Ci a ion: de-la-Hue a-Sainz, S.;
Balles e os, A.; Co de o, N.A.
Quan um Re i als in Cu ed
G aphene Nano lakes. Nanoma e ials
2022,12, 1953. h ps://doi.o g/
10.3390/nano12121953
Academic Edi o : A hu P Baddo
Recei ed: 23 Ap il 2022
Accep ed: 3 June 2022
Published: 7 June 2022
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nanoma e ials
A icle
Quan um Re i als in Cu ed G aphene Nano lakes
Se gio de-la-Hue a-Sainz 1, Angel Balles e os 1and Nicolás A. Co de o 1,2,3,*
1Physics Depa men , Uni e sidad de Bu gos, E-09001 Bu gos, Spain; [email p o ec ed] (S.d.-l.-H.-S.);
[email p o ec ed] (A.B.)
2In e na ional Resea ch Cen e in C i ical Raw Ma e ials o Ad anced Indus ial Technologies (ICCRAM),
Uni e sidad de Bu gos, E-09001 Bu gos, Spain
3Ins i u e Ca los I o Theo e ical and Compu a ional Physics (IC1), E-18016 G anada, Spain
*Co espondence: nco de [email p o ec ed]
Abs ac :
G aphene nanos uc u es ha e a ac ed a lo o a en ion in ecen yea s due o hei
uncon en ional p ope ies. We ha e employed Densi y Func ional Theo y o s udy he mechanical
and elec onic p ope ies o cu ed g aphene nano lakes. We explo e hexagonal lakes elaxed wi h
di e en bounda y condi ions: (i) all a oms on a pe ec sphe ical sec o , (ii) only bo de a oms o ced
o be on he sphe ical sec o , and (iii) only e ex a oms o ced o be on he sphe ical sec o . Fo
each case, we ha e analysed he beha iou o cu a u e ene gy and o quan um egene a ion imes
(classical and e i al) as he sphe ical sec o adius changes. Re i al ime p esen s in one case a
di e gence usually associa ed wi h a phase ansi ion, p obably caused by he pseudomagne ic ield
c ea ed by he cu a u e. This could be he i s case o a phase ansi ion in g aphene nanos uc u es
wi hou he p esence o ex e nal elec ic o magne ic ields.
Keywo ds: g aphene; cu a u e; quan um e i als; DFT; phase ansi ion
1. In oduc ion
The expe imen al isola ion o a single g aphi ic laye (now known as g aphene) by
means o he so-called “Sco ch ape me hod” by Geim and No oselo [
1
] has undoub edly
opened a new ield in science. The p oo is ha in ecen yea s, o e 1% o all he scien i ic
publica ions included in he Web o Science
™
global ci a ion da abase [
2
] a e ela ed o his
one-a om- hick sys em.
The ou s anding p ope ies o his 2D ma e ial ha e led o many applica ion p oposals.
To ci e jus a ew o hem: nanocomposi es o bone issue enginee ing [
3
], composi es
o mul i unc ional applica ions [
4
], lub ica ion [
5
,
6
], sola cells [
7
], ul acapaci o s [
8
,
9
],
ba e ies [
10
], senso s [
11
], ca alysis [
12
], nanomedicine [
13
], uel cells [
14
,
15
] o e en ene gy
ha es ing [16].
Quan um e i als consis in he empo al pe iodic econs uc ion o a wa e packe
in sys ems wi h a commensu able disc e e spec um [
17
]. They ha e ecen ly a ac ed
a en ion due o hei possible in e es in quan um de ices [
18
–
21
]. Quan um e i als ha e
been s udied in in ini e g aphene unde magne ic ields [
22
–
24
] o ci cula ly pola ized
adia ion [25], as well as in g aphene ings [26] and g aphene quan um do s [27].
One o he common misconcep ions abou g aphene is ha i is la . In ac , Peie ls
ins abili y [
28
,
29
] c ea es ipples in ee-s anding g aphene [
30
–
35
]. A omis ic Mon e Ca lo
simula ions based on a e y accu a e many-body in e a omic po en ial o ca bon [
36
] ga e
ipples wi h a size dis ibu ion ha peaked a ound 80 Å, in ag eemen wi h expe imen s
ha yield esul s in he 50–100 Å ange [
31
]. Besides, when g aphene is g own o deposi ed
on a subs a e, nanobubbles wi h diame e s be ween 80 Å and 1000 Å appea [37–41].
Rec angula g aphene lakes ei he ee [
42
] o subjec ed o plana bending [
43
] ha e
been analysed in he li e a u e. Conical g aphene ings in a magne ic ield ha e also been
s udied, bu only in he con inuum limi app oxima ion in he icini y o he Di ac poin s
Nanoma e ials 2022,12, 1953. h ps://doi.o g/10.3390/nano12121953 h ps://www.mdpi.com/jou nal/nanoma e ials
Nanoma e ials 2022,12, 1953 2 o 13
and e y a om he e ex [
44
,
45
]. Bo h in-plane and ou -o -plane bending esul in he
appea ance o a pseudo-magne ic ield.
We p esen in his a icle he i s ( o he bes o ou knowledge) calcula ion o quan um
e i als using Densi y Func ional Theo y (DFT) and use i o s udy he in e play be ween
cu a u e and egene a ion imes in a non-plana g aphene lake.
2. Ma e ials and Me hods
Since Wallace did he i s calcula ion o he elec onic p ope ies o g aphene using he
igh -binding me hod [
46
], many heo e ical models ha e been used o s udy his sys em:
molecula mechanics, molecula dynamics, semiempi ical me hods, Densi y Func ional
Theo y, Ha ee–Fock (HF), pos -HF including co ela ion, Mon e Ca lo simula ions, hyb id
me hods, con inuum models, e c. We ha e used he Densi y Func ional Theo y (DFT)
o malism [
47
] wi hin he Local Densi y App oxima ion (LDA) [
48
,
49
] as implemen ed in
he Gaussian 09 [
50
] and Gaussian 16 [
51
] sui es o p og ams. We ha e selec ed DFT o i s
balance be ween accu acy and compu a ional e o , and ha e chosen LDA because i gi es
be e esul s han g adien co ec ed app oxima ions (GGAs) o g aphi ic sys ems [
52
–
54
]
and because i has been p e iously used o success ully s udy he in e ac ion be ween
ca bon nanos uc u es and se e al small molecules and a oms [
55
–
60
] as well as— e y
ecen ly—g aphene nano ibbons [
61
]. We ha e selec ed he 6-31G** basis se [
62
–
67
] ha
adds o he 6-31G se d- ype and p- ype Ca esian–Gaussian pola iza ion unc ions and is
commonly used o ca bon nanos uc u es calcula ions.
In o de o a oid non- i ial magne ic g ound s a es induced by asymme ic edge
ex ensions [
68
] we ha e used he hexagonal lake wi h zig-zag bo de s passi a ed wi h
hyd ogen a oms shown in Figu e 1. The size o he lake (10 concen ic hexagonal ings
comp ising 600 C a oms and 60 H a oms wi h a sepa a ion o app oxima ely 47 Å be ween
opposi e e ices) has been selec ed as a easonable comp omise be ween he size o
expe imen ally measu ed ipples and compu a ional cos .
Figu e 1. G aphene lake used in he calcula ions (image gene a ed using GausView 6 [69]).
Nanoma e ials 2022,12, 1953 3 o 13
We ha e s udied (quasi-)sphe ical nano lakes o di e en cu a u e adii wi h he
h ee di e en bending possibili ies depic ed in Figu e 2 o a adius o 40 Å: o cing all
600 ca bon
a oms o lie on a sphe ical su ace (uppe panel), ixing only he 60 bo de a oms
on he su ace and allowing he es o elax (middle panel) and o cing only he 12 e ex
a oms o belong o he sphe e and no imposing any condi ion on he es (lowe panel).
Since ca bon a oms end o be in a sp
2
hyb idiza ion s a e, he nano lake ends o la en, as
he es ic ions a e less demanding.
Figu e 2.
Op imized geome ies o a (quasi-)sphe ical g aphene lake o adius 40 Å (image gene a ed
using GausView 6 [69]).
3. Resul s and Discussion
We ha e s udied bo h cu a u e ene gy and quan um e i als in his nanos uc u e.
The i s one o checking i he mac oscopic con inuum limi model ha p edic s ha
cu a u e ene gy is p opo ional o he Gaussian cu a u e (i.e., he in e se o he adius
squa ed) [
70
] holds a his scale. The second one o ying o ind ends in egene a-
ion imes.
3.1. Cu a u e Ene gy
We p esen in Figu e 3 he dependence o he ene gy o he nano lake
E
wi h he
in e se squa ed adius o he sphe e
1
R2
. Since we ha e aken he la con igu a ion as he
ene gy o igin, his plo ep esen s he cu a u e ene gy. Fo he case o a pe ec sphe ical
su ace ha co esponds o ou ixed su ace calcula ions (shown in ed in he igu e),
1
R2
is p ecisely he Gaussian cu a u e. Fo he ixed bo de s (depic ed in blue) and ixed
e ices (pain ed in g een) cases, cu a u e is no longe cons an , bu we can ake he
alue o
1
R2
o he ixed a oms as an es ima e o he cu a u e o compa ison pu poses.
Logically, o a ixed alue o
R
he geome y wi h only he e ices ixed is ene ge ically
mo e a ou able han ha wi h he bo de s ixed and his in u n is mo e s able han he one
Nanoma e ials 2022,12, 1953 4 o 13
wi h all he a oms o ced o lie on a sphe ical su ace. As expec ed, o he h ee geome ies
he cu a u e ene gy inc eases wi h cu a u e bu he end o he ixed su ace case is
somewha unexpec ed. The mac oscopic con inuum esul
E∝1
R2
would lead o a s aigh
ed line, bu ha does no seem o be he case.
Figu e 3.
Ene gy o a (quasi-)sphe ical g aphene lake as a unc ion o i s cu a u e. Fla con igu a ion
is aken as ene gy o igin. Lines a e me ely guides o he eye.
To check i he mac oscopic esul holds a leas o low cu a u es, we p esen in
Figu e 4a log–log plo o he cu a u e ene gy e sus adius.
E=k1
R2⇒log E=logk−2log R, (1)
and he ed g aph should p esen a cons an slope equal o
−
2. This is clea ly no he case
and he con inuum app oxima ion is no alid o so small a lake.
Figu e 4.
Log–log plo o he ene gy o a sphe ical g aphene lake as a unc ion o i s cu a u e. Fla
con igu a ion is aken as ene gy o igin. Lines a e me ely guides o he eye.
This de ia ion om he con inuum beha iou is undoub edly due o he di e ence
be ween he classical physics laws go e ning he mac oscopic wo ld and he quan um ones
Nanoma e ials 2022,12, 1953 5 o 13
uling a he nanoscale. This di e ence ansla es in o wo dis inc aspec s. The equilib ium
geome y dic a ed by bo h se s o laws is di e en and, e en o he same con igu a ion,
hey lead o di e en ene gies. To check which one is mo e impo an in his case, we
ha e made a se ies o non-sel -consis en calcula ions wi h a hyb id Molecula Mechanics
(MM)/DFT model. We ha e i s used LDA calcula ions o de e mine a “semiclassical”
Hooke po en ial o he C–C bond. To his end, we ha e used a g aphene lake simila
o he one in Figu e 1, bu wi h only 7 concen ic hexagonal ings. We ha e changed he
leng h o he cen al C–C bond and calcula ed he ene gy o he sys em o bo h s e ched
and comp essed bond leng hs using LDA. We ha e i ed he ene gies o a pa abolic cu e
and de e mined a C–C pai po en ial o ca bon nano lakes. We ha e w i en an in-house
code o de e mine he semiclassical equilib ium geome y o he sphe ical nano lakes. This
geome y is hen used in a single-poin calcula ion o de e mine he LDA ene gy o he
nano lake. We will label he esul s ob ained by his p ocedu e as “ ixed su ace (MM)”.
We ha e included in Figu e 3 he ene gies esul ing om his hyb id app oach in g ay.
The ene gies a e bigge ha hose co esponding o he sel -consis en LDA calcula ion (in
ed) since, acco ding o he a ia ional p inciple, using a wa e unc ion di e en o ha o
he undamen al s a e leads o a highe ene gy. The g ea e he cu a u e, he bigge he
ene gy inc ease.
In o de o check i he con inuum model is alid o his app oach, we ha e also
included he co esponding esul s in Figu e 4. The g aph is now nea ly a s aigh line,
bu he e is s ill some non-linea i y. Equa ion (1) seems o be app oxima ely alid, bu he
ac o in on o log Rdepends sligh ly on R, adop ing he o m
E=k1
Rn⇒log E=logk−nlog R. (2)
We p esen in Figu e 5 he alue o
n
as
R
changes calcula ed using a 3-poin ini e
di e ences me hod.
Figu e 5.
Value o
n
in Equa ion (2) as a unc ion o he adius o he sphe ical ca bon nano lake
calcula ed wi h he hyb id MM/DFT app oxima ion.
I is possible o i he esul s in his igu e using he simples Padé app oximan
n=a0+a1R
−1+b1R. (3)
The esul o his i ing is p esen ed in Table 1.
Nanoma e ials 2022,12, 1953 6 o 13
Table 1. Resul s o he i ing o he da a in Figu e 5 o he Padé app oximan in Equa ion (3).
a0a1/Å−1b1/Å−1
−0.1818 0.0443 0.0219
We can use his Padé app oximan o calcula e he asymp o ic beha iou o n.
lim
R→∞n=a1
b1=2.02. (4)
This esul is e y close o 2, which is he alue p edic ed in he con inuum model.
The e o e, he semiclassical MM/DFT app oach ends o he classical con inuum mac o-
scopic limi , p o ing ha he main quan um con ibu ion o he de ia ion om his model
is due o he small change in he equilib ium geome y.
3.2. Quan um Re i als
We can w i e he ini ial s a e o a ime-independen Hamil onian
ˆ
H
as a linea combi-
na ion o i s eigen unc ions:
|Ψ(0)i=
∞
∑
n=0
an|uni, (5)
whe e ana e cons an s and |unia e he eigen unc ions wi h ene gies En,
ˆ
H|uni=En|uni. (6)
The empo al e olu ion o his s a e can be w i en as
|Ψ( )i=
∞
∑
n=0
an|unie−i
¯hEn . (7)
Since we ha e calcula ed he ene gy o cu ed g aphene nano lakes using DFT (i.e.,
sol ing he Kohn–Sham equa ions o he sys em) we know hei ene gy spec um and
can he e o e s udy he ime e olu ion o a wa e packe in hese sys ems. Le us conside
a supe posi ion o eigens a es o he Hamil onian concen a ed a ound a cen al ene gy
le el
n0
cha ac e ised by an ene gy
En0
. We can pe o m a Taylo expansion o he ene gy
spec um a ound En0:
En=En0+E0
n0(n−n0) + 1
2!E00
n0(n−n0)2+1
3!E000
n0(n−n0)3+. . . . (8)
Taking in o accoun Equa ions (7) and (8),
|Ψ( )i=
∞
∑
n=0
an|unie−i
¯hhEn0+E0
n0(n−n0)+ 1
2! E00
n0(n−n0)2+1
3! E000
n0(n−n0)3+...i . (9)
Each e m in he exponen ial (excep he i s one ha is jus a global phase) de ines a
cha ac e is ic ime scale:
TCl ≡2π¯h
|E0
n0|is called classical ime, (10)
TRe ≡2π¯h
|E00
n0|/2 is called e i al ime, and (11)
TSup ≡2π¯h
|E000
n0|/6 is called supe - e i al ime (12)
(see [17] o u he de ails).
Nanoma e ials 2022,12, 1953 7 o 13
We a e going o conside a Gaussian ini ial wa e packe ,
an=1
σ√πe−(n−n0)2
2σ2, (13)
whe e, we ha e selec ed he cen al le el as he ou h le el abo e he Highes Occu-
pied Molecula O bi al (HOMO)
n0=HOMO +
4 and, in o de o ge a sha ply concen-
a ed packe
σ=
0.7 so ha only i e le els a ound
n0
ha e a signi ica i e con ibu ion
(an>0.001).
The easies way o isualising wa e packe egene a ion is making use o he squa ed
modulus o he so called au oco ela ion unc ion ha measu es he o e lap o he wa e
packe a imes 0 and :
|A( )|2=|hΨ(0)|Ψ( )i|2. (14)
A ypical case is p esen ed in Figu e 6.
|A( )|2
oscilla es e y as , eaching a maximum
e e y
TCl
inside an en elope wi h
TRe
pe iodici y.
TSup
is usually much la ge ha
TRe
and
we will no conside i in his wo k.
0TCl TRe
0
1/4
1/2
3/4
1
Time
A( )2
Figu e 6.
A simple example o he ime e olu ion o he squa ed modulus o he au oco ela ion
unc ion in blue wi h i s uppe en elope in o ange. Classical ime is ma ked in ed and e i al ime
in g een.
In p inciple, calcula ing
TCl
is s aigh o wa d. One only has o sea ch o he i s
maximum o
|A( )|2
. De e mining
TRe
is no so easy. In his case, i is necessa y o calcula e
he en elope o he unc ion and calcula e i s i s maximum. Howe e , o some cu a u es,
he si ua ion is no as clea as he one depic ed in Figu e 6. I
TRe
is no much bigge han
TCl
,
bo h imes in e e e, he pa e n is mo e complica ed and i is di icul o de e mine bo h
o hem, especially
TCl
. I is hen desi able o ha e ano he way o calcula ing hese imes.
The solu ion is employing Equa ions (10) and (11). To do ha , we ha e calcula ed an
in e pola ing unc ion by adjus ing a pa abolic cu e o e e y h ee consecu i e le els in
he spec um and used i o calcula e he i s wo de i a i es o he ene gy wi h espec
o he le el ha appea s in hose equa ions. The e o e, we ha e wo di e en ways o
calcula ing hese imes. We will call he i s one nume ical (num.) and he second one
analy ical (an.).
Expe imen s o measu ing quan um egene a ion imes a e based on he use o wo
consecu i e lase pulses [
17
]. The i s one, called pump, c ea es he ini ial wa e packe ,
while he second one, called p obe, measu es i s ime e olu ion. By changing he delay
Nanoma e ials 2022,12, 1953 8 o 13
be ween he wo pulses, di e en ime scales can be explo ed. This pump–p obe scheme was
p oposed by Albe , Ri sch, and Zolle [
71
], and was ini ially used o s udy a oms [
72
,
73
].
In his case, he second lase pulse was used o ionize he sys em and he pho oioniza ion
signal was measu ed. The me hod was modi ied by Zewail (who was awa ded he 1999
Nobel P ize in Chemis y o his s udies in his ield) and his g oup o s udy ew-a om
molecules, using he p obe pulse o ge he sys em o an uppe luo escen s a e and measu e
he luo escence [
74
,
75
]. Mo e ecen ly, he scheme has been adap ed so ha he second
pulse pho oexci es he sample and he di e en ial ansmission spec a is analysed. This
app oach has made possible o s udy he wa e packe e olu ion o CdSe quan um do s
wi h a mean diame e o 6.4 nm (sligh ly la ge han he diame e o he ca bon nano lake
we ha e selec ed) [76].
3.2.1. Classical Time
We p esen in Figu e 7classical imes de e mined bo h nume ically (poin s) and
analy ically (lines) o he ou geome ies conside ed: ixed su ace ( ed), ixed bo de s
(blue), ixed e ices (g een) and ixed su ace (MM) (g ay). Analy ical es ima ions a e
p esen ed as lines o cla i y pu poses, bu hey can only be calcula ed a he same poin s
as nume ical ones. We ha e simply linked wo consecu i e poin s wi h a s aigh line.
Figu e 7.
Classical imes o a cu ed g aphene nano lake. Poin s co espond o nume ical alues
while do ed lines ep esen analy ical ones.
The ag eemen be ween nume ical and analy ical es ima ions is good o all geome ies
o low cu a u es (below app oxima ely 10
−4Å−2
), bu in wo o he h ee sel -consis en
calcula ions, his ag eemen is los o high cu a u es. The eason is ha , as we will see in
he ollowing subsec ion,
TRe
dec eases as he cu a u e inc eases and i becomes less ha
one o de o magni ude bigge han TCl o he ixed su ace and ixed bo de cases.
Recalling Equa ion (10),
TCl
is p opo ional o he in e se o he i s de i a i e o he
ene gy wi h espec o he le el index. This means classical ime is ela ed o he sp eading
o he ene gy spec um. TCl dec eases as he sepa a ion among ene gy le els inc eases.
I we look a he analy ical es ima ion o he classical imes co esponding o he ixed
su ace and ixed bo de cases, hey g ow wi h cu a u e, while in he ixed e ices case
his ini ial endency b eaks beyond 1.5
×
10
−4Å−2
and he g aph becomes nea ly la . We
Nanoma e ials 2022,12, 1953 9 o 13
ha e o emembe ha his hi d case co esponds o he geome y wi h lowe es ic ions
and he sys em can adap i sel be e o he de o ma ion o i s ixed poin s. This means ha
bo h o al ene gy (see Figu e 3) and ene gy le el sepa a ion a e less sensi i e o cu a u e.
I we conside he nume ical es ima ions o he h ee sel -consis en geome ies,
TCl
emains essen ially la since he sligh dec ease in ene gy spec um sp eading is
compensa ed by he in e e ence om
TRe
. In he ixed su ace case his in e e ence
o e comes he endency o TCl o g ow and, in ac , i dec eases o high cu a u es.
Finally, i we concen a e on he non-sel consis en ixed su ace (MM) case, he be-
ha iou is simple . Nume ical and analy ical es ima ions ag ee pe ec ly excep o he high
cu a u e egime because, as we will see in he nex subsec ion,
TRe
is much highe han
TCl
. The sp eading o he ene gy spec um dec eases mono onically wi h cu a u e and
classical ime inc eases in a linea way.
3.2.2. Re i al Time
We show in Figu e 8 e i al imes de e mined bo h nume ically (poin s) and analy i-
cally (lines) o he ou geome ies conside ed: ixed su ace ( ed), ixed bo de s (blue),
ixed e ices (g een) and ixed su ace (MM) (g ay). Analy ical es ima ions a e p esen ed
as lines o cla i y pu poses, bu hey can only be calcula ed a he same poin s as nume ical
ones. In his case, we ha e connec ed wo consecu i e poin s wi h a smoo hed line.
Figu e 8.
Re i al imes o a cu ed g aphene nano lake. Poin s co espond o nume ical alues
while do ed lines ep esen analy ical ones.
In all cases, nume ical and analy ical es ima es pe ec ly ag ee since supe e i al imes
a e much highe han e i al ones and do no in e e e wi h hem.
Acco ding o Equa ion (11),
TRe
is p opo ional o he in e se o he second de i a i e
o he ene gy wi h espec o he le el index. This means e i al ime is ela ed o he
non-linea i y o he ene gy spec um.
TRe
inc eases as he spec um ge s close o being
linea (i.e., ene gy le els end o be equally spaced).
Re i al imes o he ixed su ace and ixed bo de s cases a e e y simila and dec ease
mono onically wi h cu a u e.
TRe
o he ixed e ices case coincides wi h hem up o
1.5
×
10
−4Å−2
and om ha poin on i emains essen ially cons an . The si ua ion is
