
phenyl side-g oup indica e he 3 di e en posi ions a which he ibbon can o m co a-
len bonds. STM images (le ): igh STM image is a bond- esol ed STM (BR-STM)
image aken wi h a CO- ip. (d) [18]-annulene GNR. Illus a ion ( op): shaded phenyl
side-g oups indica e he o iginal posi ion o phenyl ings be o e mig a ion akes place.
STM images (bo om): igh STM image is BR-STM. (e) NPG. Illus a ion ( op): he
h ee possible bonding con igu a ions, namely pa a-pa a,me a-me a and pa a-me a, a e
depic ed, and he chi ali y S/R is indica ed on op o each ibbon. Reac ion names and
ac i a ion empe a u es and indica ed on each s ep. Figu e aken om Re s. [171,172].
3.2. Po ous GNRs: phenyl mig a ion and elec onic s uc u e 31
3.2 Po ous GNRs: phenyl mig a ion and elec onic s uc-
u e
3.2.1 A omic s uc u e cha ac e iza ion using STM
Fig. 3.3a shows a STM image ob ained a e he phenyl mig a ion eac ion has aken
place a T3= 450◦C on a la Au(111) su ace. Besides he o ma ion o [18]-annulene
po es a he ibbon edges (con igu a ion 1), wo o he scena ios a e also ound. When a
mig a ed phenyl does no ind ano he side-g oup in he adjacen uni -cell o which i can
couple, he phenyl ing emains ee a he edge (con igu a ion 1’). On he o he hand,
phenyl mig a ion om C2(C3) si es o C1(C4) si es in he bay a eas o he ibbon can
also occu . In ha case, he mig a ed phenyl ing uses wi h he ibbon backbone ia
G own on la Au(111)
G own on s epped Au(111)
(a)
(b)
(c)
Figu e 3.3: Expe imen al esul s o [18]-annulene-GNR o ma ion. (a) STM image o
GNRs g own on a la Au(111) su ace a e phenyl mig a ion eac ion has been he mally
induced a T3= 450◦C. The di e en p oduc con igu a ions ound in he expe imen a e
indica ed wi h numbe s in he STM image and schema ically ep esen ed on he igh side
o he panel. (b) STM image o GNRs g own on a cu ed Au(111) su aces a e phenyl
mig a ion eac ion has been he mally induced a T3= 480◦C. (c) S a is ical analysis
o he selec i i y ( op) and yield (bo om) ob ained on he la Au(111) annealead a
T3= 450◦C (g een), and he cu ed Au(111) annealead a T3= 480◦C (blue). Selec i i y
is measu ed as 1/(1 + 1′+ 2) ×100. The yield accoun s o he ela i e a io o phenyls
o ming [18]-annulene po es. Figu e aken om Re . [172].
38 Chap e 3. Po ous 1D and 2D nanoa chi ec u es
Ph-7-13-AGNR
[18]-annulene GNR
Bonding Bonding
An ibonding An ibonding
Figu e 3.9: Band s uc u e and lowes -ene gy acuum-s a e wa e- unc ions o Ph-7-13-
AGNRs (le ) and [18]-annulene GNRs ( igh ). The uni -cell o he o me was a i icially
doubled o acili a e compa ison. Vacuum s a e bands a e highligh ed wi h hicke lines.
shows he band s uc u e and lowes -ene gy bay and po e s a e wa e- unc ions o Ph-
7-13-AGNRs and [18]-annulene GNRs. The uni -cell o he o me has been a i icially
doubled o acili a e compa ison. In each o he ibbons, acuum-s a es (highligh ed wi h
hicke lines) consis o 4 bands, co esponding o wo pai s o s a es wi h bonding and
an ibonding cha ac e in he pe iodic di ec ion, labeled as “bonding” and “an ibonding”.
No e ha in he case o Ph-7-13-AGNRs his is only due o a i icial band- olding, and
bonding and an ibonding s a es belong o he same band bu ha e di e en Bloch phases.
Consequen ly, bonding and an ibonding bands a e degene a e a he X-poin . On he
con a y, he in oduc ion o po es in [18]-annulene GNRs doubles he uni -cell, and hei
acuum-s a e bands a e no degene a e a X. Besides, he wo s a es wi hin each o
hese pai s labeled as “bonding” and “an ibonding” also exhibi bonding and an ibonding
cha ac e ac oss he ibbon backbone. This is associa ed wi h a iny band spli ing which
ac ually e lec s some weak in e ac ion ac oss he ibbon backbone be ween acuum s a es
o di e en edges. Howe e , his spli ing is impe cep ible in he ene gy scale selec ed o
ep esen ing he band s uc u e.
3.3 NPG: molecula b idge enginee ing
A e ho oughly s udying he o ma ion and elec onic s uc u e o [18]-annulene GNRs,
we now u n o he case o la e al usion o Ph-7-13-AGNRs gi ing ise o NPG s uc u es
(see Fig. 3.2e). In o de o s udy he elec onic p ope ies o pp-pm- and mm-NPG, i is
essen ial o i s p o ide a de ailed desc ip ion o hei a omic s uc u e.

3.3. NPG: molecula b idge enginee ing 39
3.3.1 A omic s uc u e o molecula b idges
Bond- esol ed STM (BR-STM) images o he h ee di e en molecula b idges (see
Fig. 3.10a; see also Sec. 3.1 and Fig. 3.2e) ob ained in he syn hesis o NPG a e shown in
Fig. 3.10b. In e es ingly, he image con as depends on he ype o bond o med by he
phenylene uni s in he b idges. p-phenylenes appea as sha p con as ea u es, which is
a common cha ac e is ic o nonplana molecula g oups measu ed wi h a CO- ip in STM
[183]. On he o he hand, m-phenylenes exhibi he same con as as he C hexagons
composing he honeycomb s uc u e o he 7-13-backbone. These expe imen al esul s
a e co obo a ed by compa ison o DFT calcula ions o pp-pm- and mm-NPG s uc-
(a)
(b)
(c)
(d)
Expe imen Theo y
Figu e 3.10: (a) Schema ic ep esen a ions o pp-, pm- and mm-NPG s uc u es. The
a ows and he θand αangles ep esen he o a ion o eedom o phenyl ings. (b)
Expe imen al BR-STM images o pp-, pm- and mm molecula b idges. The a ows in-
dica e wis ed phenyl ings. (c) Phenyl- ibbon (θ, g een-pand ed-m) and in e -phenyl
(α, black) angles in he ee-s anding (solid ci cles) and Au(111)-suppo ed (open ci cles)
NPG s uc u es. (d) Side iew o ee-s anding ( op) and Au(111)-suppo ed (bo om)
NPG s uc u es, whe e he colo s indica e ou -o -plane displacemen s o each a om wi h
espec o he a e age e ical posi ion. The Au(111) subs a e is omi ed in he igu e.
Figu e aken om Re . [171].
40 Chap e 3. Po ous 1D and 2D nanoa chi ec u es
u es in a ee-s anding con igu a ion and adso bed on Au(111)1, as shown in Figs. 3.10c,d.
Speci ically, he b idge con o ma ion is cha ac e ized by wo di e en wis angles in he
elaxed s uc u es: one be ween he plane o he ou e mos benzene ing o he GNR
backbone and he adjacen phenyl ing (θ), and he second be ween he wo phenyl ings
a he b idge (α). In con as o expe imen s, calcula ions on ee s anding NPGs e eal
a nonplana con o ma ion d i en by s e ic hind ance o he h ee ypes o b idges [184],
wi h θand αangles exceeding 15◦and 30◦, espec i ely. This esul sugges s ha he
subs a e plays a c ucial ole in s abilizing he speci ic b idge con o ma ions obse ed in
BR-STM images. In ac , calcula ions o NPG adso bed on Au(111) show an o e all pla-
na iza ion o he h ee NPG s uc u es (see Fig. 3.10c,d). This is specially p ominen o
m-phenylenes, while p-phenylenes e ain a no ably il ed con o ma ion. This e ec is mos
isible in pm-b idges, bu i can also be no iced om he mo e p onounced plana i y o
mm-b idges compa ed o pp-b idges. In pa icula , i is wo h no ing ha he in e phenyl
wis angle αis as high as ∼30◦in pp-NPG. The selec i e plana iza ion o m-phenylenes
ia in e ac ion wi h he subs a e and he concomi an wis o p-phenylenes cons i u es
an impo an s uc u al ea u e wi h c ucial consequences in he elec onic p ope ies o
NPG, as we will explain in he ollowing.
3.3.2 Band s uc u e and in e ibbon coupling
Aimed a unde s anding he in luence o he chemical bonding con igu a ion and he
con o ma ion o molecula b idges on he elec onic s uc u e o he syn hesized NPGs,
we pe o m DFT simula ions o ee-s anding pp-, pm- and mm-NPGs. Pe iodic NPG
s uc u es a e a anged in he XY plane, whe e Y and X axes co espond o he g ow h
di ec ion o he ibbons and he ans e se di ec ion, espec i ely. The ec angula uni -
cells a e 48.14 ˚
A×8.69 ˚
A, 48.05 ˚
A×8.69 and 48.01 ˚
A×8.69 o he pp-, pm- and
mm-NPG, espec i ely, and a acuum egion o 20 ˚
A is used in he e ical Z-di ec ion
in all o hem. The uni -cells consis o 104 C and 36 H a oms. In o de o disen angle
he ole played by he bonding con igu a ion om ha o i s speci ic con o ma ion (i.e.
he wis angle be ween he phenyl ings), we s a by conside ing plana geome ies wi h
ixed e ical a omic coo dina es. A omic posi ions and la ice ec o s a e hen elaxed in
he XY plane wi h o ce and p essu e ole ances o 0.01 eV/˚
A and 0.25 GPa, espec i ely.
The a omic s uc u e o elaxed uni -cells a e p o ided in Fig. 3.11a. Exchange-co ela ion
is ea ed in he GGA-PBE app oxima ion [138]. Co e elec ons a e simula ed using
T oullie -Ma ins pseudopo en ials [177], while alence elec ons a e ep esen ed using a
linea combina ion o localized NAOs. We employ DZP o bi als, wi h an ene gy shi
pa ame e o 0.01 Ry [148]. The eal-space in eg a ion g id is de ined by a cu o ene gy
o 400 Ry, while we use a Monkho s -Pack g id o 4 ×15 k-poin s o BZ sampling [181].
E ec o chemical bond
The calcula ed band s uc u es o he h ee plana NPGs a e shown in Fig. 3.11a. The
case o pp-NPG quali a i ely esembles ha o NPG in he absence o phenyl side-g oups
1DFT calcula ions shown in Fig. 3.10 we e pe o med by D . Ane Sa asola I˜niguez. The es o he
simula ions in his hesis we e pe o med by Xabie Diaz de Ce io Palacio, unless o he wise s a ed.
3.3. NPG: molecula b idge enginee ing 41
[111]. In pa icula , i exhibi s a ini e spli ing o degene a e alence and conduc ion
single- ibbon bands. This beha iou is due o he in e ibbon elec onic coupling, and
indica es a ini e ansmission p obabili y in he ans e sal di ec ion. This band spli -
ing is also esponsible o a sligh educ ion o he bandgap wi h espec o he pa en
Ph-7-13-AGNR. On he con a y, he ene gy spli ing o he on ie bands is d as ically
educed upon he inclusion o m-bonds in he b idges, demons a ing an e ec i e quench-
ing o in e ibbon c oss alk. Acco dingly, he band s uc u e o bo h pm- and mm-NPG
closely mimic ha o single-Ph-7-13-AGNR a ound he on ie band onse s, wi h neg-
ligible dispe sion in he ans e sal X-di ec ion. The ac ha a single m-phenylene is
enough o almos comple ely p e en in e ibbon coupling is in good ag eemen wi h a
p e ious wo k on polyphenylene chains, whe e elec on p opaga ion is g ea ly hampe ed
ac oss single m-connec ions [185]. Simila conclusions we e also ex ac ed om heo e -
ical s udies on NPG s uc u es wi h 7-13-AGNRs bonded by single m-phenylene uni s
[161]. In pa icula , he elec onic decoupling be ween adjacen ibbons is a ibu ed o
des uc i e quan um in e e ence ocu ing in m-bonds, as ex ensi ely explo ed in expe i-
men al [186,187] and heo e ical [184,188–191] in es iga ions o elec on anspo ac oss
molecula junc ions.
Mo eo e , quan i a i e in o ma ion on he in e ibbon coupling is ex ac ed om he
ene gy esol ed momen um spli ing o alence and conduc ion on ie bands ∆k. This
(a)
(b)
Figu e 3.11: (a) Band s uc u e o plana pp-(g een), pm-(black), and mm-NPG ( ed),
ep esen ed in di ec ions pa allel (Γ →Y) and pe pendicula (Γ →X) o he ibbon
g ow h di ec ion. A omis ic models o he elaxed s uc u es a e p o ided below each
band s uc u e. (b) Ene gy- esol ed in e channel coupling coe icien κc= ∆k/4 ex ac ed
om he momen um spli ing o he longi udinal bands o each NPG s uc u e. The black
ho izon al line a E=EVBM −0.2 eV indica es he ene gy posi ion a which κcis analyzed
as a unc ion o he phenylene wis -angle in Fig. 3.12. Figu e aken om Re . [171].
42 Chap e 3. Po ous 1D and 2D nanoa chi ec u es
quan i y is di ec ly ela ed o he in e channel coupling coe icien , κc= ∆k/4, which
cha ac e izes he Talbo in e e ence e ec [113]. Fig. 3.11b shows κcas a unc ion o he
ene gy o he h ee di e en NPG s uc u es. The con as be ween pp-NPG and pm- and
mm-NPG is e iden , showing ha he b idge bonding con igu a ion ac s as chemical knob
o he in e ibbon c oss alk. Mo e speci ically, pp-b idges lead o quali a i ely highe
momen um spli ing o e he whole ene gy ange spanned in he igu e. Fo example,
−0.2 eV below he VBM, κcis educed by mo e han a ac o o 5 by in oducing a single
me a-connec ion in he molecula b idge.
E ec o a omic con o ma ion in pp-NPG
The combined expe imen al and DFT cha ac e iza ion o he a omic s uc u e in Fig. 3.10
e eals he endency o p-phenylenes o be wis ed wi h espec o he plana ibbon back-
bone. Tha indica es ha he elec onic p ope ies o NPG migh be u he uned by
manipula ing his con o ma ional deg ee o eedom [192,193]. To analyze he ela ion
be ween b idge con o ma ion and in e ibbon coupling, we pe o m addi ional DFT calcu-
la ions on ee-s anding pp-NPG. S a ing om he plana geome y shown in Fig. 3.11a,
we sys ema ically inc ease he wis angle o he wo phenyl ings in he b idge by he
same amoun and in opposi e di ec ion (i.e. α= 2θ). No u he elaxa ion o he ge-
ome y is pe o med in o de o isola e he e ec o he wis angle on he elec onic
s uc u e. The band s uc u es co esponding o θ= 0◦,20◦and 45◦(α= 0◦,40◦and 90◦,
espec i ely) o a ions a e shown in Fig. 3.12. The esul s show ha he phenylene wis
angle ac s as a con o ma ional knob o he in e ibbon c oss alk. In pa icula , he band
spli ing exhibi s a g adual educ ion o he in e ibbon coupling wi h inc easing θ. No e
ha he θ= 45◦case almos exac ly beha es as an a ay o isola ed Ph-7-13-AGNRs wi h
negligible ans e sal g oup eloci y.
Because he wis angle can be con inuously a ied, i is in e es ing o ack he in e -
ibbon c oss alk in he whole ange o wis angles. Fig. 3.12b (bo om panel, g een do s)
shows he momen um spli ing o pp-NPG longi udinal bands a an ene gy o −0.2 eV be-
low he VBM o 25 di e en wis angles θ(α) uni o mly dis ibu ed be ween 0◦(0◦) and
90◦(180◦). In con as o he ab up on/o swi ching o e ed by he chemical knob, he
con o ma ional knob p o ides con inuous con ol on he momen um spli ing. In pa icu-
la , κcis maximized in he plana con igu a ion (θ= 0◦) and ini ia es a smoo h dec ease
as he phenyl ings a e wis ed. I eaches a minimum a θ≈41.75◦, whe e i becomes
negligible. Ne e heless, he e olu ion o κcis non-mono onic, and i expe iences a sligh
g ow h o e he nex ∼15◦. This end is again e e sed a ound θ= 60◦, whe e κcbegins
a new dec ease ha culmina es a θ≈82.5◦.
This speci ic beha iou can be unde s ood by mapping he molecula b idge o a simple
π-o bi al TB model in he wo-cen e app oxima ion, whe e he ou phenyl ings o he
b idge a e subs i u ed by a linea chain o ou π-o bi als. The hopping be ween wo
o bi als in he chain a ies wi h he wis angle (θ) be ween hem as ∼ ×cos θ. Thus, he
e olu ion o he o al hopping ac oss he pp-b idge, and he e o e he momen um spli ing,
can be app oxima ed as κc(θ)∼κc(0◦)×cos2(θ)×cos(2θ), whe e we only conside ed
hoppings be ween NNs and he ac ha α= 2θ. The esul ob ained om his model by
i ing κc(0◦) o he DFT esul is ep esen ed by he blue cu e in Fig. 3.12b, showing
3.3. NPG: molecula b idge enginee ing 43
(b)
EVBM - 0.2eV
(a)
Figu e 3.12: (a) Band s uc u e o pp-NPG s uc u es wi h plana con o ma ion (θ=
0◦) and wi h phenyl ings wis ed by θ= 20◦and θ= 45◦. (b) Co esponding o al
ene gy ( op) and in e channel coupling coe icien κc= ∆k/4 e alua ed a E=EVBM −
0.2 eV (as indica ed in panel (a)) (bo om) as a unc ion o he phenyl- ibbon (θ) and
in e phenyl (α) wis angles. Thick g ay and yellow lines indica e he angles calcula ed o
he ee-s anding and he Au(111)-suppo ed s uc u es. The inse in op panel p o ides
an schema ic ep esen a ion o phenylene o a ion in he b idges and he co esponding θ
and α wis angles. Figu e aken om Re . [171].
good quali a i e ag eemen wi h DFT simula ions. In pa icula , he i s minimum a
θ= 45◦co esponds o an in e phenyl angle o α= 90◦, o which he hopping be ween he
wo cen al phenyl ings is ully quenched due o a comple e loss o local π-conjuga ion.
No e ha he hopping be ween he wo cen al phenyl ings is symme ic wi h espec
o α→180◦−α, because hey g adually eco e coplana i y as he in e phenyl angle α
g ows beyond 90◦. Fo θ > 45◦, his leads o a si ua ion whe e he p og essi e b eaking
o he π-conjuga ion in opposi e ends o he b idge compe es agains he g adual eco e y
o local π-conjuga ion a he cen al phenylene uni s. The ou come o ha compe i ion
de e mines he non-mono onic beha iou o κc(θ), which e en ually d ops o ze o o
θ= 90◦, as he π-conjuga ion b eaks comple ely a he wo ends o he b idge. Despi e
he success ul explana ion p o ided by his model, DFT esul s exhibi sligh de ia ions
om his o e simpli ied pic u e (see o ins ance he small downshi o he angles a which
he spli ing is minimized). This quan i a i e disag eemen may be assigned o he ac
ha we a e neglec ing hopping p ocesses beyond NNs, as well as he e ec o he a omic
en i onmen and he σ-o bi als [192].
Based on he beha iou o he in e ibbon c oss alk as a unc ion o θ/α, he ange
o angles α∈[0◦,90◦] ep esen s he egion o in e es o uning in e ibbon coupling.
The op panel in Fig. 3.12b shows he DFT o al ene gy o ee-s anding pp-NPG as a
unc ion o he wis angle (g een solid ci cles). The o al ene gy landscape exhibi s a
∼200 meV well a ound he minimum a α= 47◦, which is wi hin he wis angle ange o

44 Chap e 3. Po ous 1D and 2D nanoa chi ec u es
in e es . The shape o he ene gy landscape a ound he minimum can be used o es ima e
he mally induced oscilla ions o α. Fo example, oom empe a u e o a ional ene gy
(1
2kBT= 12.7 meV) could induce oscilla ions o ±8◦a ound he minimum. Beyond he
icini y o he i s minimum, he o al ene gy inc eases wi h α, and does no display any
o he local minima ha a e he mally accessible. Besides, we ind om Fig. 3.10c ha
he g ound s a e in e phenyl angle in he Au(111)-suppo ed pp geome y also lies wi hin
he ange α∈[0◦,90◦], as schema ically ep esen ed by he dashed cu e in Fig. 3.12b
( op panel). Rema kably, he subs a e induced plana iza ion, which educes α om 47◦
o 29◦, causes a 44% inc ease in κc( hick g ay and yellow lines in Fig. 3.12b). This esul
sugges s he possibili y o uning in e ibbon coupling by deposi ing NPG on di e en
subs a es.
3.3.3 Quan um elec on anspo simula ions
Based on he band s uc u e analysis epo ed abo e, i is expec ed ha he chemical
and con o ma ional knobs will signi ican ly in luence he cu en p opaga ion in NPG,
pa icula ly he elec onic Talbo in e e ence pa e n eme ging a la ge leng h scales
[113,161]. We add ess his ques ion by pe o ming elec on anspo simula ions o
NPG de ices o ealis ic size. Cu en -poin injec ion and p opaga ion in o he de ice
is simula ed by combining NEGFs wi h TB Hamil onians h ough he TRANSIESTA
u ili y TBTRANS. The la e a e ob ained by p ojec ing uni -cell DFT Hamil onians on
a educed subse o a omic o bi als ha is ele an o ep oduce he elec onic s uc u e
in he ene gy ange o in e es . This mul i-scale app oach allow us o ea de ices in he
∼100 nm scale while p ese ing DFT-le el o accu acy [113,161,162]. We cons uc his
model using he SISL py hon so wa e[163] o e ain s,px,py,pzo bi als in he C a oms
in he b idges and pzand pola iza ion Pdxz and Pdyz o bi als in he es o C a oms.
Including s,px,pyo bi als enables an accu a e desc ip ion o he elec onic s uc u e o
pp molecula b idges wi h non-plana con o ma ions. The eliabili y o he model is es ed
by compa ing he co esponding band s uc u es o hose ob ained wi h DFT, as shown
in Fig. 3.13. The selec ed educed basis se p o ides excellen ag eemen wi h ull DFT
simula ions o longi udinal bands wi hin |E−EF|<1 eV. Impo an ly, he e olu ion o
longi udinal band spli ing wi h phenyl o a ion is e y well cap u ed by he inclusion o
s,pxand pyo bi als.
We hen build de ices consis ing o 42.759 nm ×64.950 nm NPG lakes (9 ×75 NPG
uni -cells, ∼70,200 a oms). Cu en is injec ed om a me allic ip in poin -con ac o a
single a om in he middle o a NPG ibbon. We use he wide-band limi o app oxima e
he me allic ip ia an on-si e imagina y sel -ene gy (SE) iΓ in he con ac a om, whe e
Γ ep esen s he decay a e in o he ip. This app oach has demons a ed o be a eliable
wo ka ound o explici ly modelling he ip [125,161]. The de ice is u he equipped
wi h op and bo om d ain elec odes desc ibed by NPG SEs and le and igh 7.5 nm
CAPs ha mimic open bounda y condi ions, a oiding undesi ed backsca e ing o he
ini e de ice edges [113,123,161]. A ull schema ic ep esen a ion o he anspo se -up
showing all he di e en componen s is p o ided in Fig. 3.14.
The eal-space cu en - low o elec ons injec ed om he ip in o he de ice can be
isualized compu ing bond- ansmissions (see Me hods in Chap e 2) [113,123,161,162,
3.3. NPG: molecula b idge enginee ing 45
pa a-pa a
pa a-pa a pa a-me a me a-me a
s + px + py + pzpz + Pdxz + Pdyz
s + px + py + pzpz + Pdxz + Pdyz
(a)
(b)
Figu e 3.13: DFT s educed TB Hamil onians. (a) Band s uc u e o plana pp-
,pm-, and mm-NPG as calcula ed wi h DFT (black solid lines) and he educed TB
Hamil onian ( ed dashed lines). A schema ic op- iew ep esen a ion o he uni -cell
s uc u e and he basis o bi als employed o cons uc ing educed TB Hamil onians is
gi en below hei co esponding band s uc u e. (b) Band s uc u e o pp-NPG wi h
b idge phenylenes wis ed by θ= 0◦(plana con igu a ion, same as panel (a)), θ= 20◦
and θ= 40◦, as calcula ed wi h DFT (black solid lines) and he educed TB Hamil onian
( ed dashed lines). A schema ic side- iew ep esen a ion o he uni -cell s uc u e and he
basis o bi als employed o cons uc ing educed TB Hamil onians a e gi en below hei
co esponding band s uc u e.
194,195]. Fig. 3.15 shows bond- ansmission maps e alua ed a −0.2 eV below he VBM
o di e en de ices buil om pp-NPGs wi h di e en wis angles (a), as well as plana
pm- (b) and mm-NPG (c). The ac ha on ie bands in plana pp-NPG dispe se mo e
s ongly in he longi udinal (Y) han in he ans e sal (X) di ec ion (Fig. 3.11) leads o a
p edominan ly longi udinal aniso opic p opaga ion. Ne e heless, simila o o he NPG
s uc u es [113,161,162], ini e in e ibbon coupling is esponsible o he eme gence o a
Talbo in e e ence pa e n in he la ge scale p opaga ion (Fig. 3.15a). This beha iou is
ab up ly modi ied by he in oduc ion o m-phenylene uni s in he b idges (Fig. 3.15b,c).
46 Chap e 3. Po ous 1D and 2D nanoa chi ec u es
iΓ
(a) (b)
Tip
CAP
Sel -ene gy
Figu e 3.14: T anspo se -up. (a) Illus a ion ep esen ing he use o an on-si e imagi-
na y SE o simula e cu en injec ion om a me allic ip in o NPG. (b) An NPG de ice
equipped wi h op and bo om SEs (yellow a ea), and le and igh CAPs ( ed a ea).
The posi ion o he ip is indica ed wi h a yellow do .
The d as ic supp ession o in e ibbon coupling (Fig. 3.11) p e en s elec ons om leaking
o adjacen ibbons e en in he ∼100 nm scale, and cu en low is almos comple ely
con ined o a single GNR. In con as , he pp-b idge con o ma ion e eals a mo e g ad-
ual modi ica ion o elec on p opaga ion, in line wi h he momen um spli ing analysis
(Fig. 3.12b). As shown in Fig. 3.15a, when he in e phenyl wis angle inc eases he
p opaga ion becomes mo e aniso opic and he wa eleng h o he Talbo s anding wa e-
pa e n inc eases (see pp(40◦)), slowly app oaching he limi in which cu en is con ined
o a single GNR (see pp(80◦)).
3.4. Summa y 47
(a)
(b)
(c)
Bond ansmission
con o ma ional
chemical
Figu e 3.15: La ge-scale bond- ansmission maps e alua ed a E=EVBM −0.2 eV o
(a) pp-NPG wi h in e phenyl wis angles α= 0◦(θ= 0◦, coplana ), α= 40◦(θ= 20◦)
and α= 80◦(θ= 40◦), (b) plana pm-NPG, and (c) plana mm-NPG. Figu e aken om
Re . [171].
3.4 Summa y
In summa y, we ha e heo e ically s udied expe imen ally syn hesized [18]-annulene GNRs
and NPG s uc u es, which a e ab ica ed om he same pa en Ph-7-13-AGNR ia wo
di e en on-su ace chemis y eac ions. By pe o ming a DFT-based ene ge ics analysis,
we ha e p oposed a eac ion pa h leading o he mig a ion o phenylene side-g oups in
Ph-7-13-AGNRs, which is he key dis inc i e s ep enabling he o ma ion o [18]-annulene-
GNRs. In pa icula , we highligh he c ucial ole o su ace Au ada oms in acili a ing he
ini ial s eps o he eac ion. Addi ionally, ou calcula ions e eal ha closing he bays o
Ph-7-13-AGNRs in o [18]-annulene po es signi ican ly a ec s he ene ge ics o SAMO-like
acuum s a es.
On he o he hand, he la e al coupling o Ph-7-13-AGNRs leads o he ab ica ion
o NPG s uc u es exhibi ing in e ibbon phenylene b idges wi h pa a-pa a,pa a-me a
and me a-me a chemical bonding con igu a ions. DFT-simula ed band s uc u es e eal
wo con ol knobs o a y he in e ibbon elec onic coupling and he esul ing elec onic
aniso opy. While he b idges con aining m-bonds ac as chemical knobs ha ab up ly
supp ess he coupling be ween adjacen nano ibbons, he con o ma ional knob in pp-NPG
54 Chap e 4. Twis ing nanopo ous g aphene on g aphene
(a) (b)
Figu e 4.2: (a) TB s DFT band s uc u e o he aligned NPG/g aphene bilaye in he
AB s acking con igu a ion. (b) TB s DFT DOS o a NPG/g aphene bilaye wis ed by
θ= 21.78◦. Black cu es a e he o al DOS, while ed and blue cu es a e he p ojec ed
DOS (pDOS) on NPG and g aphene, espec i ely. Solid lines co espond o DFT and
dashed lines o he TB model.
size. The TB model and elec onic s uc u e calcula ions ha e been se up using he SISL
py hon package [163].
DFT calcula ions
DFT simula ions a e pe o med by ea ing exchange-co ela ion ene gies wi hin he
GGA-PBE amewo k [138]. Co e elec ons a e add essed by no m-conse ing T ouille -
Ma ins pseudopo en ials [177], while a linea combina ion o a omic o bi als is used o
alence elec ons. We employ a DZP basis se , wi h he basis o bi als ange de ined by
a 0.01 Ry ene gy shi [148]. The BZ is sampled using Monkho s -Pack k-g ids wi h
5×19 k-poin s in aligned bilaye s and 11 ×7k-poin s in he θ= 21.78◦case [181]. The
eal-space g id is de ined by a 400 Ry mesh-cu o . The employed TB model does no
conside e ec s de i ed om a omic s uc u e elaxa ions, and op imizing he geome y
in he DFT simula ions would mos likely in oduce quan i a i e e ec s ha will hampe
a p ope i ing o he model pa ame e s. Geome y and la ice elaxa ions a e he e o e
no conside , and he C-C dis ance and in e laye sepa a ion a e ixed o a= 1.42 ˚
A and
d= 3.35 ˚
A, espec i ely. The aligned NPG/g aphene bilaye o ms a ec angula 31.97
˚
A×8.52˚
A uni -cell, while he la ice ec o s o he bilaye wis ed by θ= 21.78◦a e
a1= (−22.14,55.38) ˚
A and a2= (−20.91,138.45) ˚
A. The i s consis s o 184 C and 20 H
a oms, while he la e has 1288 C and 140 H a oms. We include a acuum egion o 30
˚
A in he di ec ion pe pendicula o he bilaye s in o de o a oid spu ious e ec s be ween
pe iodic images.

4.2. Elec on anspo simula ions 55
4.2 Elec on anspo simula ions
4.2.1 De ice se -up and simula ion de ails
Based on he abo e-desc ibed TB model, we simula e quan um anspo as a unc ion
o he in e laye wis angle on NPG/g aphene de ices o ealis ic size (∼100 nm) in
poin con ac wi h a me allic ip. The calcula ions a e pe o med wi hin he NEGF
o malism as implemen ed in he TRANSIESTA u ili y TBTRANS [157]. Con a y o
he me hod applied in Chap e 3, we do no employ sel -ene gies (SEs) o he bilaye s
as elec odes o d ain he cu en injec ed om he me allic ip. On he one hand,
he non- ec angula uni -cells o wis ed NPG/g aphene bilaye s can ake e y di e en
shapes o di e en wis angles, which would p e en us om building de ice geome ies
sys ema ically. Besides, gene a ing SE objec s o he huge supe cell sizes (∼1000 a oms)
conside ed he e would ha e a high compu a ional cos , making he app oach p ac ically
un easible. Fo hese easons, we conside a single-elec ode se -up composed o a ini e
ci cula bilaye disk in poin -con ac o a me allic ip in he wide-band limi . The ip is
in con ac wi h a single si e in he NPG laye and is simula ed ia an on-si e imagina y
SE e m, as desc ibed in Chap e 3and Re . [161]. In pa icula , we conside con ac
o a si e in he cen e o a ibbon and closes o he o a ion cen e o he bilaye . This
choice o con ac posi ion ensu es ha , o all wis angles, he cu en is injec ed in a
egion wi h local AB s acking, p o iding consis ency among de ices. The ci cula bilaye
disk has a diame e o 70 nm and is su ounded by an iso opic 10 nm wide complex
adso bing po en ial (CAP) ing (see Chap e 2) which p e en s backsca e ing o he ini e
de ice edges and mimics open-bounda y condi ions by adso bing p opaga ing elec ons.
10 nm
50 nm
CAP
(a) (b)
Tip
Y (longi udinal)
X ( ans e sal)
Z
X
Y
Figu e 4.3: T anspo se -up. (a) Illus a ion ep esen ing he cu en injec ion om a
me allic ip in con ac o a single NPG a om. (b) Schema ic ep esen a ion o he whole
de ice se -up, consis ing o a ci cula NPG/g aphene lake equipped wi h a su ounding
CAP ( ed a ea). The yellow do indica es he posi ion o he ip.
56 Chap e 4. Twis ing nanopo ous g aphene on g aphene
NPG
(a) (b)
Figu e 4.4: (a) Bands s uc u e o single-laye NPG, as ob ained om he single-laye
TB Hamil onian used in his chap e . The ed ho izon al line indica es he ene gy a
which bond- ansmissions a e e alua ed in panel (b). (b) Bond- anmissions in a single-
laye NPG de ice, whe e we use a ci cula ≈130 nm NPG lake su ounded by a 10
nm-wide su ounding CAP, as explained in he ex and illus a ed in Fig. 4.3. Bond
ansmissions a e e alua ed a E= 0.5 eV, as indica ed in panel (a).
A schema ic ep esen a ion o he anspo se -up is p o ided in Fig. 4.3.
Impo an ly, he use o a CAP as he only d ain o elec ons is in con as o he
accu a e SE me hod used o simula e open-bounda y condi ions along he anspo di-
ec ion in Chap e 3. In o de o es he eliabili y o ou app oach, we i s simula e he
elec on p opaga ion in a single-laye NPG using he anspo se -up shown in Fig. 4.4.
The simula ed bond- ansmissions e alua ed a an ene gy wi hin he longi udinal bands
o NPG (0.5 eV) exhibi he expec ed la ge-scale Talbo in e e ence pa e n, in excellen
ag eemen wi h Re . [113] which combines SEs and CAPs as illus a ed in Fig 3.14 o
Chap e 3. We ex end his es o aligned NPG/g aphene bilaye s, by compa ing bond
ansmissions ob ained using a su ounding CAP (Fig. 4.3) o hose ob ained combining
SEs and CAPs (Fig 3.14). These wo app oaches a e in e y good quali a i e ag ee-
men , as demons a ed by hei bond ansmissions in bo h NPG and g aphene laye s in
Fig. 4.5. Al hough we es ic ou p oblem o commensu a e in e laye o a ions, he use
o su ounding CAPs allows o simula e la ge-scale elec on anspo p ope ies o any
a bi a y commensu a e o incommensu a e wis -angle, as opposed o elec ode SEs ha
equi e he sca e ing egion o be pe iodic.
4.2.2 La ge-scale cu en p opaga ion
Fig. 4.6 shows he eal-space in-plane bond- ansmission o elec ons injec ed a he po-
si ion o he ed do in NPG and p opaga ing in o he bilaye . We ep esen he bond
ansmissions e alua ed a an ene gy E=−0.4 eV, lying in he ene gy ange o lon-
gi udinal bands o single-laye NPG a which Talbo in e e ence is known o eme ge
4.2. Elec on anspo simula ions 57
G aphene
G aphene
NPG
NPG
(a) (b)
Su ounding CAP Top and Bo om SEs
+
Le and Righ CAPs
Figu e 4.5: Bond ansmissions in NPG ( op) and g aphene (bo om) e alua ed a
E=−0.4 eV, as ob ained using (a) a su ounding CAP (Fig. 4.3), and (b) a combina ion
o op and bo om SEs and le and igh CAPs (Fig 3.14). The scale o he bond-
ansmission magni ude is he same in bo h panels.
[113]. In he aligned case (θ= 0◦in AB s acking) such a cha ac e is ic in e e ence
e ec is obse ed in NPG, al hough signi ican ly smea ed ou by he p esence o he un-
de lying g aphene subs a e. Rema kably, such smea ed Talbo in e e ence pa e n is
also imp in ed in o g aphene, inducing aniso opic elec on p opaga ion in an o he wise
quasi-iso opic c ys al.
When NPG is o a ed by a small wis angle (θ= 1.92◦), he in e e ence pa e n is
hea ily pe u bed and u he smea ed ou . A he same ime, he in e laye ansmission
emains s ong, indica ing an enhanced in e laye coupling. In e es ingly, o la ge wis
angles he ansmission pa hways become s ongly asymme ic wi h espec o he con-
ac ed GNR (see θ= 5.78◦,10.53◦), which can be explained by he wis -induced b eaking
o he in-plane mi o symme y a ound i . Besides, his in oduces chi ali y in he sys-
em, as he p edominan di ec ion o p opaga ion wi h espec o he con ac ed GNR
would be e e sed o θ→ −θ. Simul aneously, he in e laye ansmission dec eases
upon inc easing he wis angle and, consequen ly, he elec onic p opaga ion is g adually
58 Chap e 4. Twis ing nanopo ous g aphene on g aphene
θ=0∘
θ=1.92∘
θ=5.78∘
θ=10.53∘
θ=21.78∘
G aphene
Bond ansmission Bond ansmission
NPG
X
Y
Figu e 4.6: Real-space bond- ansmission maps a an ene gy E=–0.4 eV in
NPG/g aphene o di e en wis angles. Top and bo om ows shows bond- ansmissions
h ough NPG and g aphene, espec i ely. The ed do indica es he posi ion in he
XY plane a which elec ons a e injec ed in NPG. Scale ba in he bo om- igh panel
(θ= 21.78◦) is 10 nm.
es ic ed o NPG. E en ually, he asymme ies disappea and he bond- ansmissions
exhibi a pe ec Talbo in e e ence pa e n along NPG o θ= 21.78◦, mimicking single-
laye -like anspo beha iou and p o iding e idence o he elec onic decoupling om
g aphene.
In o de o elucida e whe he he p edic ed anspo beha iou depends on speci ic
s uc u al de ails o he de ice, such as he ini ial s acking, o a ion cen e o he cu en
injec ion posi ion, we pe o m addi ional quan um elec on anspo simula ions o de-
ices wi h a modi ied se -up. In pa icula , s a ing om NPG/g aphene bilaye s in AA
s acking (θ= 0◦) we o a e he NPG laye wi h espec o an axis c ossing a omic si es
in bo h laye s, as shown in Fig. 4.7(a). The me allic ip is hen placed in con ac o he
NPG a om a he o a ion cen e , which ensu es injec ion in an AA si e ega dless o he
wis angle. Fig. 4.7(b) shows, o di e en wis angles, bond- ansmissions o elec ons
injec ed wi h an ene gy E=−0.4 eV. The p e iously obse ed ends a e quali a i ely
ep oduced by his se -up. Namely, in e laye ansmission is p og essi ely quenched as
he wis angle inc eases and he elec ons p opaga e asymme ically o small ini e o-
a ions (θ≲10◦). We he e o e conclude ha he obse ed anspo beha iou is obus
agains de ails o he de ice se -up.
4.3. Elec onic s uc u e: o igin o decoupling and chi al cu en s 59
(AA s acking)
θ=0∘
θ=1.92∘
θ=5.78∘
θ=10.53∘
θ=21.78∘
G aphene NPG
Bond ansmission Bond ansmission
(a) (b)
X
Y
Figu e 4.7: (a) A omic s uc u e o a wis ed NPG/g aphene bilaye ini ially in an AA
s acking con igu a ion, whe e he black do indica es he selec ed o a ion cen e . (b)
Real-space bond ansmission maps a an ene gy E=–0.4 eV o di e en wis angles o
he NPG/g aphene bilaye shown in (a). The ed do indica es he injec ion poin in he
NPG laye , which coincides wi h he o a ion cen e (black do in panel (a)). Scale ba
is 10 nm.
4.3 Elec onic s uc u e: o igin o decoupling and chi al
cu en s
4.3.1 Momen um-space analysis
We ha e ocused ou analysis o la ge-scale quan um elec on anspo simula ions on he
e olu ion o he Talbo in e e ence pa e n and i s in e laye ansmission as a unc ion
o he wis angle. In o de o achie e deepe unde s anding o he obse ed anspo be-
ha iou , i is key o ack he e olu ion o he longi udinal on ie bands o NPG and hei
coupling o g aphene Di ac cones in he band s uc u e. Howe e , wis ed commensu a e
NPG/g aphene bilaye s gi e ise o skewed supe cells ha b eak he mi o -symme y
o he aligned geome y, and i is no s aigh o wa d o sys ema ically de ine a con e-
nien BZ pa h o in es iga e he NPG bands in he bilaye . Besides, he high numbe o
single-laye uni -cell epe i ions equi ed o build he new pe iodic supe cell o he bilaye
leads o a la ge inc ease o he numbe o s a es upon band- olding in o a smalle BZ.
Consequen ly, sys ema ically disen angling he band s uc u e o wis ed NPG/g aphene
bilaye s becomes a cumbe some ask.
He e we p ojec he band s uc u e o NPG/g aphene bilaye s on NPG, and we un old
hem o he BZ o he la e , mo e pa icula ly o he X→Γ→Ypa h commonly
employed o s udy i s cha ac e is ic elec onic aniso opy. A de ailed desc ip ion o he
me hod is gi en in Fig. 4.8. This simple econs uc ion o he NPG band s uc u e allows
us o access he e olu ion o i s on ie s a es unde he in luence o he unde lying
g aphene. Fig. 4.9 shows he band s uc u e o wis ed NPG/g aphene bilaye s, p ojec ed
on NPG and un olded o i s BZ, o he wis angles conside ed in anspo simula ions
(see Fig. 4.6). The band s uc u e o single-laye NPG is p o ided o compa ison in
he panel co esponding o θ= 0◦(g ay dashed lines). The aligned case (θ= 0◦, AB
s acking) exhibi s a s ong hyb idiza ion be ween he longi udinal bands o NPG and he

60 Chap e 4. Twis ing nanopo ous g aphene on g aphene
Γ
Y
X
bz (NPG)
BZ (bilaye )
qi

ki(qi)
Sample con enien pa h in bz:
qi
Diagonalize monolaye NPG Hamil onian o each :
HNPG
qi
HNPG(qi)→({ϕnqi},{ϵnqi})
Repea monolaye NPG cell o ma ch NPG laye geome y in he
NPG/g aphene bilaye . Apply Bloch heo em o acco dingly.
{ϕnqi}
Fold o BZ:
qi
qi→
ki
Diagonalize bilaye Hamil onian o each :
H

ki
H(
ki)→({Ψm
ki},{ϵm
ki})
Ob ain he NPG con ibu ion o each bilaye eigens a e by
p ojec ing on NPG eigens a es:
PNPG(m,qi)=∑
n
|⟨ϕn,qi|Ψm,
ki(qi)⟩|2
Plo he bilaye ene gy-momen um dispe sion in he
selec ed bz pa h by weigh ing he eigen alues acco ding o hei
NPG con ibu ion .
ϵm,
ki(qi)
PNPG(m,qi)
Me hod o un olding o p ojec ed band s uc u e
olding
Figu e 4.8: Schema ics o he me hod o un old he p ojec ed band s uc u e
o NPG/g aphene bilaye s. (a) Rep esen a ion o he B illouin zone o a wis ed
NPG/g aphene bilaye (BZ, black) and single-laye NPG (bz, ed). (b) Flowcha o
he me hod.
Di ac cones o g aphene, upon olding o he g aphene K(K′) poin s o ±2/3(π/Lx) in o
he ΓXline, whe e Lx= 13√3ais he la ice cons an o NPG in he di ec ion ac oss
he ibbons. The in e laye coupling leads o a ini e con ibu ion o NPG o g aphene-
like s a es e en nea he Fe mi le el, and o he opening o a iny bandgap in he Di ac
cone. The la e e ec is enabled by he small band misalignmen be ween he laye s (see
Sec. 4.1.2) which, simila ly o AB bilaye g aphene [211,212], can be uned using ex e nal
elec ic ields pe pendicula o he su ace [210]. Besides, longi udinal NPG bands exhibi
signi ican ene gy eno maliza ion, e ec i ely educing he NPG bandgap a he Γ poin .
Rema kably, in e -GNR band spli ing in he ΓYdi ec ion emains weak, i.e. longi udinal
s a es e ain hei aniso opic cha ac e .
The band s uc u e ge s no ably modi ied by wis ing o θ= 1.92◦. On he one
hand, g aphene-like ea u es a e no longe isible. On he o he hand, longi udinal NPG
bands a e s ongly pe u bed, exhibi ing nume ous a oided c ossings and weakly dispe s-
ing b anches p edominan ly a ound he onse o he on ie s a es. Rema kably, some o
hese b anches appea a ene gies as low as E≈ −78 meV in he occupied egion, well
4.3. Elec onic s uc u e: o igin o decoupling and chi al cu en s 61
P ojec ion on NPG
Figu e 4.9: Band s uc u e o NPG/g aphene a di e en θ, p ojec ed on NPG and
un olded o i s B illouin zone. Un olded bands a e ep esen ed in di ec ions ac oss (X→
Γ) and along (Γ →Y) he NPG ibbon axis. The band s uc u e o single-laye NPG is
plo ed as a e e ence in he panel co esponding o θ= 0◦(g ay dashed lines).
inside he single-laye bandgap. As he wis angle keeps inc easing, a oided c ossings
become weake and a e es ic ed o highe ene gies, e en ually leading o NPG single-
laye -like dispe sion a θ= 21.78◦. This e olu ion o he band s uc u e demons a es an
e ec i e elec onic decoupling be ween NPG and g aphene o inc easing alues o θ, and
sa is ac o ily explains he g adual supp ession o in e laye ansmission and simul ane-
ous eco e y o single-laye -like Talbo in e e ence ob ained om ou quan um elec on
anspo simula ions.
Howe e , he esul s shown in Fig. 4.9 do no explain he o igin o asymme ic in-plane
p opaga ion (see Fig. 4.6). The eason is ha ou un olding me hod p o ides he p ojec ed
band s uc u e in he 1s BZ (1BZ) o single-laye NPG, which is no a ue physical BZ
o he sys em, bu a con enien ly selec ed auxilia y ecip ocal-space egion ha allows a
meaning ul ep esen a ion o NPG bands. As such, p ojec ed band s uc u es un olded o
ha auxilia y BZ a e no expec ed o espec he symme ies o single-laye NPG, unless
he in e ac ion wi h g aphene becomes negligible. Consequen ly, he in o ma ion p o ided
in Fig. 4.9 is incomple e ega ding he speci ic band dispe sion in di e en di ec ions.
We hus ex end ou me hod o he whole 2D momen um-space and plo Fe mi su aces
co esponding o he band s uc u e o NPG/g aphene bilaye s p ojec ed on NPG and
un olded o i s 1BZ. Fig 4.10 shows hese Fe mi su aces e alua ed a an ene gy o E=
−0.4 eV, which allows o es ablish a di ec connec ion o he elec on anspo simula ions
shown in Fig. 4.6. A AB s acking (θ= 0◦), he Fe mi su ace p ojec ed on NPG con ains
g aphene-like b anches eme ging om he olding o Di ac cones. The la e appea as
wo ci cles cen e ed on k= (±2π
3Lx,0) and a e coupled o each o he media ed by he
hyb idiza ion wi h NPG bands. This is accompanied by an addi ional pai o NPG-
like b anches a la ge ky. These co espond o aniso opic longi udinal bands, whose
spli ing in he ky-axis dec eases as kxapp oaches he BZ bounda y a π/Lx, whe e
hey a e degene a e. Impo an ly, he Fe mi su ace o med by his se o b anches is
cha ac e ized by he mi o symme y o he aligned geome y, hus exhibi ing symme ic
band dispe sion wi h espec o he Y-axis (i.e. he axis pa allel o he GNR g ow h
di ec ion).
As a combina ion o in e laye coupling and wis -induced mi o -symme y-b eaking,
he Fe mi su ace appea s s ongly and asymme ically (wi h espec o he Y-axis) mod-
62 Chap e 4. Twis ing nanopo ous g aphene on g aphene
P ojec ion on NPG
Figu e 4.10: Fe mi su aces o NPG/g aphene a E=–0.4 eV and di e en θ, p ojec ed
on o he elec onic s a es o NPG and un olded o i s B illouin zone. X(Y)-axis co e-
sponds o he di ec ion ac oss (along) he NPG ibbon axis.
i ied a small θ(see θ= 1.92◦and 5.78◦). As θkeeps inc easing, he Fe mi su ace o
single-laye NPG is g adually es o ed, wi h weake a oided c ossings asyme ically dis-
ibu ed wi h espec o he Y-axis (see θ= 10.53◦). This e ec b eaks he symme y o
he dispe sion ela ion in he di ec ion ac oss GNRs, and is he key ea u e leading o he
asymme ic p opaga ion shown in Fig. 4.6. Fo θ= 21.78◦, he in e laye coupling is so
weak ha he mi o -symme y-b eaking becomes i ele an and a single-laye anspo
beha iou is es o ed.
Despi e demons a ing an e ec i e elec onic decoupling upon wis ing, Figs. 4.9 and
4.10 do no answe he ques ion on why he in e ac ion be ween NPG and g aphene is
quenched. In o de o answe his ques ion, i is ins uc i e o p o ide a con inuum de-
sc ip ion o he unneling p ocesses be ween Bloch s a es in di e en laye s. In pa icula ,
we ollow an app oach ha has been widely used o de i e con inuum models o TBLG
[51,151,225–228]. We s a by conside ing a sys em composed o any wo pa allel laye s
sepa a ed by a cons an in e laye dis ance. The wo laye s di e om each o he , hei
la ice cons an s and a omic indices being di e en . In a single-o bi al TB app oxima ion,
a Bloch s a e o he op laye wi h band index nand c ys al momen um kis w i en as
|ψb
nk⟩=1
√NX
i
a(nk)
iX
Rb
eik·(Rb+τb
i)|Rb+τb
i⟩,(4.7)
whe e Rba e la ice ec o s, τb
iis he subla ice ec o indica ing he posi ion o o bi al
iinside he uni cell, |Rb+τb
i⟩is an o bi al cen e ed on posi ion Rb+τb
i, and Nis he
numbe o uni cells in he laye . Simila ly, a Bloch s a e on he op laye wi h band index
mand c ys al momen um pis gi en by
|ψ
mp⟩=1
√MX
i
c(mp)
iX
R
eip·(R +τ
i)|R +τ
i⟩.(4.8)
The o a ion o he op laye wi h espec o he bo om laye is desc ibed by a o a ion
ma ix Mθ.|R +τ
i⟩is hen eplaced by |R′ +τ′
i⟩, and eal-space la ice and posi ion
ec o s and c ys al momen um a e eplaced by ′ =Mθ and p′=Mθp, espec i ely.
By using p imes o indica e o a ed op laye a iables, he hopping e m desc ibing
he coupling be ween a Bloch s a e wi h band index mand c ys al momen um p′in he
4.3. Elec onic s uc u e: o igin o decoupling and chi al cu en s 63
op laye , and a Bloch s a e wi h band index nand c ys al momen um kin he bo om
laye is gi en by
T(b )
nm (k,p′) = ⟨ψb
nk|ˆ
Hin e |ψ
mp′⟩
=1
√NM X
i,i′
(a(nk)
i)∗c(mp′)
i′X
Rb,R′
e−ik·(Rb+τb
i)eip′·(R′ +τ′
i′)×⟨Rb+τb
i|ˆ
Hin e |R′ +τ′
i′⟩.
(4.9)
Acco ding o he wo-cen e app oxima ion employed in ou TB model
⟨Rb+τb
i|ˆ
Hin e |R′ +τ′
i′⟩= in e (Rb+τb
i−R′ −τ′
i′),(4.10)
and
T(b )
nm (k,p′) = 1
√NM X
i,i′
(a(nk)
i)∗c(mp′)
i′X
Rb,R′
e−ik·(Rb+τb
i)eip′·(R′ +τ′
i′)
× in e (Rb+τb
i−R′ −τ′
i′),
(4.11)
whe e in e ( ) is gi en by Eq. 4.6. Because in e ( ) is a smoo h unc ion o he plana
p ojec ion o he in e a omic dis ance, we can Fou ie expand i in he con inuum o
momen um:
in e (Rb+τb
i−R′ −τ′
i′) = 1
NΩbX
q
qeiq·(Rb+τb
i−R′ −τ′
i′)
=1
NΩbX
q∈BZ X
G
q+Gei(q+G)·(Rb+τb
i−R′ −τ′
i′),
(4.12)
whe e qis he Fou ie ans o m o in e ( ), momen um qin he bo om ow uns o e
he 1BZ o he bo om laye , Gis a bo om laye ecip ocal la ice ec o , and Ωbis he
a ea o he bo om laye uni -cell. Inse ing Eq. 4.12 in o Eq. 4.11 yields
T(b )
nm (k,p′) = 1
√NM
1
NΩbX
i,i′
(a(nk)
i)∗c(mp′)
i′X
qX
G
q+Gei(q+G−k)·τb
iei(p′−q−G)·τ′
i′
×X
Rb
ei(q+G−k)·RbX
R′
ei(p′−q−G)·R′ .
(4.13)
Then, using he ela ions
X
Rb
ei(q+G−k)·Rb=NX
Gb
δk+Gb,q+G
X
R′
ei(p′−q−G)·R′ =MX
G′
δp′+G′ ,q+G,(4.14)
70 Chap e 4. Twis ing nanopo ous g aphene on g aphene
ge g adually con ined o a single laye and he IPR dec eases.
4.4 Spec oscopic signa u es o in e laye coupling
Di e en e ec s o he in e laye coupling, such as wis -dependen an Ho e singula i ies
and moi ´e s a es, a e commonly measu ed in TBLG [231–233] and o he an de Waals
he e os uc u es [218] using dI/dV poin -spec oscopy and mapping in STM. Simila ly,
we expec ha he esul s p esen ed in his chap e could be explo ed using dI/dV cha -
ac e iza ion. In pa icula , in a STM expe imen , we expec he laye placed u he om
he ip o ep esen a negligible con ibu ion o he di e en ial conduc ance, due o he
as decay o he elec onic densi y o e he ela i ely la ge in e laye dis ance. Assuming
ha he STM ip app oaches he sample om abo e, he measu emen s will be domina ed
by he pDOS o NPG [234,235]. Consequen ly, we look o spec oscopic signa u es o he
in e laye coupling by compu ing he pDOS on NPG, as shown in Fig. 4.16. A θ= 0◦,
he hyb idiza ion signi ican ly modi ies he pDOS o NPG wi h espec o he single-
Twis angle
Figu e 4.16: Densi y o s a es o NPG/G aphene p ojec ed on NPG a di e en θ. The
DOS o single-laye NPG is shown as a e e ence in all cases (g ay shaded cu e). The
pDOS is no malized wi h espec o he numbe o a oms in each uni -cell. Red and blue
a ows in he spec um co esponding o θ= 1.92◦indica e he ene gies a which local
DOS is ep esen ed in Fig. 4.17,E=−78 meV and E=−400 meV, espec i ely.

4.4. Spec oscopic signa u es o in e laye coupling 71
laye beha iou (g ay shaded cu e). In pa icula , new peaks a ise inside he bandgap
o single-laye NPG and a ound i s on ie band onse s, wi h elec onic densi y slowly
decaying owa ds he Fe mi le el. This esul is in good ag eemen wi h he ema kable
ene gy eno maliza ion o on ie NPG-like s a es and he concomi an NPG con ibu ion
o Di ac-like s a es obse ed in he p ojec ed band s uc u e (Fig. 4.9). When a small
wis angle (see θ= 1.92◦) is in oduced be ween he laye s, he peaks eme ging in he
aligned case spli in a bunch o new addi ional esonances, o igina ing om he mul iple
a oided c ossings in he band s uc u e (Fig. 4.9). These esonances g adually mo e ou
om he single-laye NPG bandgap as θis u he inc eased, and he spec um e en ually
esembles ha o single-laye NPG.
Addi ional in o ma ion abou he spa ial dis ibu ion o hyb id bilaye s a es can be
ex ac ed by ep esen ing he local DOS (LDOS) e alua ed a ele an ene gy posi ions.
Fig. 4.17(b) shows he LDOS on NPG o NPG/g aphene a θ= 1.92◦, e alua ed a he
ene gies indica ed by blue (E=−400 meV) and ed (E=−78 meV) a ows in Fig. 4.16.
These co espond o he ene gies selec ed in ou elec on anspo simula ions (Fig. 4.6)
and o he lowes ene gy shoulde obse ed in he pDOS (Fig. 4.16), espec i ely. A
E=−400 meV, he LDOS on he NPG laye e y closely esembles he VB o single-laye
NPG, ep esen ed in Fig. 4.17(a), and does no display any app eciable modula ion despi e
he s ong in e laye hyb idiza ion a such ene gy (Fig. 4.15). On he con a y, a E=−78
meV, he elec onic densi y exhibi s a ma ked long- ange modula ion in he scale o he
moi ´e pa e n o he bilaye , while conse ing an NPG-like cha ac e . In pa icula , he
densi y a his low-ene gy esonance locally anishes a ound he domain walls sepa a ing
egions o local AB and BA s acking ha a e pe pendicula o he nano ibbon axis.
E = -400 meV
E = -78 meV
E = -195 meV
Monolaye NPG Twis ed NPG/g aphene bilaye
Y
X
(a) (b)
Figu e 4.17: (a) LDOS map o single-laye NPG e alua ed a he VB onse ene gy
(E=−195 meV). The LDOS o single-laye NPG does no change app eciably o e he
ene gy ange E∈[−500,−195] meV. (b) LDOS on NPG o NPG/g aphene a θ= 1.92◦
e alua ed a E=−400 meV and E=−78 meV. These ene gies a e indica ed in Fig 4.16
by blue and ed a ows, espec i ely. The scale ba in he igh panel is 1 nm and he
NPG geome y is o e laid in he bo om igh co ne o all panels.
72 Chap e 4. Twis ing nanopo ous g aphene on g aphene
4.5 E ec o ou -o -plane co uga ion in he in e laye
coupling
As discussed in Sec ion 4.1.1, la ice and ou -o -plane a omic elaxa ions play a signi ican
ole in he elec onic p ope ies o TBLG a small wis angles [220–222]. While in-plane
moi ´e pa e n econs uc ions may be neglec ed, ou -o -plane a omic co uga ion migh be
signi ican in NPG/g aphene bilaye s wis ed by θ= 1.92◦[222]. To es ima e he in luence
o ou -o -plane dis o sions on he in e laye elec onic coupling, we build a simpli ied model
based on he phenomenology obse ed in TBLG. In pa icula , he in e laye dis ance in
TBLG is enhanced in egions o AA s acking, while i emains unchanged in egions o
AB s acking [220,221]. Thus, we p opose o co ela e he in e laye dis ance wi h he
local s acking o de a each a om o he NPG laye . Fo a NPG/g aphene bilaye wis ed
by θ= 1.92◦, we quan i y he local s acking o de sio each a om iin he NPG laye by
si=1−max(di, dN)
a×1−min(di, dN)
a×1−min(di, dN)
a.(4.20)
He e, diis he plana p ojec ion o he dis ance om a om iin NPG o he closes a om in
he g aphene laye , while dNis he a e age o he plana p ojec ion o he dis ance om
he NNs o a om iin NPG o hei closes a oms in he g aphene laye . ais he in-plane
C-C dis ance, and siis bounded be ween 0 and 1. I he local s acking esembles a pe ec
AB con igu a ion, si≈0, while a pe ec local AA s acking co esponds o si≈1. We
hen co ela e he local s acking o de wi h he local co uga ion by applying an ou -
o -plane displacemen o ∆zi=si× ×d0on each a om i, whe e is he p opo ion
by which he e e ence in e laye dis ance d0= 3.35 ˚
A is inc eased in he posi ion o
maximum co uga ion. This p ocedu e leads o a pe iodic co uga ion o he bilaye , as
shown in Fig. 4.18a,b o NPG/g aphene wi h θ= 1.92◦ wis angle, which in u n esul s
in a pe iodic modula ion o he in e laye hopping in eg als.
Using his model, we compu e he IPR o di e en deg ees o co uga ion (up o 8%
o he in e laye dis ance o he plana bilaye ) o NPG/g aphene wi h θ= 1.92◦. As a
e e ence, changes in he in e laye dis ance o ∼7−8 % ha e been p edic ed o g aphene
bilaye s wis ed by θ < 3◦[220,221]. The esul s shown in Fig 4.18c indica e ha he
IPR and, hus, he in e laye hyb idiza ion is educed upon inc easing he co uga ion.
Howe e , e en in he case o 8% co uga ion, he hyb idiza ion a he ene gies o in e es
is only educed by ∼10%. Fu he mo e, he IPR a E = -0.4 eV emains abo e he alue
o he aligned NPG/g aphene bilaye a all deg ees o co uga ion. Hence, we can assume
ha he ou -o -plane dis o sions do no quali a i ely modi y he main conclusions o his
chap e . In ac , he ac ha co uga ion induces a educ ion o he in e laye coupling
ein o ces he abili y o decoupling NPG and g aphene by wis ing.
4.6. Summa y 73
AA
AA
AA
AB
AB
AB
AB
AB
(a) (b)
(c)
Figu e 4.18: E ec o co uga ion in NPG/g aphene bilaye s wis ed by θ= 1.92◦. (a)
A omic s uc u e showing he moi ´e pa e n, whe e AA and AB egions a e indica ed.
(b) Colo map showing he in e laye dis ance in each a om o he NPG laye . (c) IPR o
di e en alues o he co uga ion. The ho izon al blue dashed line indica es he IPR o
aligned NPG/g aphene (AB s acking) a an ene gy E = -0.4 eV.
4.6 Summa y
In his chap e , we ha e combined a simple TB model and NEGFs o demons a e ha
he elec onic and anspo p ope ies in NPG/g aphene bilaye s can be e icien ly uned
ia an in e laye wis angle. A small angles (θ≲11◦), elec onic cu en s injec ed in
NPG exhibi s ong asymme ic pe u ba ions in he Talbo in e e ence pa e n, which
is ansmi ed o he unde lying g aphene laye . This beha iou e eals a signi ican in e -
laye coupling, as con i med by he s ong hyb idiza ion o single-laye elec onic bands.
On he con a y, la ge wis angles (θ≳11◦) e ec i ely decouple he laye s and lead o a
single-laye -like p opaga ion in NPG, which is cha ac e ized by a pe ec Talbo in e e -
74 Chap e 4. Twis ing nanopo ous g aphene on g aphene
ence pa e n. The p og essi e supp ession o he in e laye hyb idiza ion wi h inc easing θ
is explained by means o wis -induced momen um sepa a ion be ween single-laye eigen-
s a es o NPG and g aphene. We also p esen he IPR as a use ul me hod o quan i y he
hyb idiza ion s a e o he bilaye s.
F om a compu a ional pe spec i e, we ha e demons a ed he e ec i eness o using
su ounding CAPs, ins ead o compu a ionally hea ie and mo e cons aining SEs, in
o de o mimic open bounda y condi ions in elec on anspo simula ions. Besides,
we ha e made he o he wise cumbe some dispe sion ela ion o wis ed NPG/g aphene
accessible by implemen ing a me hod o un olding hei laye -p ojec ed band s uc u es.
Finally, we show ha he pDOS and LDOS p o ide spec oscopic signa u es o he
wis -dependen elec onic coupling be ween NPG and g aphene, which could be p obed
in u u e STM expe imen s. Rema kably, ecen expe imen al ad ances enable accu a e
measu emen s o momen um- esol ed elec on bands ia he Quan um Twis ing Mic o-
scope (QTM), which may be used o con i m he coupling mechanism exposed in his
chap e [236].
CHAPTER 5
Disen angling decep i e o bi al con inemen a he
edges and po es o ca bon-based nanoa chi ec u es
As we ha e shown in Chap e 3, he elec onic s uc u e o ca bon-based nanoa chi ec-
u es can be expe imen ally in es iga ed wi h a omic-le el p ecision ia STS. Fo example,
dI/dV poin -spec oscopy and mapping ha e been ex ensi ely used o de e mine he elec-
onic bandgap o a wide a ie y o sys ems and cha ac e ize he ene gy and spa ial dis-
ibu ion o hei on ie elec onic o bi als [83,85,86,96]. Howe e , STS measu emen s
can also yield ambiguous esul s ha lead o e oneous in e p e a ions o he elec onic
p ope ies.
In 7-AGNRs, o ins ance, S¨ode e al. e ealed ha he VB and he CB+1 wa e
unc ions appea ed con ined o he acuum egion along he edges in dI/dV maps (see
Fig. 5.1a) [237]. Pa adoxically, DFT p edic ed hese s a es o be delocalized o e he
Exp. dI/dV
(b)
Theo y
(a)
Exp. dI/dV
Theo y
VB CB
VB CB
Figu e 5.1: Elec onic o bi al con inemen in AGNRs and ZGNRs. (a) STS measu e-
men s in a 7-AGNR e eals alence s a es ha a e localized in he ibbon edges, while he
co esponding DFT-calcula ed wa e unc ion is delocalized o e he en i e ibbon back-
bone. Top igu e aken om Re . [82]. Rep oduced wi h pe mission om Sp inge Na u e.
Bo om igu es ep in ed wi h pe mission om Re . [237]. Copy igh 2015 by he Ame -
ican Physical Socie y. (b) Bo h expe imen al dI/dV maps and DFT calcula ions e eal
he exis ence o low-ene gy s a es inhe en ly localized in he edges o ZGNRs. Figu e
aken om Re . [83]. Rep oduced wi h pe mission om Sp inge Na u e.

76 Chap e 5. Decep i e o bi al con inemen in ca bon-based nanoa chi ec u es
en i e 7-AGNR backbone. The appa en con inemen o VB and CB+1 wa e unc ions was
a ibu ed o hei pa icula oscilla ion pa e n ac oss and along he backbone. Regions
o posi i e and nega i e wa e unc ion weigh would lead o a local cancella ion o he
STS signal, leading o he obse a ion o a decep i e dis ibu ion o he elec onic densi y
[237].
dI/dV mapping has also e ealed a s ong localiza ion o he VB and CB a he edges
o ZGNRs (see Fig. 5.1b) [83]. In con as o 7-AGNRs, elec onic localiza ion in ZGNRs
is unambiguously a ibu ed o hei so-called edge s a es. These a e on ie elec onic
s a es ha a e inhe en ly localized a he zigzag edges [57,238], and hei expe imen al
obse a ion has been co obo a ed by DFT simula ions [83].
Besides, a simila elec onic con inemen is also cha ac e is ic o bay and po e s a es
(b)
2.2 V
3.2 eV
Exp. dI/dV
Theo y
Exp. dI/dV
(d)
STM
Exp. dI/dV
Theo y
Exp. BR-STM
(a)
(c)
STM
Exp. dI/dV
Figu e 5.2: Elec onic o bi al con inemen in ca bon-based nanoa chi ec u es exhibi ing
no ched edges and nanopo es. (a) In a po ous nanog aphene, he expe imen al dI/dV
signal ob ained a 2.4 V is s ongly con ined o he nanopo e. This measu emen was
assigned o a acuum po e s a e p edic ed by DFT a 3.25 eV. Rep in ed wi h pe mission
om Re . [239]. Copy igh 2018 Ame ican Chemical Socie y. (b) In NPG, he s ong
con inemen o dI/dV ea u es o he po e egions a 2.2V was a ibu ed o he exis ence
o in insically localized po e s a es, which we e p edic ed a 3.2 eV by DFT. Figu e aken
om Re . [111]. Rep in ed wi h pe mission om AAAS. (c) dI/dV maps e eal elec onic
s a es con ined o he bays o a GNR wi h no ched edges. Such a localized signal was
assigned o elec onic s a es delocalized o e he en i e ca bon-ne wo k. Figu e aken
om Re . [240]. Rep oduced wi h pe mission om Sp inge Na u e. (d) The elec onic
densi y appea s con ined o he co es o a che on GNR. This measu emen was a ibu ed
o he o ma ion o a s anding wa e pa e n by he unde lying Au(111) su ace s a e.
Rep oduced om Re . [182] wi h pe mission om he Royal Socie y o Chemis y.
5.1. Syn hesis o gul -GNRs and NPG 77
o igina ing om IPSs (see Fig. 5.2a,b). These a e inhe en ly localized a he bays and
po es o nanos uc u es [111,239], as we ha e desc ibed o he Ph-7-13-AGNRs and
[18]-annulene GNRs s udied in Chap e 3. Ne e heless, compa ed o he localized dI/dV
signals obse ed a ene gies a ound he bandgap in 7-AGNRs and ZGNRs, combined STS
and DFT cha ac e iza ion has de e mined ha bay and po e s a es appea a much highe
ene gies.
Mo e gene ally, localiza ion e ec s like hose desc ibed abo e ha e been ou inely
obse ed in many o he on-su ace syn hesized ca bon-based nanoa chi ec u es exhibi ing
no ched edges and nanopo es (see Fig. 5.2c,d). Con a y o bay and po e s a es, hese
localized STS signals ha e been usually obse ed a low ene gies a ound he on ie s a e
onse s. Rega ding hei o igin, i has been o en assigned o s a es whose wa e unc ions
a e ac ually delocalized o e he en i e ca bon-ne wo k acco ding o heo e ical p edic ions
(Fig. 5.2c)[175,240–246]. Addi ionally, elec onic localiza ion has also been ela ed o
he unde lying Au(111) subs a e, alleging ha he su ace s a e o ms a s anding wa e
pa e n a he co es o che on-GNRs [182] (Fig. 5.2d).
The exis ence o mul iple, appa en ly plausible in e p e a ions sugges s ha a be e
unde s anding o edge and nanopo e-localized STS signals is s ill lacking. In his chap e ,
we p o ide a ho ough DFT-based in es iga ion o he ecu en expe imen al obse a-
ion o elec onic localiza ion, using as e e ence sys ems on-su ace syn hesized gul - ype
GNR (g-GNR) and NPG s uc u es. We ind ha he o bi al con inemen obse ed in
dI/dV maps is caused by he decay o delocalized sample wa e unc ions in o he acuum,
and is enhanced wi h inc easing ip-sample sepa a ion. Suppo ed by addi ional simula-
ions o ela ed sys ems, we show ha his e ec , de e mined by he speci ic edge and
nanopo e geome y, is p esen in a wide ange o ca bon-based nanoa chi ec u es. Fu -
he mo e, we con i m he alidi y o DFT-LDOS map simula ions in he Te so -Hamman
app oxima ion o add ess decep i e o bi al con inemen .
5.1 Syn hesis o gul -GNRs and NPG
We s a by b ie ly desc ibing he ab ica ion and a omic s ucu e o he sys ems ha
se e as e e ence o ou s udy. The on-su ace syn hesis o g-GNRs has been pe o med
s a ing om he p ecu so 4’,5”-dib omo1,1’:2’,1”:2”,1”’-qua e phenyl (DBQP) monome
and ollowing he mally-ac i a ed polyme iza ion (T= 250◦C) and cyclodehyd ogena ion
(T= 500◦C) eac ions on Au(111), as shown in Fig. 5.3a. Fig. 5.3b exhibi s he non-
con ac AFM (nc-AFM) image o a g-GNR, which consis s o a 3-ZGNR backbone dec-
o a ed wi h pe iodic single-phenyl- ing p o usions in a s agge ed a angemen a bo h
sides o he ibbon. Fu he annealing he subs a e o T= 600◦C igge s he la e al
dehyd ogena i e c oss-coupling o g-GNRs in o NPG s uc u es. The esul ing 2D s uc-
u es consis o pe iodic ou -o -phase a ays o small nanopo es whe e a single phenyl
ing is missing (≈4.7˚
A in diame e ) (Fig. 5.3c). No ched gul edges and nanopo es make
hese pla o ms ideal o he s udy o elec onic con inemen in well-de ined acuum e-
gions. Acco dingly, nex sec ions ocus on he s udy o he elec onic s uc u e o g-GNRs
and NPG using STS cha ac e iza ion and DFT simula ions.
78 Chap e 5. Decep i e o bi al con inemen in ca bon-based nanoa chi ec u es
(b)
(c)
250 ºC
Au(111)
500 ºC
600 ºC
NPG g-GNR
(a)
Figu e 5.3: (a) Schema ic ep esen a ion o he syn he ic pa h ollowed o ab ica e g-
GNRs and NPG. (b,c) Nc-AFM images o g-GNR and NPG. Figu e aken om Re .[180].
5.2 Elec onic s uc u e o gul -GNRs
5.2.1 Expe imen al STS cha ac e iza ion
Fig. 5.4a shows dI/dV poin -spec oscopy measu emen s pe o med wi h a CO- ip on
di e en posi ions o a g-GNR. The spec a exhibi s a semiconduc ing bandgap, wi h VB
and CB onse s a −1V and 1.7V, espec i ely. In e es ingly, he CB onse is ema kably
mo e in ense when measu ed in he gul s ( ed cu es), while he VB onse is enhanced in
he gul s and p o uding phenyls (blue cu es). This dis inc beha iou is clea ly e lec ed
by he elec onic densi y dis ibu ion in dI/dV maps aken a ound he VB and CB onse
ene gies (Fig. 5.4b). While occupied s a es (-1.3 V and -1.1 V) a e mo e in ensely loca ed
along he edges, unoccupied s a es (1.9 V and 2.1 V) appea as in ense conduc ance
ea u es con ined o he gul s, no ably ex ending in o he la e al acuum egion and wi h
negligible weigh in he ca bon-ne wo k. As desc ibed abo e, his obse a ion esembles
expe imen al dI/dV maps o o he ca bon-based nanoa chi ec u es ound in he li e a u e,
whe e elec onic densi y con inemen has been a ibu ed o a s anding wa e pa e n
o med by he Au(111) su ace s a e [182], bay s a es o igina ed om IPS [111], delocalized
molecula o bi als (MOs) [240,242,244], o a supe posi ion o non-in e ac ing molecula
s a es [247].
5.2.2 DFT simula ions
In o de o disen angle he elec onic s uc u e o g-GNRs and cla i y he o igin o he
obse ed elec onic localiza ion, we pe o m DFT calcula ions o ee-s anding pe iodic
g-GNRs. The uni -cell consis s o 24 C and 8 H a oms wi h a 7.54 ˚
A la ice pa ame e
5.2. Elec onic s uc u e o gul -GNRs 79
(b)
(a)
III III IV
Figu e 5.4: Expe imen al STS cha ac e iza ion o g-GNRs. (a) dI/dV poin -
spec oscopy o a g-GNR measu ed wi h a CO- ip. The inse shows a BR-STM image o
a g-GNR. Spec oscopy is measu ed a he di e en posi ions indica ed by solid and open
do s in he inse BR-STM image. (b) dI/dV maps o a g-GNR measu ed wi h a CO- ip
a he selec ed bias ol ages indica ed in each map. Figu e aken om Re .[180].
along he Y-di ec ion (Fig. 5.5a). Vacuum spacings o 40 ˚
A a e in oduced in ans e sal
(X) and e ical (Z) di ec ions in o de o a oid spu ious e ec s om pe iodic images.
All a omic coo dina es a e op imized un il o ces below 0.01 eV/˚
A a e achie ed, and he
1D la ice is elaxed below a p essu e h eshold o 0.25 GPa. Co e elec ons a e ea ed
using no m-conse ing T oullie -Ma ins pseudopo en ials [177], while alence elec ons
a e desc ibed using a linea combina ion o NAOs. We employ a DZP basis se wi h a
0.01 Ry ene gy shi [148], which is u he expanded wi h slowly decaying 3sand 3p
o bi als in o de o accoun o he possible eme gence o SAMOs o IPSs in gul s and
nanopo es, as explained in Chap e 3. Exchange-co ela ion ene gies a e app oxima ed
by he GGA-PBE unc ional [138]. A 400 Ry cu o is employed o de ine he eal-space
g id, while he BZ is sampled using a Monkho s -Pack g id wi h 51 k-poin s [181].
The DFT calcula ed band s uc u e o he g-GNR, shown in Fig. 5.5b, e eals a wide
1.84 eV bandgap ha con i ms he semiconduc ing cha ac e o he ibbon. The bandgap
is bounded by pai s o dispe si e on ie bands. In s icking con as o expe imen al
obse a ions, Γ-poin alence and conduc ion on ie s a es appea delocalized o e he
en i e ca bon-ne wo k and do no ex end in o he acuum egion along he edges, as
depic ed in Fig. 5.5c. Pe haps he only ag eemen be ween expe imen and heo y is ound
o he VB. The dI/dV map a -1.1 V shows pai s o lobes loca ed in he p o uding phenyl
ings, in p e y good ag eemen wi h simila ea u es o he calcula ed wa e unc ion.
Besides, as in 7-13-AGNRs [111] and Ph-7-13-AGNRs (Chap e 3), he band s uc u e
86 Chap e 5. Decep i e o bi al con inemen in ca bon-based nanoa chi ec u es
(b) (c)
GNR
ip
z = 2 Å z = 5 Å
ip
(a)
X
Y
Figu e 5.10: Wa e unc ion decay in a ini e 24-uni -cell-long 7-AGNRs. (a) A omic
s uc u e o he ini e 7-AGNR conside ed in he simula ions. (b,c) DFT Kohn-Sham
Γ-poin wa e unc ions co esponding o VB-1, VB, CB and CB+1 s a es e alua ed a
dis ances o (a) z= 2 ˚
A and (b) z= 5 ˚
A abo e he ibbon su ace. Figu e aken om
Re .[180].
co esponding o Gdec eases wi h z. Consequen ly, o la ge enough ip-sample dis ances,
only he lowes Fou ie componen s Gwill su i e in he expansion, and he wa e unc ion
will p esen a slowly oscilla ing appea ance. Likewise, because he in a-uni -cell nodal
s uc u e o he wa e unc ion is desc ibed by he expansion o e G, hose wa e unc ions
cha ac e ized by complex and apidly oscilla ing nodal s uc u es, domina ed by as
Fou ie componen s, will be measu ed less in ensely han smoo he wa e unc ions. This
well-known e ec leads o he ac ha STM is mo e sensi i e o s a es close o he Γ
poin (low k∥) [254], bu i also has a signi ican in luence on he esolu ion o dI/dV
poin -spec oscopy and mapping [237,255].
Wi h he aim o illus a ing he signi icance o Eq. 5.1, we i s explo e he ex ensi ely
s udied 7-AGNR, whe e he de ec ion o he CB onse emained elusi e o yea s due
o i s speci ic o bi al decay [237]. In o de o compa e ou esul s o hose epo ed in
Re . [237], we conside a ini e 24-uni -cell-long 7-AGNR in ou simula ions, as shown in
Fig. 5.10a. We i s pe o m DFT calcula ions o he molecula o bi als co esponding
o he VB-1, VB, CB and CB+1. Conside ing ha he cu o adius o he 2pzo bi al

5.4. Wa e unc ion decay: o igin o decep i e o bi al con inemen 87
employed in he calcula ion is ∼2.9˚
A, we use he wa e unc ions e alua ed a z= 1 ˚
A
as e e ences, whe e he o bi al is physically accu a e. Tha e e ence wa e unc ion is
hen ex apola ed o la ge dis ances om he ibbon. The ex apola ion is pe o med, as
implemen ed in he SIESTA STM ool, which Fou ie expands he wa e unc ion e alua ed
a he e e ence plane (z= 1 ˚
A) and applies he z-dependen Fou ie il e ing de ined in
Eq. 5.1. Fig. 5.10b,c shows he co esponding ex apola ed wa e unc ions a dis ances
o z= 2 ˚
A and z= 5 ˚
A om he plane. The esul s a e in good ag eemen wi h hose
epo ed by S¨ode e al. [237]. In pa icula , all s a es a e delocalized o e he en i e
ibbon backbone and exhibi simila in ensi ies a z= 2 ˚
A. Howe e , because all s a es
oscilla e s ongly in he di ec ion ac oss he ibbon backbone, hei wa e unc ions end
o decay owa ds he acuum egion nex o he ibbon edges and anish in he backbone.
In addi ion, VB-1 and CB oscilla e apidly also along he g ow h di ec ion o he ibbon
+
-
0
VB-1 VB CB CB+1
z≈ 2.0 Åz≈ 3.0 Åz≈ 4.0 Åz≈ 5.0 Åz≈ 6.0 Å
Inc easing ip-sample dis ance
Figu e 5.11: Wa e unc ion decay in g-GNRs. DFT Kohn-Sham Γ-poin wa e unc ions
co esponding o VB-1, VB, CB and CB+1 s a es e alua ed a dis ances o z= 2,3,4,5
and 6 ˚
A abo e he ibbon su ace. Figu e aken om Re .[180].
88 Chap e 5. Decep i e o bi al con inemen in ca bon-based nanoa chi ec u es
and, consequen ly, hei in ensi y a z= 5 ˚
A is s ongly supp essed in compa ison o VB
and CB+1, which do no oscilla e along he ibbon backbone. Consequen ly, he VB-1
and CB can be ha dly de ec ed using STS a ypical ip-sample dis ances, while he VB
and CB+1 a e ypically measu ed con ined o he ibbon edges [237].
In he case o g-GNRs, due o hei no ched edge s uc u e, his decay e ec exhibi s
i s own pa icula i ies. No e ha , in o de o a oid ini e size e ec s, we simula ed
pe iodic g-GNRs as desc ibed in Sec. 5.2.2. Fig. 5.11 shows he decay o DFT-calcula ed
Γ-poin wa e unc ions o VB-1, VB, CB and CB+1 e alua ed a a ious dis ances om
he su ace. As he dis ance inc eases, all hese delocalized wa e unc ions p og essi ely
anish in he ibbon backbone and decay owa ds he acuum egion along he edges.
+
-
0
VB-1 VB CB CB+1
z≈ 2.0 Åz≈ 3.0 Åz≈ 4.0 Åz≈ 5.0 Åz≈ 6.0 Å
Inc easing ip-sample dis ance
Figu e 5.12: Wa e unc ion decay in NPG. DFT Kohn-Sham Γ-poin wa e unc ions
co esponding o VB-1, VB, CB and CB+1 s a es e alua ed a dis ances o z= 2,3,4,5
and 6 ˚
A abo e he ibbon su ace. Figu e aken om Re .[180].
5.5. Gene aliza ion o o he po ous ca bon-based nanoa chi ec u es 89
A la ge ip-sample dis ances, VB and CB, which oscilla e along he ibbon backbone,
exhibi a pai o opposi e sign lobes in each p o uding phenyl ing. In con as , VB-1
and CB+1 exhibi a con inuous ea u e ha pe ec ly ollows he shape o he edge a
bo h sides o he g-GNR.
In he case o NPG, he po ous s uc u e signi ican ly in luences he decay e ec . The
decay o DFT-calcula ed Γ-poin wa e unc ions o VB-1, VB, CB and CB+1 e alua ed
a a ious dis ances om NPG is shown in Fig. 5.12. Simila o g-GNRs, wa e unc ion
in ensi ies g adually anish in he 2D ca bon-ne wo k as zinc eases. In he absence o
a 1D edge, howe e , wa e unc ions decay owa ds he nanopo es, e en ually exhibi ing
maximun in ensi y wi hin hem. I is wo h no ing ha he in ensi y o he VB wa e
unc ion is signi ican ly educed compa ed o he es o s a es. Such wa e unc ion
oscilla es along he ibbon backbone, while i exhibi s a change o sign om one g-GNR
o he nex , i.e. i is an isymme ic be ween adjacen ibbons. This esul s in a s ongly
oscilla ing wa e unc ion in bo h di ec ions o he NPG, which decays as e han he
es o s a es in he e ical di ec ion. The es o s a es a e ei he symme ic be ween
adjacen ibbons o do no oscilla e along he ibbon backbone, and a e he e o e no ably
mo e in ense a la ge dis ances om he su ace. In pa icula , CB+1 does no change
sign wi hin he nanopo e, closely esembling heo e ical LDOS and expe imen al dI/dV
maps.
The decay ea u es p esen ed in Fig. 5.11 and Fig. 5.12 explain he localiza ion o
elec onic densi y ob ained om he LDOS maps, and a e he key ac o leading o de-
cep i e o bi al con inemen in expe imen al STS measu emen s. Impo an ly, such s ong
localiza ion a ises om he combina ion o wo ac o s: (1) he decay o he g-GNR and
NPG o bi als in he e ical di ec ion, and (2) he con inemen o he o bi als in o 1D
and 0D acuum egions (e.g., edges, gul s and nanopo es), as de e mined by he exac
edge s uc u e o he sys em. While he i s is a gene al and well-known e ec epo ed
in many su aces and molecula assemblies, he second is expec ed o be pa icula ly el-
e an in g aphene-based 1D and 2D nanos uc u es, o many o hei edge and nanopo e
mo phologies. Nex sec ion will se e us as a suppo o he la e s a emen .
5.5 Gene aliza ion o o he po ous ca bon-based nanoa -
chi ec u es
In o de o demons a e he gene al alidi y o he esul s p esen ed in he p e ious sec ion,
we nex show ha he decay o he wa e unc ion owa ds he acuum can also explain
simila localiza ion e ec s eme ging in o he ypes o ca bon-based nanos uc u es. In
pa icula , we s udy GNRs unc ionalized wi h double- oid po es and di acancies.
5.5.1 Double- oid GNRs
We pe o m DFT simula ions o an 18-AGNR wi h double- oid nanopo es pe iodically
embedded along he ca bon-backbone (see inse in Fig. 5.13a). The ibbon is pe iodic in
he Y-di ec ion, wi h a la ice cons an o 17.39 ˚
A, and he uni -cell consis s o 132 C and
26 H a oms. We employ a 51 k-poin Monkho s -Pack g id o he BZ sampling [181].
The es o simula ion pa ame e s a e hose used in p e ious sec ions o his chap e .
90 Chap e 5. Decep i e o bi al con inemen in ca bon-based nanoa chi ec u es
a b
VB-1 VB
CB CB+1
z≈ 6.0 Åz≈ 6.0 Å
1.00
0.75
0.50
0.25
0.00
-0.25
-0.50
-0.75
-1.00
E-EF (eV)
ΓY
3.71 pA
-2.56 pA
LDOS max.
min.
3.71 pA
-2.56 pA
LDOS max.
min.
(a)
(b)
VB
CB
CB+1
VB-1
Figu e 5.13: (a) DFT band s uc u e o double- oid 18-AGNRs. The ene gy anges used
o LDOS map simula ions in (b) a e indica ed by pink shaded egions. Inse schema ically
ep esen s he sys em geome y, wi h ed dashed lines indica ing he uni -cell. (b) DFT-
LDOS maps a a dis ance o z= 6 ˚
A abo e he ibbon su ace and e alua ed in he ene gy
egions indica ed in (a), which include VB-1, VB, CB and CB+1 onse s. Figu e aken
om Re .[180].
+
-
0
VB-1 VB CB CB+1
z≈ 2.0 Å
z≈ 6.0 Å
Figu e 5.14: Wa e unc ion decay in double- oid 18-AGNRs. DFT Kohn-Sham Γ-poin
wa e unc ions co esponding o VB-1, VB, CB and CB+1 s a es e alua ed a dis ances
o z= 2 and z= 6 ˚
A abo e he ibbon su ace. Figu e aken om Re .[180].
Fig. 5.13 shows he co esponding (a) band s uc u e and (b) LDOS map simula ions
o VB-1, VB, CB, and CB+1. The band s uc u e exhibi s a ∼0.95 eV bandgap. The
5.5. Gene aliza ion o o he po ous ca bon-based nanoa chi ec u es 91
bandgap is bounded by dispe sing on ie s a es, which coun e in ui i ely lead o s ongly
localized DFT-LDOS ea u es in nanopo es and edges o ip-sample dis ances o z= 6˚
A
(see Fig. 5.13b).
Fig. 5.14 shows he VB-1, VB, CB and CB+1 wa e unc ions a he Γ-poin a dis-
ances close (z= 2˚
A) and a (z= 6˚
A) om he su ace. All s a es exhibi an in insic
delocaliza ion o e he en i e ca bon-ne wo k a z= 2˚
A, while hei con inemen in o he
nanopo es is e iden a z= 6˚
A. This esul co ela es e y well wi h he DFT-LDOS maps
shown in Fig. 5.13, and co obo a es he alidi y o he wa e unc ion decay mechanism
epo ed abo e in he case o double- oid po es.
5.5.2 Di acancy GNRs
We ha e also s udied he wa e unc ion decay in a 8-ZGNR deco a ed wi h pe iodically
embedded di acancy po es. These a e a anged acco ding o a 14.98 ˚
A la ice pa ame e
in he Y di ec ion, wi h 94 C and 16 H a oms in each uni -cell. A 51 k-poin Monkho s -
Pack g id is employed o he BZ sampling [181]. The es o simula ion pa ame e s a e
hose used in p e ious sec ions o his chap e . The a omic s uc u e is shown in Fig. 5.15
and Fig. 5.16a. Due o he small size o he di acancy, s e ic e ec s d i e he H a oms in
he po es as well as he nea by C a oms in o an ou -o -plane con igu a ion, as shown by
he co uga ion o he GNR in Fig. 5.15. [175].
The DFT-calcula ed band s uc u e and LDOS maps o he di acancy-8-ZGNR a e
depic ed in Fig. 5.16. When elec on spins a e no conside ed, ZGNRs hos doubly degen-
e a e ze o-ene gy la bands, whose wa e unc ions a e in insically localized a he ibbon
edges [57,238]. I is well-known ha upon inclusion o an on-si e Coulomb epulsion in
TB models o aking in o accoun spins in DFT calcula ions, a bandgap opens in ZGNRs
[60,238]. Howe e , because we a e jus in e es ed in he wa e unc ion decay o s a es ha
a e in insically delocalized o e he en i e ca bon-backbone, we do no conside he spin
X
Y
Z
Y
Y
X
Y
Z
Y
Y
Z
Y
(a) (b)
Figu e 5.15: Side (a) and op (b) iews o a di acancy-8-ZGNR uni -cell. The colo code
ep esen s he displacemen (co uga ion) in he Z-di ec ion o each a om wi h espec o
he a e age Z-coo dina e. Figu e aken om Re .[180].

92 Chap e 5. Decep i e o bi al con inemen in ca bon-based nanoa chi ec u es
a b
VB-1 VB
CB CB+1
z≈ 6.0 Åz≈ 6.0 Å
1.00
0.75
0.50
0.25
0.00
-0.25
-0.50
-0.75
-1.00
E-EF (eV)
ΓY
cc
3.71 pA
-2.56 pA
LDOS max.
min.
3.71 pA
-2.56 pA
LDOS max.
min.
a b
VB-1 VB
CB CB+1
z≈ 6.0 Åz≈ 6.0 Å
1.00
0.75
0.50
0.25
0.00
-0.25
-0.50
-0.75
-1.00
E-EF (eV)
ΓY
cc
LDOS max.
min. LDOS max.
min.
(a)
(b)
FBs
VB
CB
CB+1
VB-1
Figu e 5.16: (a) DFT band s uc u e o di acancy-8-ZGNRs. The ene gy anges used
o LDOS map simula ions in (b) a e indica ed by pink shaded egions. Side image is
an schema ic ep esen a ion o he sys em geome y, wi h ed dashed lines indica ing he
uni -cell. (b) DFT-LDOS maps a a dis ance o z= 6 ˚
A abo e he ibbon su ace and
e alua ed in he ene gy egions indica ed in (a), which include VB-1, VB, CB and CB+1
onse s. The in insically localized ze o-ene gy la bands a e no conside in he LDOS
analysis. Figu e aken om Re .[180].
+
-
0
VB-1 VB CB CB+1
z≈ 2.0 Å
z≈ 6.0 Å
Figu e 5.17: Wa e unc ion decay in di acancy-8-ZGNRs. DFT Kohn-Sham Γ-poin
wa e unc ions co esponding o VB-1, VB, CB and CB+1 s a es e alua ed a dis ances
o z= 2 and z= 6 ˚
A abo e he ibbon su ace. Wa e unc ions o in insically localized
ze o-ene gy la bands a e no conside . Figu e aken om Re .[180].
5.6. Summa y 93
deg ee o eedom in ou calcula ions. Acco dingly, we ob ain a me allic band s uc u e
wi h a pai o app oxima ely la ze o-ene gy bands (FBs). These a e no conside in ou
analysis, as hey a e in insically localized a he ibbon edges. In con as , we compu e
LDOS maps a ene gy in eg a ion windows co esponding o dispe si e VB-1, VB, CB
and CB+1 a a dis ance o z= 6 ˚
A abo e he ibbon su ace. As in p e ious cases,
he elec onic densi y anishes in he inne pa o he ca bon-ne wo k and ge s con ined
o edges and di acancies. The ac ha he LDOS is asymme ic in he H-passi a ed
di acancies is due o he a o emen ioned co uga ion a ound hem, which enhances he
in ensi y a ound he a oms ha a e close o he ip. The LDOS localiza ion can be
unde s ood by looking a he decay o he wa e unc ions om z≈2˚
A o z≈6˚
A, as
shown in Fig. 5.17.
5.6 Summa y
We ha e add essed he o igin o he elec onic con inemen a he edges and nanopo es
o a ious ca bon-based nanoa chi ec u es, as equen ly obse ed in STS expe imen s.
Based on he Te so -Hamann heo y o STM, we ha e ep oduced dI/dV maps o g-GNR
and NPG s uc u es, linking localized LDOS ea u es o in insically delocalized elec onic
s a es. This decep i e con inemen e ec is a ibu ed o he il e ing o high momen um
componen s in he Fou ie expansion o GNR and NPG wa e unc ions as hese a e
e alua ed a inc easing dis ances om he sample. The e ec becomes c i ical a la ge ip-
sample dis ances, and i is s ongly in luenced by he speci ic edge and nanopo e s uc u e
o he sys em unde conside a ion. Impo an ly, DFT-LDOS maps allow us o co ec ly
in e p e he expe imen al dI/dV measu emen s aken in g-GNRs and NPG. Addi ional
calcula ions also show ha his e ec holds o o he edge and nanopo e geome ies.
We hus expec decep i e o bi al con inemen o be p esen in he as majo i y o STS
expe imen s ha a e pe o med in ca bon-based nanoa chi ec u es employing ypical ip-
sample dis ances. This e ec p o ides an al e na i e, plausible explana ion o many
localiza ion e ec s epo ed in he li e a u e.
CHAPTER 6
Conclusions & ou look
This hesis p o ides an ex ensi e heo e ical s udy o elec onic and anspo phenomena
in a omically p ecise, po ous ca bon-based nanoa chi ec u es. In o de o a ionalize
expe imen al esul s and p edic elec onic p ope ies om a omic o de ice scale, we
ha e app op ia ely combined DFT, TB and G een’s unc ions me hods, which ha e been
desc ibed in Chap e 2.
In Chap e 3, we ha e ocused on he s udy o on-su ace syn hesized po ous GNR
and NPG s uc u es. Pe o ming a DFT-based ene ge ics analysis we ha e un eiled he
mechanism leading o he mig a ion o phenyl side-g oups in Ph-7-13-AGNRs, which
e en ually leads o he o ma ion o [18]-annulene po es in he ibbons. We ha e shown
ha he phenyl mig a ion canno be ini ia ed wi hou he p esence o Au(111) su ace
ada oms, which is in line wi h he c ucial ole o su ace ada oms as ca alyze s o o he
OSS eac ions highligh ed in he li e a u e. Combined wi h expe imen al STM imaging,
ou heo e ical esul s demons a e ha phenyl mig a ion can be inco po a ed in o he
oolbox o OSS eac ions in o de o p oduce con ollable in e nal ans o ma ions in
ca bon-based nanos uc u es. This may ul ima ely en ich he ca alogue o a omically
p ecise ca bon-based nanoa chi ec u es by p omo ing he syn hesis o mo e sophis ica ed
geome ies. Besides, we ha e shown ha he o ma ion o [18]-annulene po es lea es he
bandgap o he ibbon unchanged, while he acuum s a es localized a he bays and po es
unde go a signi ican ene gy shi .
Ins ead, i Ph-7-13-AGNRs a e la e ally used, NPG s uc u es con aining molecu-
la b idges wi h ei he pa a-pa a,pa a-me a, and me a-me a bonding con igu a ions a e
syn hesized. Ou DFT simula ions demons a e ha he me a bonding con igu a ions
ac as chemical knobs inducing an ab up enhancemen o he elec onic aniso opy in
NPG. In addi ion, he wis ed con o ma ion e ealed in pa a-pa a b idges is shown o
beha e as a g adual con o ma ional knob o he in e ibbon coupling and he elec onic
aniso opy. Based on p uned TB Hamil onians ex ac ed om DFT calcula ions, we ha e
pe o med NEGF simula ions o he elec onic anspo in ealis ic de ice scales, p e-
dic ing ich quan um elec on anspo phenomena in NPG. While he chemical knob
ab up ly quenches he Talbo in e e ence e ec by con ining la ge-scale cu en s o a sin-
gle GNR, he con o ma ional knob can be used o ine une he wa e leng h o he Talbo
s anding wa e pa e n. In e es ingly, hese esul s demons a e he po en ial o molecula
b idge enginee ing o op imize elec onic aniso opy in NPG, and p o ide new pa hways
owa ds con olling nanoscale cu en s and hei cohe en beha iou in no el elec onic
nanode ices.
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