Anha monici y Re eals he Tunabili y o he Cha ge Densi y Wa e
O de s in Monolaye VSe2
Adol o O e o Fumega,*Josu Diego, Vic o Pa do, San iago Blanco-Canosa, and Ion E ea*
Ci e This: Nano Le . 2023, 23, 1794−1800
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sı Suppo ing In o ma ion
ABSTRACT: VSe2is a laye ed compound ha has a ac ed g ea
a en ion due o i s p oximi y o a e omagne ic s a e ha is
quenched by i s cha ge densi y wa e (CDW) phase. In he
monolaye limi , un ela ed expe imen s ha e epo ed di e en
CDW o de s wi h di e en ansi ion empe a u es, making his
monolaye e y con o e sial. He e we pe o m i s -p inciples
nonpe u ba i e anha monic phonon calcula ions in monolaye
VSe2in o de o es ima e he CDW o de and he co esponding
ansi ion empe a u e. They e eal ha monolaye VSe2de elops
wo independen cha ge densi y wa e o de s ha compe e as a
unc ion o s ain. Va ia ions o only 1.5% in he la ice pa ame e
a e enough o s abilize one o de o he o he . Mo eo e , we
analyze he impac o ex e nal Lenna d-Jones in e ac ions, showing
ha hese can ac oge he wi h anha monici y o supp ess he CDW o de s. Ou esul s sol e p e ious expe imen al con adic ions,
highligh ing he high unabili y and subs a e dependency o he CDW o de s o monolaye VSe2.
KEYWORDS: anha monic e ec s, cha ge densi y wa e, compe ing o de s, 2D ma e ials, an de Waals in e ac ions, s ain
Two-dimensional (2D) ma e ials a e an ideal pla o m o
a i icially enginee he e os uc u es wi h new unc ion-
ali ies due o he weak an de Waals bonding be ween laye s.
1
Monolaye s hos ing symme y-b oken phases, such as supe -
conduc i i y,
2,3
magne ism,
4−8
e oelec ici y,
9,10
cha ge den-
si y wa es (CDWs),
11,12
o mul i e oici y,
13,14
ep esen he
mos in e es ing building blocks o design no el phases o
ma e . One o he main challenges in he ask o enginee ing
no el unc ional ma e ials wi h b oken-symme y monolaye s
is o o e come he es ic ions imposed by he educed
dimensionali y,
15,16
which may p e en he o ma ion o hese
phases, and he compe i ion be ween o de ed phases due o
he sub le in e play o di e en in e ac ions.
17,18
Fo ins ance,
CDW phases ha e been epo ed o des oy
19,20
o p omo e
21
2D e omagne ism. VSe2is a pa adigma ic example o his as,
despi e some ea ly claims,
22
i is now clea bo h expe imen ally
and heo e ically ha he CDW o de quenches he eme gence
o i ine an e omagne ism.
19,20,23−28
In i s bulk o m, VSe2
de elops a commensu a e 4 ×4×3 CDW phase below 110
K.
29
The CDW phase opens pseudogaps a he Fe mi le el
impeding he eme gence o e omagne ism.
19
Inelas ic X- ay
sca e ing expe imen s and nonpe u ba i e anha monic
phonon calcula ions ha e p o en ha he CDW ansi ion is
d i en by he collapse o a low-ene gy acous ic mode and ha
he elec on−phonon coupling is he o igin o he ins abili y,
30
as sugges ed as well by o he quan i a i e models.
31
These
anha monic calcula ions ha e shown ha an de Waals
in e ac ions a e essen ial o mel he CDW and ob ain a cha ge
densi y wa e empe a u e (TCDW) in ag eemen wi h expe i-
men s. This sugges s ha he CDW in he monolaye may also
be cha ac e ized by simila phonon so ening e ec s bu wi h
limi ed in luence o an de Waals in e ac ions.
The main p oblem in he monolaye o VSe2is ha he
CDW is no ully unde s ood ye , as un ela ed expe imen s
ha e epo ed dis inc CDW o de s wi h di e en ansi ion
empe a u es. A nonmono onic e olu ion o TCDW as a
unc ion o he numbe o laye s has been epo ed in e s
32−34 bu e aining an in-plane 4 ×4 modula ion. A
me as able phase wi h modula ion has also been
iden i ied o he ew-laye case.
35
In he pu ely 2D limi
di e en CDW o de s wi h nonequi alen modula ions ha e
been ound. A 4 ×4 o de was obse ed in VSe2 ilms g own
on bilaye g aphene on op o SiC and on highly o ien ed
py oly ic g aphi e (HOPG) wi h a TCDW o K and a
la ice pa ame e o a= 3.31 ±0.05 Å.
27
On he con a y, a
modula ion has been obse ed in VSe2samples
g own on se e al subs a es by molecula beam epi axy by
di e en g oups, wi h a consis en TCDW = 220 K.
20,23
Some
Recei ed: No embe 21, 2022
Re ised: Feb ua y 22, 2023
Published: Feb ua y 24, 2023
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o he o de s ha e also been epo ed: a combina ion o
and wi h a TCDW ∼135 K
25,26
and a 4 ×1
modula ion wi h TCDW ∼350 K.
26,36
These expe imen al
con adic ions poin o he p esence o di e en compe ing
CDW o de s, which can lead o di e en low- empe a u e
phases depending on he subs a e.
26,36
By calcula ing he ha monic phonons o he VSe2monolaye
wi hin densi y unc ional heo y (DFT), heo e ical s udies
ha e also desc ibed he compe i ion o di e en CDW o de s
and how s ain can in luence he g ound s a e.
37
Ha monic
phonon calcula ions, howe e , canno explain ha abo e TCDW
he 1T phase is he g ound s a e. In he p esence o compe ing
o de s, only calcula ions conside ing anha monici y can
disen angle wha is he CDW o de and he ansi ion
empe a u e, as i has al eady been shown in di e en
ansi ion me al dichalcogenides (TMDs).
30,38−41
The e o e,
in o de o un eil he in insic CDW o de s o monolaye VSe2
and how hey a e a ec ed by ex e nal ields, a DFT s udy
including anha monici y is equi ed.
In his wo k, we p esen a heo e ical analysis o he CDW
o de s a ising in monolaye VSe2using nonpe u ba i e
anha monic phonon calcula ions based on he s ochas ic sel -
consis en ha monic app oxima ion (SSCHA).
42−45
This
o malism has been c ucial o unde s and and cha ac e ize
he CDWs in se e al TMDs
30,38−41
as i o e comes he
limi a ions o he ha monic analysis, allowing o de e mine he
dependence o he CDW o de as a unc ion o empe a u e.
We demons a e he eme gence and compe i ion o wo
in insic CDW o de s in monolaye VSe2as o a la ice
pa ame e o a= 3.35 Å a o de domina es wi h
TCDW = 217 K, while o sligh ly smalle a= 3.30 Å he 4 ×4
o de p e ails wi h TCDW = 223 K. Mo eo e , he non-
pe u ba i e anha monic p ocedu e allows us o demons a e
ha he CDW can be supp essed by he inclusion o Lenna d-
Jones ene gy e ms, which migh appea na u ally o may be
a i icially induced by he in e play be ween he monolaye and
a pa icula subs a e.
We s a ou analysis pe o ming ha monic phonon
calcula ions on monolaye VSe2. The no mal s a e (NS) uni
cell is shown in Figu e 1a. The alue o he expe imen ally
epo ed la ice pa ame e a= 3.31 ±0.05 Å is in a he good
ag eemen wi h he heo e ical one o 3.35 Å ob ained a he
Pe dew−Bu ke−E nze ho
46
le el wi hou conside ing he
ze o-poin mo ion.
23,27
The e o e, in his s udy we pe o m
calcula ions o wo la ice pa ame e s a= 3.35 Å and a= 3.30
Å, which p o ide a good ep esen a ion o he expe imen al
ange. Densi y unc ional pe u ba ion heo y (DFPT)
47
is
used o compu e he ha monic phonon band s uc u e o bo h
la ice pa ame e s (see he Suppo ing In o ma ion o a
de ailed desc ip ion o he calcula ions). Bo h ha monic
phonon bands (see Figu e 1b) show wo dominan ins abili ies
a and . These a e associa ed wi h
he wo in insic CDW o de s o monolaye VSe2, wi h
modula ions shown in Figu e 1c,d, ha lowe i s Bo n−
Oppenheime ene gy. The ins abili y a q1is associa ed wi h a
supe cell, while he one a q2leads o a 4 ×4
modula ion. Bo h so ened phonon modes ha e an ou -o -
plane componen in he displacemen ec o s, as i is also he
case o he CDW ins abili y in he bulk o m o his compound.
In spi e o p o iding he wo in insic CDW o de s, ha monic
calcula ions do no su ice o p edic which o hese CDW
o de s is he dominan one o he associa ed ansi ion
empe a u e o each la ice pa ame e . In ac , he small
change in he la ice pa ame e does no impac he weigh o
he ins abili ies. Ou nonpe u ba i e anha monic calcula ions
based on a ee ene gy o malism wi hin he SSCHA can gi e
he answe o hese ques ions (see he Suppo ing In o ma ion
o a de ailed desc ip ion o he SSCHA me hod and he
echnical aspec s o hese calcula ions).
Figu e 2 shows he empe a u e e olu ion o he phonon
band s uc u e ob ained wi h he SSCHA me hod o he wo
la ice pa ame e s a= 3.35 Å (in ed in he op panels o Figu e
2) and a= 3.30 Å (in blue in he lowe panels o Figu e 2). A
high enough empe a u e (T= 250 K) he 1T NS phase
(Figu e 1a) is dynamically s able o bo h la ice pa ame e s as
shown in Figu es 2a,d, showing ha anha monici y mel s he
CDW phase as i happens in o he TMDs.
30,38−41
By
dec easing he empe a u e, he phonon modes associa ed
Figu e 1. (a) No mal-s a e s uc u e o monolaye VSe2wi h la ice
pa ame e a. V (Se) a oms a e depic ed in g ay (g een). (b)
Ha monic phonon band s uc u es o monolaye VSe2as a unc ion o
he la ice pa ame e , le ( igh ) panel o a= 3.35 Å (a= 3.30 Å).
Two dominan ins abili ies a and can be
iden i ied. (c,d) In insic CDW o de s wi h and 4 ×4
modula ions associa ed wi h he ins abili ies a q1and q2can be
iden i ied in he ha monic phonon band s uc u es. The displacemen
ec o s associa ed wi h each CDW o de a e plo ed as b own a ows.
Planes pe pendicula o he z-di ec ion o V (Se) we e plo ed in g ay
(g een) o a be e cha ac e iza ion o he displacemen ec o s.
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wi h he CDW ins abili ies a q1and q2so en. In pa icula , o
a= 3.35 Å a 200 K (Figu e 2b) we can obse e ha he mode
a becomes uns able, e en i he one a q2 emains
s able. This esul indica es ha o his la ice pa ame e he
CDW o de domina es. Howe e , o a= 3.30 Å a
200 K (Figu e 2e) he phonon mode a is
uns able, bu no a q1. The e o e, o he smalle la ice
pa ame e , he 4 ×4 CDW o de is he dominan one. A low
enough empe a u es (Figu es 2c, ) bo h q- ec o s show
uns able modes. Howe e , no e ha his si ua ion is no
indica ing ha a low empe a u es bo h CDW o de s coexis ,
al hough i is a clea signa u e ha he anha monic ee ene gy
landscape becomes mo e complex. Once one o he CDW
o de s becomes s able when he empe a u e is dec eased, he
sys em collapses o i , and he analysis in e ms o he
anha monic phonons o he NS phase is no longe use ul o
desc ibe he e olu ion o each o he CDW phases a low
empe a u es. Ne e heless, he anha monic phonons a low
empe a u e shown in Figu e 2c, con i m ha bo h q1and q2
a e he in insic CDW o de s o VSe2 ha can be accessed
h ough a ansi ion om he NS phase.
To analyze in mo e de ail he compe i ion be ween he wo
CDW o de s as a unc ion o he la ice pa ame e , Figu e 3a
shows he empe a u e e olu ion o he equency o he
phonon mode ha so ens a q1and q2. Fo he la ge la ice
pa ame e , a= 3.35 Å, he equency a q1becomes nega i e
(imagina y) a highe empe a u e han a q2, and hence he
CDW o de is he dominan (le panel in Figu e 3a
in ed). The opposi e beha io is obse ed o he small la ice
pa ame e a= 3.30 Å. The equency a q2becomes nega i e a
highe empe a u e han ha a q1, and hence he 4 ×4 CDW
o de is dominan ( igh panel in Figu e 3a in blue). F om
Figu e 3a we can ob ain he ansi ion empe a u e o each
la ice pa ame e : o a= 3.35 Å he o de eme ges
a TCDW = 217 K, while o a= 3.30 Å he 4 ×4 o de a ises a
TCDW = 223 K. Impo an ly, ou anha monic calcula ions
including he ze o poin ene gy
44
p edic an associa ed in-plane
p essu e o 0.7 GPa o a= 3.35 Å and 1.3 GPa o a= 3.30 Å.
Conside ing ha i s in-plane p essu e is lowe , hese esul s
poin ou ha he in insic CDW o de in monolaye VSe2is
wi h a TCDW = 217 K, which is in pe ec ag eemen
wi h he expe imen s on e s 20 and 23 and ha he 4 ×4
o de , which is he in-plane p ojec ion o he bulk 4 ×4×3
CDW o de , appea s only unde s ain. Ou esul s p o ide an
explana ion o he di e en CDW o de s obse ed o small
a ia ions (∼1.5%) o he la ice pa ame e .
23,27
No e ha ,
e en ually, o he modula ions could appea in monolaye VSe2
as expe imen ally epo ed.
25,26,36
Howe e , ou esul s show
ha he and 4 ×4 modula ions a e he in insic
CDW o de s and poin ou ha hose di e en modula ions
a e a consequence o he in e play be ween he highly
dynamically uns able NS o VSe2monolaye and he pa icula
subs a e.
Ha ing es ablished he compe i ion be ween he wo
in insic CDW o de s o monolaye VSe2as a unc ion o
he la ice pa ame e , we s udy now he o igin o hese CDW
o de s. In o de o do so, we use DFPT o compu e bo h he
nes ing unc ion η(q) (Figu e 3b), which is gi en by
(1)
and he phonon line wid h associa ed wi h he elec on−
phonon in e ac ion (see Figu e 3c).
(2)
In eqs 1 and 2ϵnkis he ene gy o band nwi h wa enumbe
k,ϵF he Fe mi ene gy, and Nis he numbe o kpoin s in he
sum o e he i s B illouin zone (1BZ). The nes ing unc ion
Figu e 2. Nonpe u ba i e anha monic calcula ions o he NS o monolaye VSe2. (a−c) Fo la ice pa ame e a= 3.35 Å and empe a u es o 250,
200, 100 K espec i ely. (d− ) Fo la ice pa ame e a= 3.30 Å and empe a u es 250, 200, 150 K. The phonon mel ing occu s i s a he q1(q2)
poin o a= 3.35 Å (a= 3.30 Å) as shown in panels (b) and (e).
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peaks o qindica es ha nes ed egions o he Fe mi su ace
connec and, hus, e eal i he ins abili y eme ges om a
pu ely elec onic ins abili y. The equa ion o γμ(q) is e y
simila o he nes ing unc ion, bu he alue is weigh ed by he
mode μand momen um qdependen elec on−phonon ma ix
elemen s and, hus, e eals i he ins abili y eme ges
om elec on−phonon in e ac ions. The elec on−phonon
line wid h is independen o he phonon equency wμ(q) as
he elec on−phonon ma ix elemen s scale as wμ(q)−1/2. I is
wo h no ing ha bo h quan i ies need o be compu ed in
o de o es ablish he o igin o he CDW o de s. Fo 2D
sys ems like monolaye VSe2, despi e he exis ence o nes ing
condi ions a he qpoin s associa ed wi h he CDW o de s, a
pu ely elec onic pic u e does no su ice o p oduce a CDW
o de and elec on−phonon in e ac ions play a key ole o
d i e he ansi ion.
48
The e o e, he di ec compa ison
be ween he nes ing unc ion and he elec on−phonon line
wid h allows us o es ablish which is he main d i ing o ce o
he CDW o de s in monolaye VSe2.
We can see in Figu e 3b ha , o bo h la ice pa ame e s he
nes ing unc ion does no show any s ong peak a he CDW
ec o s despi e he exis ence o small shoulde s nea q1and q2.
In a di e en way, Figu e 3c shows ha he phonon line wid h
coming om he elec on−phonon in e ac ion ab up ly peaks
a bo h q1and q2 o bo h la ice pa ame e s, meaning ha in
all cases he enhancemen comes om he mode and
momen um dependence o he elec on−phonon ma ix
elemen s. The e o e, he wo in insic CDW o de s de eloped
by monoloye VSe2a e d i en by he elec on−phonon
coupling, in ag eemen wi h he heo e ical p edic ions o
2D sys ems.
48
Besides, he elec on−phonon in e ac ion also
plays a key ole in he CDW ansi ion in 3D sys ems, as in
bulk 1T-VSe2, in which case he p esence o nes ing is
symbolic.
30
The analysis abou he s abili y o he di e en CDW o de s
as a unc ion o s ain was pe o med wi h a nonlocal an de
Waals densi y exchange-co ela ion unc ional.
49
The eason
o his is ha his unc ional allows us o p ope ly desc ibe
bo h bulk and monolaye limi s o VSe2, opposi ely o he
widely used GGA-PBE unc ional (see he Suppo ing
In o ma ion o a mo e de ailed desc ip ion o he elec ion
o he exchange co ela ion unc ional o s udy he CDW
o de s o VSe2). In he la e case, a huge o e es ima ion o
TCDW occu s in bulk due o he lack o an de Waals
in e ac ions ha cause a mel ing o he CDW phase.
30
Mo i a ed by hese esul s in he bulk, we explo e he e he
e ec ha ex e nal an de Waals in e ac ions may cause in he
CDW o de s o monolaye VSe2. These an de Waals
in e ac ions migh na u ally appea by p oximi y e ec be ween
he analyzed monolaye and o he laye s, such as he subs a e
o in an de Waals he e os uc u es. We will in oduce hese
ex e nal in e ac ions in a phenomenological way. This
app oach allows us o de i e gene al conclusions o he pu e
e ec o an de Waals in e ac ions on CDW o de s, i.e.,
ac o ing ou e ec s ha could appea in pa icula subs a es,
such as cha ge ans e .
40
Fi s , o illus a e he e ec o ex e nal an de Waals o ces
on he CDW, we make use o a simple one-dimensional
double-well ou h o de po en ial
(3)
whe e Aand Ba e he coe icien s o he di e en powe s, is
he posi ion o he a oms, and 0is he equilib ium a omic
posi ion in he high empe a u e phase. The CDW occu s
when he ee ene gy calcula ed wi h his po en ial is lowe a
− 0≠0 han a − 0= 0. Ob iously he lowe and wide he
minimum o he well he mo e p obable i is o ind a b oken-
symme y CDW o de . We can now add o he po en ial V( ) a
ELJ( ) Lenna d-Jones ene ge ic con ibu ion o mimic he ole
played by ex e nal an de Waals in e ac ions
(4)
whe e Cis he coe icien ha con ols he s eng h o he
Lenna d-Jones in e ac ions, cis a cu o adius ha p e en s a
di e gence a = 0, and D=C/ cis simply a cons an ha ixes
he po en ial o be equal o 0 a 0. The e ec o he Lenna d-
Jones in e ac ions on V( ) can be seen in Figu e 4a o
di e en alues o C. We can obse e ha an inc ease in he
s eng h o he Lenna d-Jones in e ac ions makes he po en ial
shallowe . The e o e, Lenna d-Jones in e ac ions end o
quench he low- empe a u e CDW phase and p omo e he
high- empe a u e symme ic phase.
Figu e 3. (a) Tempe a u e e olu ion o he equencies o he
so ened modes a q1and q2as ob ained in he SSCHA calcula ions.
(b) Nes ing unc ion η(q). I does no clea ly peak a q1and q2. (c)
Phonon line wid h γμ(q) gi en by he elec on−phonon in e ac ion.
Sizable peaks appea a he CDW q- ec o s. All o he esul s o a=
3.35 Å (a= 3.30 Å) a e shown in ed (blue) in he le ( igh ) panels.
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We can con i m his simple pic u e in he pa icula case o
monolaye VSe2by including an ene gy e m like he one
shown in eq 4 h ough he G imme’s semiempi ical app oach
in ou SSCHA calcula ions on op o he PBE unc ional.
50
Figu e 4c shows he ha monic phonon band s uc u e
including ene gy con ibu ions om Lenna d-Jones in e ac-
ions. (These calcula ions a e o a= 3.35 Å, bu quali a i ely
he esul s hold o any o he la ice pa ame e . The s eng h o
he Lenna d-Jones in e ac ions was se o he one conside ed
by de aul by he G imme’s semiempi ical co ec ion
implemen ed in he QUANTUM ESPRESSO package
51,52
)
The ins abili y a q1and q2is educed compa ed o he
ha monic bands wi hou he Lenna d-Jones con ibu ion (see
he Suppo ing In o ma ion o a be e isualiza ion o his
e ec a he ha monic le el.). In his si ua ion, anha monic
e ec s a e able o supp ess bo h CDW o de s by s abilizing he
so ened phonons e en a 0 K, as shown in Figu e 4(c). This
e ec can be also analyzed in Figu e 4(b), whe e he e olu ion
o he equencies o he so ened modes a q1and q2as a
unc ion o empe a u e is shown. The equencies emain
s able a any empe a u e. The e o e, his demons a es ha
he combina ion o Lenna d-Jones in e ac ions and anha mo-
nici y can des oy he CDW o de s, s abilizing he NS phase a
low empe a u es. No e ha he e we ha e conside ed a
s eng h o he Lenna d-Jones in e ac ions ha quenches bo h
CDW o de s. Howe e , his s eng h could be modula ed by
he pa ame e Cin eq 4, p o iding simply a dec ease o TCDW,
bu no a supp ession o he CDW o de , as epo ed o
bulk.
30
This e ec may also be ela ed wi h he enhancemen o
he CDW o de in he 2D limi ,
34
in which his kind o
in e ac ions dec eases. The e o e, no only s ain bu also
Lenna d-Jones in e ac ions can explain he huge a iabili y o
ansi ion empe a u es and CDW o de s epo ed in expe i-
men s whe e VSe2is g own on di e en subs a es.
23,25−27
Finally, no e ha monolaye VSe2has a ac ed g ea
a en ion due o i s p oximi y o an i ine an e omagne ic
s a e, which is supp essed by he p esence o CDW o de s.
19,20
Ou analysis sugges s ha a e omagne ic s a e in monolaye
VSe2may be possible i i s in insic CDW o de s a e
supp essed by ex e nal Lenna d-Jones in e ac ions. This uning
could be implemen ed by subs a e enginee ing o by a i icial
design o an de Waals he e os uc u es. In pa icula ,
combining compounds ha display CDW o de s wi h
e oelec ic ma e ials, which p o ide s ong dipola in e -
ac ions, migh allow an elec ic con ol o CDW phases and he
eme gence o o he compe ing o de s. This mechanism o une
o des oy CDW o de s is gene ic and could be ex ended o
o he simila sys ems, o e ing a no el pla o m o enginee ing
new unc ional ma e ials.
In conclusion, o sol e p e ious expe imen al con adic ions
ound o he CDW o de s o monolaye VSe2, in his wo k we
analyze he CDW o de s o monolaye VSe2using non-
pe u ba i e anha monic phonon calcula ions ha allow us o
de e mine he CDW o de s o his sys em and hei
co esponding ansi ion empe a u es. We analyze he e ec
o s ain and ex e nal an de Waals in e ac ions, since hese
wo pa ame e s a e in insic o any expe imen . We
demons a e he compe i ion be ween wo in insic CDW
o de s as a unc ion o he la ice pa ame e . Va ia ions o 1.5%
in he la ice pa ame e a e enough o d i e he sys em om
he o he 4 ×4 o de . T ansi ion empe a u es on
he o de o 220 K ha e been ound o bo h CDW o de s, in
e y good ag eemen wi h expe imen s. We show ha ex e nal
Lenna d-Jones in e ac ions end o weaken o e en supp ess
he CDW o de s. These esul s oge he help o unde s and
he g ea a iabili y o CDW o de s and associa ed TCDW’s
ound in he expe imen s o monolaye VSe2. Mo eo e , hey
pa e he way o une CDW o de s occu ing in an de Waals
ma e ials, hus p omo ing compe ing o de s ha migh a ise in
hese sys ems.
■ASSOCIATED CONTENT
*
sı Suppo ing In o ma ion
De ailed explana ion o he compu a ional me hods, he
elec ion o he exchange co ela ion unc ional and he e ec
o ex e nal Lenna d-Jones in e ac ions a he ha monic le el.
The Suppo ing In o ma ion is a ailable ee o cha ge a
h ps://pubs.acs.o g/doi/10.1021/acs.nanole .2c04584.
De ailed explana ion o he compu a ional me hods, he
elec ion o he exchange co ela ion unc ional and he
e ec o ex e nal Lenna d-Jones in e ac ions a he
ha monic le el (PDF)
■AUTHOR INFORMATION
Co esponding Au ho s
Adol o O e o Fumega −Depa men o Applied Physics,
Aal o Uni e si y, 02150 Espoo, Finland; o cid.o g/0000-
0002-3385-6409; Email: [email p o ec ed]
Ion E ea −Fisika Aplika ua Saila, Gipuzkoako Ingenia i za
Eskola, Uni e si y o he Basque Coun y (UPV/EHU),
20018 San Sebas ián, Spain; Cen o de Física de Ma e iales
(CSIC-UPV/EHU), 20018 San Sebas ián, Spain; Donos ia
Figu e 4. (a) V( ) po en ial as a unc ion o he a omic posi ion
desc ibed by eqs 3 and 4in a bi a y uni s (a.u.) o di e en C alues.
Pa icula alues o A=−8, B=−A/2, and c= 0.1 a e conside ed.
(b) Tempe a u e e olu ion o he so en modes’ equencies a q1and
q2when anha monici y and Lenna d-Jones in e ac ions a e included.
(c) Ha monic and anha monic phonon band s uc u es a T= 0 K o
monolaye VSe2including Lenna d-Jones con ibu ions.
Nano Le e s pubs.acs.o g/NanoLe Le e
h ps://doi.o g/10.1021/acs.nanole .2c04584
Nano Le . 2023, 23, 1794−1800
1798
In e na ional Physics Cen e (DIPC), 20018 San Sebas ián,
Spain; Email: [email p o ec ed]
Au ho s
Josu Diego −Fisika Aplika ua Saila, Gipuzkoako Ingenia i za
Eskola, Uni e si y o he Basque Coun y (UPV/EHU),
20018 San Sebas ián, Spain; Cen o de Física de Ma e iales
(CSIC-UPV/EHU), 20018 San Sebas ián, Spain;
o cid.o g/0000-0002-8659-2144
Vic o Pa do −Depa amen o de Física Aplicada,
Uni e sidade de San iago de Compos ela, 15782 San iago de
Compos ela, Spain; Ins i u o de Ma e iais iMATUS,
Uni e sidade de San iago de Compos ela, 15782 San iago de
Compos ela, Spain; o cid.o g/0000-0002-4713-3519
San iago Blanco-Canosa −Donos ia In e na ional Physics
Cen e (DIPC), 20018 San Sebas ián, Spain;
IKERBASQUE, Basque Founda ion o Science, 48013
Bilbao, Spain
Comple e con ac in o ma ion is a ailable a :
h ps://pubs.acs.o g/10.1021/acs.nanole .2c04584
No es
The au ho s decla e no compe ing inancial in e es .
■ACKNOWLEDGMENTS
We acknowledge he compu a ional esou ces p o ided by he
CESGA and he Aal o Science-IT p ojec . A.O.F. acknowl-
edges he inancial suppo ecei ed h ough he Academy o
Finland P ojec No. 349696. J.D. hanks he Depa men o
Educa ion o he Basque Go e nmen o a p edoc o al
ellowship (G an No. PRE-2020-1-0220). We hank he
Minis y o Science and Educa ion o Spain o inancial
suppo h ough he p ojec s PGC2018-101334-A-C22,
GC2018-101334-B-C21, PID2021-122609NB-C22. I.E. ac-
knowledges unding om he Depa men o Educa ion,
Uni e si ies and Resea ch o he Eusko Jau la i za, and he
Uni e si y o he Basque Coun y UPV/EHU (G an No.
IT1527-22).
■REFERENCES
(1) Geim, A. K.; G igo ie a, I. V. Van de Waals he e os uc u es.
Na u e 2013,499, 419−425.
(2) Ugeda, M. M.; B adley, A. J.; Zhang, Y.; Onishi, S.; Chen, Y.;
Ruan, W.; Ojeda-A is izabal, C.; Ryu, H.; Edmonds, M. T.; Tsai, H.-
Z.; e al. Cha ac e iza ion o collec i e g ound s a es in single-laye
NbSe2. Na . Phys. 2016,12, 92−97.
(3) Vano, V.; Ganguli, S. C.; Amini, M.; Yan, L.; Khos a ian, M.;
Chen, G.; Kezilebieke, S.; Lado, J. L.; Lilje o h, P. E idence o nodal -
wa e supe conduc i i y in monolaye 1H-TaS2wi h hidden o de
luc ua ions. a Xi e-p in s 2021,DOI: 10.48550/a Xi .2112.07316.
(4) Lee, J.-U.; Lee, S.; Ryoo, J. H.; Kang, S.; Kim, T. Y.; Kim, P.;
Pa k, C.-H.; Pa k, J.-G.; Cheong, H. Ising-Type Magne ic O de ing in
A omically Thin FePS3. Nano Le . 2016,16, 7433−7438.
(5) Huang, B.; Cla k, G.; Na a o-Mo a alla, E.; Klein, D. R.; Cheng,
R.; Seyle , K. L.; Zhong, D.; Schmidgall, E.; McGui e, M. A.; Cobden,
D. H.; Yao, W.; Xiao, D.; Ja illo-He e o, P.; Xu, X. Laye -dependen
e omagne ism in a an de Waals c ys al down o he monolaye
limi . Na u e 2017,546, 270−273.
(6) Gong, C.; Li, L.; Li, Z.; Ji, H.; S e n, A.; Xia, Y.; Cao, T.; Bao,
W.; Wang, C.; Wang, Y.; Qiu, Z. Q.; Ca a, R. J.; Louie, S. G.; Xia, J.;
Zhang, X. Disco e y o in insic e omagne ism in wo-dimensional
an de Waals c ys als. Na u e 2017,546, 265−269.
(7) Fei, Z.; Huang, B.; Malinowski, P.; Wang, W.; Song, T.; Sanchez,
J.; Yao, W.; Xiao, D.; Zhu, X.; May, A. F.; Wu, W.; Cobden, D. H.;
Chu, J.-H.; Xu, X. Two-dimensional i ine an e omagne ism in
a omically hin Fe3GeTe2. Na . Ma e . 2018,17, 778−782.
(8) Zhang, Z.; Shang, J.; Jiang, C.; Rasmi a, A.; Gao, W.; Yu, T.
Di ec Pho oluminescence P obing o Fe omagne ism in Monolaye
Two-Dimensional C B 3. Nano Le . 2019,19, 3138−3142.
(9) Cui, C.; e al. In e co ela ed In-Plane and Ou -o -Plane
Fe oelec ici y in Ul a hin Two-Dimensional Laye ed Semiconduc-
o In2Se3. Nano Le . 2018,18, 1253−1258.
(10) Yuan, S.; Luo, X.; Chan, H. L.; Xiao, C.; Dai, Y.; Xie, M.; Hao,
J. Room- empe a u e e oelec ici y in MoTe2 down o he a omic
monolaye limi . Na . Commun. 2019,10, 1775.
(11) Wang, Y.; Ren, J.; Li, J.; Wang, Y.; Peng, H.; Yu, P.; Duan, W.;
Zhou, S. E idence o cha ge densi y wa e wi h aniso opic gap in a
monolaye VTe2 ilm. Phys. Re . B 2019,100, 241404.
(12) Chen, P.; Pai, W. W.; Chan, Y.-H.; Takayama, A.; Xu, C.-Z.;
Ka n, A.; Hasegawa, S.; Chou, M. Y.; Mo, S.-K.; Fedo o , A.-V.;
Chiang, T.-C. Eme gence o cha ge densi y wa es and a pseudogap in
single-laye TiTe2. Na . Commun. 2017,8, 516.
(13) Song, Q.; Occhialini, C. A.; E gecen, E.; Ilyas, B.; Amo oso, D.;
Ba one, P.; Kapeghian, J.; Wa anabe, K.; Taniguchi, T.; Bo ana, A. S.;
Picozzi, S.; Gedik, N.; Comin, R. E idence o a single-laye an de
Waals mul i e oic. Na u e 2022,602, 601−605.
(14) Fumega, A. O.; Lado, J. L. Mic oscopic o igin o mul i e oic
o de in monolaye NiI2.2D Ma e ials 2022,9, 025010.
(15) Hohenbe g, P. C. Exis ence o Long-Range O de in One and
Two Dimensions. Phys. Re . 1967,158, 383−386.
(16) Me min, N. D.; Wagne , H. Absence o Fe omagne ism o
An i e omagne ism in One- o Two-Dimensional Iso opic Heisen-
be g Models. Phys. Re . Le . 1966,17, 1133−1136.
(17) Du, L.; Hasan, T.; Cas ellanos-Gomez, A.; Liu, G.-B.; Yao, Y.;
Lau, C. N.; Sun, Z. Enginee ing symme y b eaking in 2D laye ed
ma e ials. Na u e Re iews Physics 2021,3, 193−206.
(18) Li, W.; Qian, X.; Li, J. Phase ansi ions in 2D ma e ials. Na u e
Re iews Ma e ials 2021,6, 829−846.
(19) Fumega, A. O.; Gobbi, M.; D ehe , P.; Wan, W.; González-
O ellana, C.; Pena-Díaz, M.; Roge o, C.; He e o-Ma ín, J.; Ga giani,
P.; Ilyn, M.; Ugeda, M. M.; Pa do, V.; Blanco-Canosa, S. Absence o
Fe omagne ism in VSe2 Caused by I s Cha ge Densi y Wa e Phase.
J. Phys. Chem. C 2019,123, 27802−27810.
(20) Coelho, P. M.; Nguyen Cong, K.; Bonilla, M.; Koleka , S.;
Phan, M.-H.; A ila, J.; Asensio, M. C.; Oleynik, I. I.; Ba zill, M.
Cha ge Densi y Wa e S a e Supp esses Fe omagne ic O de ing in
VSe2Monolaye s. J. Phys. Chem. C 2019,123, 14089−14096.
(21) O e o Fumega, A.; Phillips, J.; Pa do, V. Con olled Two-
Dimensional Fe omagne ism in 1T−C Te2: The Role o Cha ge
Densi y Wa e and S ain. J. Phys. Chem. C 2020,124, 21047−21053.
(22) Bonilla, M.; Koleka , S.; Ma, Y.; Diaz, H. C.; Kalappa il, V.;
Das, R.; Egge s, T.; Gu ie ez, H. R.; Phan, M.-H.; Ba zill, M. S ong
oom- empe a u e e omagne ism in VSe2 monolaye s on an de
Waals subs a es. Na . Nano echnol. 2018,13, 289−293.
(23) Chen, P.; Pai, W. W.; Chan, Y.-H.; Madha an, V.; Chou, M. Y.;
Mo, S.-K.; Fedo o , A.-V.; Chiang, T.-C. Unique Gap S uc u e and
Symme y o he Cha ge Densi y Wa e in Single-Laye VSe2.Phys.
Re . Le . 2018,121, 196402.
(24) Lei, B.-H.; Singh, D. J. Iden i ica ion o a low-ene gy me as able
1T- ype phase o monolaye VSe2.Phys. Re . B 2021,104, 125430.
(25) Biswas, D.; e al. Ul a as T igge ing o Insula o −Me al
T ansi ion in Two-Dimensional VSe2. Nano Le . 2021,21, 1968−
1975.
(26) Du ji , G.; e al. Eme gence o a Me al−Insula o T ansi ion
and High-Tempe a u e Cha ge-Densi y Wa es in VSe2 a he
Monolaye Limi . Nano Le . 2018,18, 5432−5438.
(27) Feng, J.; e al. Elec onic S uc u e and Enhanced Cha ge-
Densi y Wa e O de o Monolaye VSe2. Nano Le . 2018,18, 4493−
4499.
(28) Vinai, G.; Bigi, C.; Rajan, A.; Wa son, M. D.; Lee, T.-L.;
Mazzola, F.; Modes i, S.; Ba ua, S.; Ciomaga Ha nean, M.;
Balak ishnan, G.; King, P. D. C.; To elli, P.; Rossi, G.; Panaccione,
Nano Le e s pubs.acs.o g/NanoLe Le e
h ps://doi.o g/10.1021/acs.nanole .2c04584
Nano Le . 2023, 23, 1794−1800
1799
G. P oximi y-induced e omagne ism and chemical eac i i y in ew-
laye VSe2he e os uc u es. Phys. Re . B 2020,101, 035404.
(29) Eaglesham, D. J.; Wi he s, R. L.; Bi d, D. M. Cha ge-densi y-
wa e ansi ions in 1T-VSe2. Jou nal o Physics C: Solid S a e Physics
1986,19, 359−367.
(30) Diego, J.; Said, A. H.; Maha ha, S. K.; Bianco, R.; Monacelli, L.;
Caland a, M.; Mau i, F.; Rossnagel, K.; E ea, I.; Blanco-Canosa, S.
an de Waals d i en anha monic mel ing o he 3D cha ge densi y
wa e in VSe2. Na . Commun. 2021,12, 598.
(31) Henke, J.; Flicke , F.; La e ock, J.; an Wezel, J. Cha ge o de
om s uc u ed coupling in VSe2.SciPos Phys. 2020,9, 56.
(32) Xu, K.; Chen, P.; Li, X.; Wu, C.; Guo, Y.; Zhao, J.; Wu, X.; Xie,
Y. Ul a hin Nanoshee s o Vanadium Diselenide: A Me allic Two-
Dimensional Ma e ial wi h Fe omagne ic Cha ge-Densi y-Wa e
Beha io . Angew. Chem., In . Ed. 2013,52, 10477−10481.
(33) Yang, J.; Wang, W.; Liu, Y.; Du, H.; Ning, W.; Zheng, G.; Jin,
C.; Han, Y.; Wang, N.; Yang, Z.; Tian, M.; Zhang, Y. Thickness
dependence o he cha ge-densi y-wa e ansi ion empe a u e in
VSe2. Appl. Phys. Le . 2014,105, 063109.
(34) Pász o , A.; Sca a o, A.; Ba e eau, C.; Giannini, E.; Renne , C.
Dimensional c osso e o he cha ge densi y wa e ansi ion in hin
ex olia ed VSe 2. 2D Ma e ials 2017,4, 041005.
(35) Zhang, D.; Ha, J.; Baek, H.; Chan, Y.-H.; Na e e , F. D.;
Mye s, A. F.; Schumache , J. D.; Cullen, W. G.; Da ydo , A. V.; Kuk,
Y.; Chou, M. Y.; Zhi ene , N. B.; S oscio, J. A. S ain enginee ing a
4a× √3a cha ge-densi y-wa e phase in ansi ion-me al dichalcoge-
nide 1T −VSe2.Phys. Re . Ma e ials 2017,1, 024005.
(36) Du ji , G.; Choi, B. K.; Ly, T. T.; Lam, N. H.; Jang, K.; Dung,
D. D.; Chang, Y. J.; Kim, J. Mul iple cha ge densi y wa e phases o
monolaye VSe2 mani es ed by g aphene subs a es. Nano echnology
2021,32, 364002.
(37) Si, J. G.; Lu, W. J.; Wu, H. Y.; L , H. Y.; Liang, X.; Li, Q. J.;
Sun, Y. P. O igin o he mul iple cha ge densi y wa e o de in 1T−
VSe2.Phys. Re . B 2020,101, 235405.
(38) Bianco, R.; E ea, I.; Monacelli, L.; Caland a, M.; Mau i, F.
Quan um Enhancemen o Cha ge Densi y Wa e in NbS2 in he
Two-Dimensional Limi . Nano Le . 2019,19, 3098−3103.
(39) Bianco, R.; Monacelli, L.; Caland a, M.; Mau i, F.; E ea, I.
Weak Dimensionali y Dependence and Dominan Role o Ionic
Fluc ua ions in he Cha ge-Densi y-Wa e T ansi ion o NbSe2.Phys.
Re . Le . 2020,125, 106101.
(40) Zhou, J. S.; Monacelli, L.; Bianco, R.; E ea, I.; Mau i, F.;
Caland a, M. Anha monici y and Doping Mel he Cha ge Densi y
Wa e in Single-Laye TiSe2. Nano Le . 2020,20, 4809−4815.
(41) Sky Zhou, J.; Bianco, R.; Monacelli, L.; E ea, I.; Mau i, F.;
Caland a, M. Theo y o he hickness dependence o he cha ge
densi y wa e ansi ion in 1 T-TiTe2. 2D Ma e ials 2020,7, 045032.
(42) E ea, I.; Caland a, M.; Mau i, F. Anha monic ee ene gies and
phonon dispe sions om he s ochas ic sel -consis en ha monic
app oxima ion: Applica ion o pla inum and palladium hyd ides. Phys.
Re . B 2014,89, 064302.
(43) Bianco, R.; E ea, I.; Paula o, L.; Caland a, M.; Mau i, F.
Second-o de s uc u al phase ansi ions, ee ene gy cu a u e, and
empe a u e-dependen anha monic phonons in he sel -consis en
ha monic app oxima ion: Theo y and s ochas ic implemen a ion.
Phys. Re . B 2017,96, 014111.
(44) Monacelli, L.; E ea, I.; Caland a, M.; Mau i, F. P essu e and
s ess enso o complex anha monic c ys als wi hin he s ochas ic
sel -consis en ha monic app oxima ion. Phys. Re . B 2018,98,
024106.
(45) Monacelli, L.; Bianco, R.; Che ubini, M.; Caland a, M.; E ea,
I.; Mau i, F. The s ochas ic sel -consis en ha monic app oxima ion:
calcula ing ib a ional p ope ies o ma e ials wi h ull quan um and
anha monic e ec s. J. Phys.: Condens. Ma e 2021,33, 363001.
(46) Pe dew, J. P.; Bu ke, K.; E nze ho , M. Gene alized G adien
App oxima ion Made Simple. Phys. Re . Le . 1996,77, 3865−3868.
(47) Ba oni, S.; de Gi oncoli, S.; Dal Co so, A.; Giannozzi, P.
Phonons and ela ed c ys al p ope ies om densi y- unc ional
pe u ba ion heo y. Re . Mod. Phys. 2001,73, 515−562.
(48) Johannes, M. D.; Mazin, I. I. Fe mi su ace nes ing and he
o igin o cha ge densi y wa es in me als. Phys. Re . B 2008,77,
165135.
(49) Thonhause , T.; Coope , V. R.; Li, S.; Puzde , A.; Hyldgaa d,
P.; Lang e h, D. C. Van de Waals densi y unc ional: Sel -consis en
po en ial and he na u e o he an de Waals bond. Phys. Re . B 2007,
76, 125112.
(50) G imme, S. Semiempi ical GGA- ype densi y unc ional
cons uc ed wi h a long- ange dispe sion co ec ion. J. Compu .
Chem. 2006,27, 1787−1799.
(51) Giannozzi, P.; e al. QUANTUM ESPRESSO: a modula and
open-sou ce so wa e p ojec o quan um simula ions o ma e ials. J.
Phys.: Condens. Ma e 2009,21, 395502.
(52) Giannozzi, P.; e al. Ad anced capabili ies o ma e ials
modelling wi h Q uan um ESPRESSO. J. Phys.: Condens. Ma e
2017,29, 465901.
Nano Le e s pubs.acs.o g/NanoLe Le e
h ps://doi.o g/10.1021/acs.nanole .2c04584
Nano Le . 2023, 23, 1794−1800
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