Academic Edi o : An onio Polimeni
Recei ed: 4 July 2025
Re ised: 12 Augus 2025
Accep ed: 13 Augus 2025
Published: 15 Augus 2025
Ci a ion: Kwapi´nski, T.; Ku zyna, M.;
K awiec, M. S abiliza ion o Cha ge
Densi y Wa es in A omic Chains on
Xenes. Ma e ials 2025,18, 3843.
h ps://doi.o g/10.3390/
ma18163843
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Licensee MDPI, Basel, Swi ze land.
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A icle
S abiliza ion o Cha ge Densi y Wa es in A omic Chains
on Xenes
Tomasz Kwapi´nski 1,*,† , Ma cin Ku zyna 2,† and Ma iusz K awiec 1,*,†
1Ins i u e o Physics, Ma ia Cu ie-Sklodowska Uni e si y , 20-031 Lublin, Poland
2Ins i u e o Compu e Science and Ma hema ics, Ma ia Cu ie-Sklodowska Uni e si y,
20-031 Lublin, Poland; [email p o ec ed]
*Co espondence: [email p o ec ed] (T.K.); [email p o ec ed] (M.K.)
†These au ho s con ibu ed equally o his wo k.
Abs ac
We in es iga e he elec onic p ope ies o a omic chains placed on g oup-14 wo-
dimensional ma e ials, Xenes, by analyzing he local elec onic p ope ies. Ou esul s
show ha he hyb idiza ion be ween he chain and he subs a e leads o signi ican modi i-
ca ions in he local densi y o s a es a each chain si e, including peak spli ing, b oadening,
and asymme y. These e ec s a e pa icula ly p onounced o plumbene. Owing o he
subs a e’s V-shaped-like densi y o s a es, he chains exhibi s ong localiza ion e ec s and
signi ican in ensi y a ia ions in he elec onic ene gy spec um. In addi ion he p esen
analysis e eals he eme gence o cha ge densi y wa es in a omic chains, o which he
appea ance and s abili y condi ions a e iden i ied and p o ided. The cha ge densi y wa es
a e mo e p onounced and s abilized by a speci ic elec onic spec um o Xenes, allowing
hem o pene a e deepe in o he chain in e io . Ou indings con ibu e o he b oade
unde s anding o he in e ac ion be ween one-dimensional chains and wo-dimensional
Xene ma e ials, which ha e signi ican implica ions o de eloping ad anced hyb id nanos-
uc u es and nex gene a ion elec onic de ices.
Keywo ds: a omic chains; 2D su aces; densi y o s a es; cha ge wa es
1. In oduc ion
One-dimensional (1D) sys ems such as a omic chains a e undamen al building blocks
o nanoelec onics, as hese chains ep esen he hinnes possible conduc o s o elec ic
cu en . Many in iguing physical phenomena ha e been obse ed in hese sys ems,
including Lu inge liquid beha io , cha ge wa es, and elec on and spin pumps [
1
–
3
],
as well as mo e exo ic e ec s like Majo ana opological s a es [
4
,
5
], Floque opological
insula o s [
6
], and ime c ys als [
7
]. These 1D sys ems, in he o m o a ays o quan um
do s (whe e all sys em pa ame e s can be con olled using auxilia y elec odes) o a omic
chains, a e in di ec con ac wi h hei en i onmen , which is ypically a subs a e. Thus
he wa e unc ions o hese s uc u es hyb idize wi h he elec onic s a es o he subs a e,
leading o a eno maliza ion o he molecula s a es. The e o e, unde s anding he elec onic
p ope ies o such chains on di e en subs a es is c ucial o nanoelec onics.
A omic chains can be expe imen ally ab ica ed h ough he sel -assembly o a oms
on c ys alline su aces, which means he chain is placed on a subs a e. The e a e many
expe imen al examples o such a omic chains a a ious icinal and la su aces, including
Si(113)-Pb [
8
], Si(335)-Au [
9
], Si(557)-Au [
10
,
11
], o Si(553) [
12
]. I is impo an o no e
Ma e ials 2025,18, 3843 h ps://doi.o g/10.3390/ma18163843
Ma e ials 2025,18, 3843 2 o 13
ha he silicon su ace on which a omic chains a e ab ica ed is usually s abilized by gold
a oms. These gold a oms o m a double chain embedded be ween silicon a oms on each
e ace [
13
]. Such a e ace can be conside ed an e ec i e wo-dimensional (2D) su ace,
which is conduc i e, in con as o he a oms in he bulk o he sample, which exhibi ypical
h ee-dimensional (3D) semiconduc o p ope ies. As a esul , a omic chains o med on
his su ace a e p ima ily coupled o he 2D s uc u e ha cons i u es he ou e mos laye
o he subs a e.
Recen ly, o he 2D sys ems wi h ema kable elec onic p ope ies ha e a ac ed much
a en ion. This new class o 2D sys ems, known as 2D-Xenes, ea u es single laye s o a oms
a anged in a honeycomb la ice [
14
–
18
]. The p ecu so o hese ma e ials is g aphene,
isola ed in 2004 [
19
]. I consis s o ca bon a oms wi h
sp2
-hyb idized elec onic o bi als
gi ing ise o linea bands in he elec onic spec um, cha ac e is ic o Di ac pa icles.
These bands o igina e om he
π
o e lap o he
pz
o bi als. O he g oup-14 Xenes, like
silicene, ge manene, s anene, and plumbene, also exhibi he Di ac physics, e en hough
hei a omic s uc u e is no comple ely plana , like in g aphene, and he mixed
sp2/sp3
hyb idiza ion appea s. As a esul , he
π
bands change hei o bi al cha ac e , and become
subs an ially na owe han in he case o g aphene. Owing o hei ema kable elec onic
p ope ies, 2D Xenes can possibly se e as empla es o a omic chains, con olling and
uning hei p ope ies.
Cha ge densi y wa es (CDWs) in 1D can eme ge in o de ed sys ems whe e he po en-
ial is dis u bed by impe ec ions such as la ice de ec s, impu i ies, o disloca ions, as well
as in con ined a omic sys ems due o bounda y e ec s [
20
–
24
]. This phenomenon was i s
p edic ed by J. F iedel and is cha ac e ized by oscilla ions in he elec on densi y a ound
impu i y si es [
25
], and i was ini ially obse ed using scanning unneling mic oscopy
on coppe su aces [
26
], whe e s anding wa e pa e ns we e de ec ed nea a omic s eps
and poin de ec s. The decay a e o hese oscilla ions exhibi s sinusoidal oscilla ions ha
dec ease wi h dis ance om he pe u ba ion and is s ongly in luenced by he dimension-
ali y o he sys em [
20
] and can a ec bo h he s abili y and physical p ope ies o me allic
a omic chains and ul a hin me al ilms [
27
]. I was also shown ha bo h he conduc ance
oscilla ions and he localized cha ge along he chain ade away due o he in luence o he
simple (s uc u eless) subs a e elec ode [28].
CDW ha e gained inc easing a en ion o hei po en ial in u u e nanoscale and
quan um echnologies. Thei unable na u e, sensi i i y o ex e nal s imuli (such as elec ic
ields, s ain, o doping), and abili y o exhibi non-linea elec ical anspo make hem
p omising o applica ions in ul a as swi ches [
29
], non- ola ile memo y de ices, and
neu omo phic compu ing a chi ec u es [
30
,
31
]. Mo eo e , hei cohe ence and wa e-like
beha io open up possibili ies in cohe en signal p ocessing and quan um in o ma ion
pla o ms [
32
]. These unique p ope ies a e especially ad an ageous when in eg a ed
wi h 2D ma e ials and low-dimensional s uc u es, o e ing compa ibili y wi h eme ging
low-powe , high-densi y de ice pa adigms. S udying 1D sys ems, such as a omic chains
placed on 2D ma e ials, is impo an bo h om a undamen al and an applied pe spec i e.
One-dimensional sys ems exhibi s ong elec onic localiza ion, and collec i e phenom-
ena like CDWs. When coupled o 2D subs a es, such as Xenes, analogues o g aphene,
hei elec onic p ope ies can be uned ia hyb idiza ion wi h he subs a e’s elec onic
s uc u e. These 1D/2D hyb id sys ems o e a unique pla o m o enginee ing no el
quan um s a es and designing unc ional elemen s o molecula elec onics, spin onics,
and nanoelec onics. The e o e, unde s anding and con olling hese hyb id s uc u es is a
p omising di ec ion o he de elopmen o u u e nanoscale and quan um echnologies.
I is he e o e essen ial o unde ake s udies ha de e mine whe he CDW can exis
in ealis ic 1D sys ems, whe e he a omic chain is coupled o a subs a e elec ode wi h a
Ma e ials 2025,18, 3843 3 o 13
speci ic and complex densi y o s a es (DOS). The commonly used wide-band limi (WBL)
app oxima ion o elec odes is me ely a simpli ied oy model, which may be alid o 3D
me allic ma e ials bu ep esen s an o e simpli ica ion when applied o 2D a omic su aces.
In such cases, i ails o accu a ely cap u e he ue physical beha io . In pa icula , i
is inadequa e o desc ibing ma e ials wi h a linea dispe sion in hei spec al unc ion,
such as in he case o Xenes. To he bes o ou knowledge, he e a e no heo e ical wo ks
ha esol e he ques ion o whe he CDW can exis in 1D sys ems coupled o ealis ic 2D
su aces, which should be ho oughly explained.
The p ima y objec i e o ou wo k is o analyze he elec ical p ope ies o de-
posi ed a oms o a omic chains placed on 2D Xenes cha ac e ized by a linea ene gy
dispe sion a ound he Fe mi le el—such as g aphene, silicene, ge manene, s anene, and
plumbene—and
o compa e hese esul s wi h hose o a me allic subs a e ha ing a la
DOS. We an icipa e ha he he e ogeneous DOS o 2D ma e ials, wi h nume ous an Ho e
singula i ies, will be e lec ed in he local DOS s uc u e on he a oms wi hin he chain
and can in luence i s elec ical p ope ies and cha ge dis ibu ion along he sys em. In his
con ex , we in es iga e whe he CDW can de elop along he chain on 2D empla es. While
such wa es a e known o occu o chains on la su aces desc ibed by he wide-band limi
app oxima ion [
23
,
28
], hey may be signi ican ly supp essed o migh no eme ge a all on
mo e complex subs a es.
Fo a egula 3D c ys al, he ene gy-band dispe sion can be ob ained analy ically
wi hin he simple igh -binding app oach. Simila ly, o some 2D la ices (e.g., squa e,
honeycomb, Lieb, and Kagome), he ene gy dispe sion can be exp essed in e ms o ellip ic
in eg als, ea u ing a an Ho e loga i hmic singula i y in he band [
33
,
34
]. Howe e , in
ou calcula ions, we ocus on ealis ic desc ip ion o he elec onic band s uc u e wi h aid
o he densi y unc ional heo y (DFT). In his way, we a e able o handle de ails o he
elec onic s uc u e o eal Xenes in a wide ange o ene gies and conside hem as conc e e
empla es o hos adso bed a omic chains. In he p esen s udies, by u ilizing he compu ed
subs a e DOS and applying he second quan iza ion me hod oge he wi h he equa ion o
mo ion o he e a ded G eens unc ion, we de e mine he local DOS o each a om in he
chain deposi ed on he gi en Xene. Knowledge o he local DOS enables us o compu e he
cha ge occupancies on each a om and o analyze he cha ge densi y wa es along he chain.
2. Theo e ical Desc ip ion
2.1. DFT Calcula ions
Fi s -p inciples densi y unc ional heo y calcula ions we e pe o med wi hin Pe dew–
Bu ke–E nze ho (PBE) [
35
] gene alized g adien app oxima ion o he exchange–co ela ion
in e ac ion as implemen ed in VASP 6.4.2 (Vienna ab ini io simula ion package) [
36
,
37
].
The co e elec ons we e ea ed wi hin he p ojec o -augmen ed wa e me hod [
38
]. The
kine ic ene gy cu o o 550 eV o he plane wa e expansion o single pa icle wa e unc-
ion was used. The con e gence c i e ion o o al ene gy was chosen o be 10
−7
eV. The
B illouin zone was sampled by a 24
×
24
×
1 Monkho s –Pack k-poin s g id, including he
Γ
poin [
39
] du ing he geome y op imiza ion and 144
×
144
×
1 in he calcula ions o he
densi y o s a es. The Xene a omic s uc u e was modeled by a single a omic laye sepa a ed
by 10 Å wide acuum gap. The a omic posi ions we e elaxed by a conjuga e g adien
me hod un il he la ges o ce in any di ec ion was below 0.001 eV/Å.
Figu e 1p esen s he subs a e DOS o a ious 2D Xenes—g aphene, silicene, ge -
manene, s anene, and plumbene—calcula ed using DFT, shown o e a wide ene gy ange
(le panel) and a na ow ange a ound he Fe mi ene gy ( igh panel). In he le panel,
cu es B–E a e e ically shi ed o cla i y. As can be seen, each DOS ex ends o e an
ene gy ange o app oxima ely ±8 eV and exhibi s mul iple peaks esembling an Ho e
Ma e ials 2025,18, 3843 4 o 13
singula i ies obse ed in ideal squa e o hexagonal la ices. They come om weakly dispe -
si e and la elec on bands o pos -g aphene Xenes and weake
π
-
π
o e lap due o he
la ge a omic adius o Si, Ge, Sn, and Pb a oms (see Figu e 2 o Re . [
18
] and discussion
he ein). The DOS p o ile o all Xenes, excep plumbene, is cha ac e ized by an almos ze o
alue a he Fe mi le el (
EF
) and a linea dispe sion o ming a V-shaped s uc u e. In he
case o plumbene, howe e , his beha io is dis up ed by
σ
bands c ossing he Fe mi le el,
esul ing in a me allic cha ac e [
18
] and a ini e DOS a
EF
( igh panel). I is also wo h
no ing ha cu es B–E, co esponding o silicene h ough plumbene, exhibi quali a i ely
simila DOS ea u es, in con as o g aphene, which displays a dis inc DOS s uc u e
cha ac e ized by a b oad and smoo h V-shaped p o ile a he Fe mi le el and a ela i ely
small numbe o an Ho e singula i ies.
0
0.1
0.2
0.3
0.4
0.5
0.6
−8 −6 −4 −2 0 2 4 6
B: silicene
A: g aphene
C: ge manene
E: plumbene
D: s anene
DOS(E) [1/eV]
ene gy [eV]
0
0.1
−1 0 1
ene gy [eV]
Figu e 1. DFT calcula ions o he su ace DOS o a ious 2D s uc u es—g aphene, silicene, ge -
manene, s anene and plumbene (labeled A o E). Fo cla i y, in he le panel, cu es B h ough E ha e
been shi ed upwa d by 0.1, 0.2, 0.3 and 0.4, espec i ely.
2.2. TB Calcula ions
To s udy he elec onic p ope ies o a omic chains on a ious Xenes, we employ he
igh -binding echnique, modeling he sys em as a subs a e wi h a gi en DOS wi h an
a ached a omic chain. The chain o leng h
N
is composed o linea a omic si es which
a e coupled oge he by hyb idiza ion ma ix elemen s
and cha ac e ized by on-si e
elec on ene gies
εi
. The model Hamil onian in he s anda d second-quan ized o m can be
exp essed as a sum o he ollowing e ms:
H=
N
∑
i=1
εia†
iai+∑
k
ε
ka†
ka
k+
N−1
∑
i=1
i,i+1a†
iai+1+
N
∑
i=1∑
k
Vi,
ka†
kai+H.c. (1)
In he abo e equa ion, he i s e m ep esen s he on-si e elec on ene gy wi hin
he chain, while he second e m co esponds o he ene gy o he su ace elec ons,
ε
k
,
cha ac e ized by wa e ec o s
k
. The ope a o s
ai
(
a†
i
) annihila e (c ea e) an elec on a
he
i
- h si e o he chain (
i=
1,
. . .
,
N
), and
a
k
(
a†
k
) a e he co esponding annihila ion
(c ea ion) ope a o s o he leads. The hi d pa o he Hamil onian desc ibes elec on
ansi ions along he chain (be ween a omic si es), and be ween he subs a e and chain si es,
espec i ely. He e, he spin index is omi ed, as we a e no in e es ed in he magne iza ion
e ec s, and bo h spin di ec ions a e conside ed independen .
Ma e ials 2025,18, 3843 5 o 13
Cha ge wa es along he chain equi e an analysis o cha ge occupa ions a each si e o
he chain, ni. These on-si e occupa ions can be de e mined based on he ela ion
ni=ZEF
−∞LDOSi(E)dE , (2)
whe e
LDOSi(E)
is he local DOS unc ion a a gi en
i
- h si e and can be ob ained
om he co esponding diagonal ma ix elemen s o he e a ded G een’s unc ion
LDOSi(E) = −1
πImG
ii(E)
. This unc ion can be de i ed using he equa ion o mo ion
echnique [
40
,
41
] om he ela ion
EG
ij(E) = ⟨[ai
,
a†
j]+⟩+⟨⟨[ai
,
H]−
;
a†
j⟩⟩E
. I allows us o
exp ess he ollowing ma ix ela ion:
ˆ
A·ˆ
G =ˆ
I
, whe e
ˆ
I
is he uni ma ix,
ˆ
G
ep esen s
a squa e N×Nma ix o G
ij(E) unc ions, and ˆ
Ais a complex ma ix wi h elemen s
Ai,j(E) = (E−εi)δi,j− i,j+1(δi,j+1+δi+1,j)−Σi,j(E). (3)
He e,
Σi,j(E) = ∑
kV∗
i,
kVj,
k(E+−ε
k)−1
, and in gene al
V∗
i,
k
elec on hyb idiza ion el-
emen s be ween chain si es and he subs a e Bloch s a es depend exponen ially on he
spa ial posi ions o a oms [
28
,
42
]. Thus, he o -diagonal elemen s o
ˆ
Σ(E)
apidly de-
c ease e en o neighbo ing a oms and a e he e o e neglec ed in ou calcula ions. All
he in o ma ion abou he su ace is con ained in he diagonal elemen s o
Σii(E)
, which
can be exp essed as
Σii(E) = Λii(E)−iΓii(E)/
2, whe e
Γii(E) =
2
π∑
k|Vi,
k|2δ(E−ε
k)
,
and o
|Vi,
k|2
independen o he wa e ec o
k
, as well as o all a oms coupled o he
same su ace, we can w i e
Γii(E) =
2
π|V
k|2DOS(E)
. The unc ion
DOS(E)
s ands o
ene gy-dependen DOS o he Xene, which in ou calcula ions is ob ained om he DFT
echnique. The second unc ion
Λii(E)
is ela ed o
Γii(E)
h ough he Hilbe ans o m
Λii(E) = 1
2πR∞
−∞Γii(E′)/(E−E′)dE′
. Thus, knowledge o he Xene DOS is su icien o
compu e bo h unc ions
Λii(E)
and
Γii(E)
, which depend on ene gy and, once subs i u ed
in o Equa ion
(3)
, allow o he calcula ion o he e a ded G een’s unc ion by in e ing
he
ˆ
A
ma ix::
G
ii(ε) = ( ˆ
A−1)ii =co ˆ
Aii/de ˆ
A
, whe e
co (ˆ
Aii)
deno es he algeb aic
complemen o
ˆ
Aii
, and
de (ˆ
A)
ep esen s he de e minan o he ma ix
ˆ
A
. Consequen ly,
we can de e mine he local DOS and cha ge occupa ions a each si e o he chain. I is wo h
no ing ha o speci ic o ms o he subs a e DOS (e.g., an Ho e ype, ec angula , o
ellip ical DOS), an analy ical o m o he Hilbe ans o m exis s, which allows o explici
analy ical exp essions o he G een’s unc ions. Also o a egula chain on a subs a e
desc ibed by ene gy-independen DOS (wide-band app oxima ion) wi h homogeneous
on-si e ene gies and uni o m a om–a om couplings, he G een unc ions can be exp essed
analy ically using Chebyshe polynomials o he second kind [
43
]. In his pape , due o he
non-analy ical subs a e DOS o 2D s uc u es, we ob ain he local DOS along he chain
and he elec on occupancy nume ically.
In ou calcula ions, we employ he ze o- empe a u e limi , and all ene gies a e ex-
p essed in uni s o
Γ0=
1 eV (whe e we use he ela ion
Γ0=
2
πV2
k/w
o a la ,
k-independen
Xene DOS o wid h
w=
20). The ene gy e e ence poin is se a he
chemical po en ial o he lead,
EF=
0. Fo pa ame e s used in he pape , he ypical
coupling s eng hs be ween a omic si es ange om 0.1 o 4 eV.
3. Resul s and Discussion
3.1. Few-A om Sys ems
Be o e analyzing he elec onic p ope ies o a omic chains, we i s examine how he
local DOS is modi ied o a single a om and o an a omic dime ( wo coupled a oms). The
uppe panel o Figu e 2p esen s he esul s o a single a om placed on a ious 2D Xenes,
anging om g aphene o plumbene, ep esen ed by cu es A–E, wi h he co esponding
Ma e ials 2025,18, 3843 6 o 13
subs a e DOS shown in Figu e 1. Addi ionally, he igu e includes a case whe e he
a om is placed on a su ace cha ac e ized by a la , s uc u eless DOS, modeled wi hin he
wide-band app oxima ion, shown as a dashed cu e.
0
0.2
0.4
0.6
0.8
N=1
LDOS(E) [1/eV]
WBL
0
0.2
0.4
0.6
0.8
−4 −3 −2 −1 0 1 2 3 4 5
B
A
C
E
D
N=2
LDOS(E) [1/eV]
ene gy [eV]
WBL
Figu e 2. Local DOS o a single a om (uppe panel,
N=
1) and o wo coupled a oms (bo om panel,
N=
2) placed on di e en 2D su aces. The co esponding su ace DOS a e shown in Figu e 1( he
colo s o he cu es and le e s A–E co espond o he su ace DOS lines). The dashed cu es ep esen
he local DOS a he a oms o he su ace DOS ob ained wi hin he wide band app oxima ion (WBL).
The o he pa ame e s a e εi=ε0=0, 12 =2, Γ=1Γ0.
Fo he la e case, he LDOS exhibi s a smoo h Lo en zian shape wi h a maximum
a
E=ε0=
0 (uppe panel). Howe e , o he 2D Xenes, he wid h o he main LDOS
peak is signi ican ly na owe , and i s in ensi y is conside ably highe compa ed o ha
ob ained om calcula ions using a la DOS su ace. This is because he a omic single-
pa icle ene gy le el lies a he sys em’s Fe mi le el, whe e he subs a e DOS o Xenes
is e y low. Consequen ly, he a omic s a e mo e closely esembles ha o an isola ed
a om, cha ac e ized by high in ensi y and na ow b oadening. Mo eo e , when he ac ual
subs a e DOS is aken in o accoun , he a omic LDOS exhibi s mul iple smalle peaks
and i egula i ies in addi ion o he main peak. These addi ional ea u es closely e lec
he s uc u e o he Xenes DOS. Fo ins ance, in he case o plumbene, p onounced peaks
appea in he subs a e DOS a app oxima ely
E=−
1.5,
+
0.6,
+
1.1,
+
1.9,
+
2.9, and
+
4.1,
wi h co esponding peaks obse ed in he a omic LDOS a simila ene gies (cu e E). I
should be no ed, howe e , ha hese addi ional LDOS peaks a e sligh ly shi ed along he
ene gy axis due o he nonze o eal pa o he sel -ene gy (in he Hilbe ans o m). As
a esul , he ac ual posi ions o he LDOS peaks o plumbene a e loca ed a
E=−
1.6,
+
0.7,
+
1.2,
+
2,
+
3, and
+
4.4, espec i ely. Ne e heless, he in ensi y o hese addi ional
peaks dec eases wi h inc easing dis ance om he main
ε0
peak, which explains why he
LDOS peaks a
E= +
3 and
+
4.4 a e only ain ly isible. No e ha he LDOS o a single
a om on g aphene—whose DOS is ela i ely smoo h nea he Fe mi ene gy—mo e closely
esembles ha p edic ed by he wide-band limi app oxima ion (black and yellow cu es).
Fo a sys em o wo coupled a oms on he su ace, he si ua ion is somewha di e en
(bo om panel o Figu e 2). In he case o a la WBL su ace, he LDOS a bo h chain si es is
iden ical and is cha ac e ized by wo dis inc peaks loca ed a ene gies E=ε0± 12 =±2,
wi h a local minimum a he Fe mi le el (black dashed line). When such an a omic dime is
placed on a subs a e wi h a ela i ely smoo h DOS, such as g aphene, he LDOS exhibi s a
s uc u e b oadly simila o ha o he WBL case, excep ha he peak a posi i e ene gies
Ma e ials 2025,18, 3843 7 o 13
becomes no iceably spli . This spli ing a ises because he g aphene subs a e has a local
DOS maximum a ound
E= +
1.8, which is close o he molecula dime peak a
E= +
2.
As a esul , bo h ea u es a e e lec ed in he LDOS cu e (yellow line). This e ec becomes
e en mo e p onounced o dime s placed on o he 2D subs a es, whe e he coexis ence
o molecula s a es and subs a e s a es a simila ene gies leads o no iceable LDOS peak
spli ing. Fo ins ance, a posi i e ene gies, plumbene exhibi s a s ong subs a e DOS
peak a
E= +
1.9. The in e ac ion be ween his subs a e s a e and he molecula dime
s a e a
E= +
2 esul s in he LDOS exhibi ing eno malized peaks a
E= +
1.5 and
+
2.5,
wi h a local minimum a
E= +
2 (pu ple line in he bo om panel). A simila LDOS
spli ing is obse ed o o he 2D subs a es, including silicene, ge manene, and s anene. A
compa able e ec also occu s a nega i e ene gies a ound he molecula peak a
E=−
2.
Mo eo e , o some subs a es, signi ican changes in peak in ensi y a e obse ed on LDOS
cu es, pa icula ly a ound
E=−
2.7 (g een and pu ple cu es o s anene and plumbene,
espec i ely). This beha io is a ibu ed o he e y low subs a e DOS in his ene gy
egion, which leads o weak dispe sion o he s a es loca ed he e. As a consequence, hese
s a es exhibi s ongly enhanced in ensi y in he LDOS.
3.2. A omic Chains on Va ious 2D Subs a es
Quan um sys ems composed o g oups o a oms a e cha ac e ized by he p esence o
molecula s a es, whose numbe usually equals he numbe o cons i uen a omic s a es.
These molecula s a es hyb idize wi h he subs a e, esul ing in each a om being desc ibed
by a local DOS ha ypically con ains as many peaks as he e a e molecula s a es, wi h
a ying in ensi ies. Simila ly, linea a ays o a oms o ming a omic chains on a su ace
exhibi analogous beha io . In his sec ion, we analyze he elec onic p ope ies o such
chains on a ious subs a es and in es iga e he possibili y o cha ge densi y wa es o ming
wi hin hese sys ems.
S ongly coupled si es. A he ou se , we conside an a omic sys em on a 2D su ace
consis ing o
N=
20 closely spaced si es, which leads o la ge o e lap in eg als o he wa e
unc ions and, consequen ly, signi ican coupling be ween neighbo ing a oms, deno ed by
. This implies ha he ene gy bandwid h o such a chain ex ends o e app oxima ely
±
2
,
he eby co e ing a subs an ial po ion o he subs a e’s DOS. Selec ed calcula ions o he
local DOS o his chain (compu ed a each a omic si e) a e p esen ed in he le panel o
Figu e 3. As be o e, a ious subs a es ha e been conside ed— om a la su ace desc ibed
wi hin he wide-band limi ( ep esen ed by he black cu es in he bo om), h ough
g aphene (yellow cu es, labeled A) o plumbene ( iole lines, labeled E). On he WBL
subs a e, he chain exhibi s 20 LDOS peaks wi h compa able dispe sion and in ensi ies
(each black cu e ep esen ing he LDOS a a di e en a omic si e). All hese cu es a e
symme ic wi h espec o he Fe mi ene gy,
EF=
0, and display spa ial symme y (i.e., he
LDOS a si es 1,2,... is iden ical o ha a si es
N
,
N−
1,..., espec i ely). In con as , o
he 2D subs a es, he LDOS unc ions exhibi dis inc cha ac e is ics. No ably, he chain
placed on g aphene shows e y high and na ow LDOS peaks nea he middle o he band,
which is associa ed wi h he e y low subs a e DOS in he icini y o he Fe mi ene gy—a
ea u e ha is also p esen , albei o a lesse ex en , in he emaining subs a es (B–E), all o
which exhibi a DOS minimum a
EF
. Mo eo e , some cu es display p onounced ene gy
asymme y, pa icula ly o he plumbene and s anene subs a es (cu es D and E). This
asymme y o igina es om he in insic asymme y o he su ace DOS in hese ma e ials,
combined wi h he no ably low DOS alues in he nega i e ene gy egion a ound
E=−
4
(see Figu e 1).
Ma e ials 2025,18, 3843 8 o 13
−0.4
0
0.4
0.8
1.2
1.6
2
−4 −2 0 2 4
A
B
C
E
D
WBL
=2, ε0=0
LDOS(E) [1/eV]
ene gy [eV]
0
2
4
6
8
−1 −0.5 0 0.5 1
=0.3, ε0=0
ene gy [eV]
−1 −0.5 0 0.5
=0.3, ε0=−0.4
ene gy [eV]
Figu e 3. Local DOS a each a omic si e o a chain o leng h
N=
20 loca ed on a ious 2D su aces
as desc ibed by he DOS shown in Figu e 1( he colo s o he cu es and he le e s A–E co espond
o he su ace DOS lines). The bo om black cu es ep esen he local DOS a he a oms o he
su ace’s WBL DOS. The le panel co esponds o s ongly coupled a omic si es,
=
2, and
ε0=
0,
Γ=
0.25. The middle and he igh panels show he esul s o weakly coupled si es,
=
0.3, and
o
ε0=
0 and
ε0=−
0.4, espec i ely, and o
Γ=
0.1. All cu es (besides A lines) a e shi ed o
be e isualiza ion.
Weakly coupled si es. In he case o chains wi h weakly coupled a oms (i.e., wi h a
small in e -si e hopping in eg al), he bandwid h o he chain’s DOS is na ow and may
encompass only a small po ion o he subs a e’s DOS. The egion a ound he Fe mi ene gy
is pa icula ly in e es ing, as he subs a e DOS exhibi s a linea dispe sion he e as shown
in Figu e 1. The LDOS unc ions o such weakly coupled chain a di e en subs a es
a e shown in Figu e 3, middle and igh panels, o e e y chain si e. No e ha when he
single-pa icle ene gy le els o he chain a e p ecisely aligned wi h he Fe mi le el (
ε0=
0,
middle panel), all LDOS cu es a e symme ic wi h espec o he Fe mi ene gy since he
subs a e DOS unc ions a e also symme ic in ha egion (compa e wi h he cu es in
he igh panel o Figu e 1). I is also wo h no ing he a ia ion in he in ensi y o he
LDOS peaks, om he highes o he g aphene subs a e (yellow lines) o he lowes o
plumbene ( iole lines). This e ec is closely ela ed o he subs a e DOS alues nea he
Fe mi ene gy— he lowe he DOS alue, he mo e he elec onic s a es esemble isola ed
a omic s a es (i.e., wi h na ow dispe sion and high in ensi y). Fu he mo e, o subs a es
wi h a e y low DOS a ound he Fe mi le el, he molecula s a es isible in he chain LDOS
a e well sepa a ed (small dispe sion), whe eas, o example, on a WBL subs a e (black
lines) o on plumbene ( iole lines), he LDOS cu es display only small local peaks, and
due o he o e lapping o hese s a es, no all a e clea ly e ealed.
In he igh panel o Figu e 3, we also analyze he case whe e he single-pa icle
ene gies o he chain a e sligh ly shi ed ela i e o he sys em’s Fe mi ene gy, se ing
εi=ε0=−
0.4. In his scena io, he LDOS cu es o he chain on a su ace wi h a
s uc u eless DOS (modeled wi hin he WBL) a e iden ical o hose shown in he middle
panel, excep o a uni o m shi along he ene gy axis by
−
0.4. In con as , signi ican
Ma e ials 2025,18, 3843 9 o 13
di e ences a e obse ed o he 2D Xene subs a es. No ably, he LDOS cu es o each
Xene exhibi a p onounced asymme y— he high-in ensi y peaks on he igh -hand side o
he plo s esul om he low subs a e DOS in ha ene gy egion (nea he Fe mi ene gy)
as discussed ea lie . Con e sely, in he ene gy ange a ound
E≃ −
1, he subs a e DOS is
conside ably highe , which leads o a b oadening and pa ial o e lap o he LDOS peaks o
he chain. As a esul , he LDOS in ha egion ( he le side o he plo s) appea s ela i ely
smoo h. The mos p onounced di e ences in he LDOS cu es occu be ween he g aphene
subs a e (yellow lines) and he plumbene subs a e ( iole lines), e lec ing he espec i e
DOS p o iles o hese subs a es in he conside ed ene gy ange.
Cha ge wa es in a omic chains. Knowledge o he local DOS enables he calcula ion
o cha ge occupancies (Equa ion
(2)
) and analysis o cha ge densi y wa e dis ibu ions
along he chain. Figu e 4p esen s hese CDWs along a chain o
N=
30 si es o exempla y
silicene, g aphene, and la DOS subs a es, shown in he le , middle, and igh panels,
espec i ely. The uppe cha ge cu es (black lines a he op o he panels) co espond o
he single-pa icle ene gy le el in he chain se o
ε0=−
4, while he subsequen cu es
ep esen inc emen al inc eases in ene gy, culmina ing a
ε0= +
4 o he lowes black
cu es. Analyzing hese plo s e eals a dis inc pa e n o med by he indi idual cha ge
cu es, cha ac e ized by speci ic concen a ions (seen as da ke a eas) ha indica e he
eme gence o cha ge wa es in such sys ems. Fo ins ance, he ed cu es (highligh ed o
ε0=
3.24) display cha ac e is ic F iedel oscilla ions, wi h maximal ampli ude a he chain
edges ha g adually decays owa d he cen e . The cha ge dis ibu ion exhibi s p onounced
le – igh symme y wi h espec o he chain cen e , as well as pa icle–hole-like symme y
ela i e o
ni=
0.5. Consequen ly, cha ge oscilla ions wi h iden ical pe iodici y eme ge o
bo h posi i e and nega i e ε0.
0
0.2
0.4
0.6
0.8
1 silicene
ni
g aphene WBL
0.1
0.2
0.3
0.4
0 5 10 15 20 25 30
ni
i
0 5 10 15 20 25 30
i (N=30)
0 5 10 15 20 25 30
M=3
M=4
M=5
M=6
i
Figu e 4. Cha ge occupancies a e e y chain si e
i
o he chain leng h
N=
30, o a ious 2D su aces:
silicene, g aphene, and la WBL ( om le o igh panels). The uppe panels show
ni
cu es o
di e en chain on-si e ene gies om
ε0=−
4 (uppe cu es) up o
ε0=
4 (bo om cu es) and he
ed-colo cu es ep esen he case o
ε0=
3.24. The bo om panels show occupancy cu es o
special alues o
ε0
, which co espond o he speci ic oscilla ion pe iod
M=
3,
εi= =
2,
M=
4,
ε0=√2 =
2.83,
M=
5,
ε0= (√5+
1
)/
2
=
3.24 and
M=
6,
ε0=√3 =
3.46, as is indica ed in he
igh bo om panel. The o he pa ame e s a e =2, Γ=0.25Γ0.
Cha ge wa es a e cha ac e is ic o a egula chain on he WBL su ace, o which
we ha e a clea unde s anding o when such wa es occu and hei oscilla ion pe-
