Ci a ion: de-la-Hue a-Sainz, S.;
Balles e os, A.; Co de o, N.A. Elec ic
Field E ec s on Cu ed G aphene
Quan um Do s. Mic omachines 2023,
14, 2035. h ps://doi.o g/10.3390/
mi14112035
Academic Edi o : Aiqun Liu
Recei ed: 1 Sep embe 2023
Re ised: 28 Oc obe 2023
Accep ed: 29 Oc obe 2023
Published: 31 Oc obe 2023
Copy igh : © 2023 by he au ho s.
Licensee MDPI, Basel, Swi ze land.
This a icle is an open access a icle
dis ibu ed unde he e ms and
condi ions o he C ea i e Commons
A ibu ion (CC BY) license (h ps://
c ea i ecommons.o g/licenses/by/
4.0/).
mic omachines
A icle
Elec ic Field E ec s on Cu ed G aphene Quan um Do s
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, 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),
Un e sidad de Bu gos, 09001 Bu gos, Spain
3Ins i u e Ca los I o Theo e ical and Compu a ional Physics (IC1), 18016 G anada, Spain
*Co espondence: nco de [email p o ec ed]
Abs ac :
The ecen and con inuous esea ch on g aphene-based sys ems has opened hei usage
o a wide ange o applica ions due o hei exo ic p ope ies. In his pape , we ha e s udied he
e ec s o an elec ic ield on cu ed g aphene nano lakes, employing he Densi y Func ional Theo y.
Bo h mechanical and elec onic analyses o he sys em ha e been made h ough i s cu a u e ene gy,
dipola momen , and quan um egene a ion imes, wi h he in ensi y and di ec ion o a pe pendicula
elec ic ield and lake cu a u e as pa ame e s. A s abilisa ion o non-plana geome ies has been
obse ed, as well as opposi e beha iou s o bo h classical and e i al imes wi h espec o he
di ec ion o he ex e nal ield. Ou esul s show ha i is possible o modi y egene a ion imes using
cu a u e and elec ic ields a he same ime. This ine con ol in egene a ion imes could allow o
he s udy o new phenomena on g aphene.
Keywo ds: g aphene; nano lake; elec ic ield; quan um e i al; DFT
1. In oduc ion
The e is no doub ha he e has been a long-las ing wa e o ad ancemen s and inno-
a ion e ol ing a ound g aphene, wi h s ange phenomena and po en ial applica ions ap-
pea ing on a yea ly basis. Almos e e y ield seems o ha e an applica ion whe e he ex ao -
dina y p ope ies o g aphene (ei he p is ine o oxidised) can shine, om
enginee ing [1–4]
o ca alysis [
5
–
8
], medicine [
9
–
13
], sensing [
14
–
16
], op ics/elec omagne ics [
17
–
20
], hy-
d aulics [21,22], and ene gy managemen [23–25], o ci e a ew examples.
In spi e o he mechanical p ope ies o g aphene being qui e ema kable, he elec onic
ones a e by a he main ac o esponsible o i s a ac i eness as a esea ch opic and he
basis o se e al g oundb eaking applica ions. The peculia band s uc u e o g aphene,
calcula ed by Wallace many decades ago [
26
] by an app oxima ed ye s ill use ul me hod,
p esen s he elec ons as massless Di ac quasipa icles [
27
], allowing hem o each high
speeds [28]
ha esul in i s inc edible elec ical conduc i i y [
29
,
30
]. This, alongside he
ballis ic anspo [
31
,
32
] obse ed on non-pe u bed sys ems, makes g aphene a g ea
eplacemen o common conduc i e ma e ials. Also, due o i s nea - ela i is ic beha iou ,
g aphene is o en used as an expe imen al analogue in o de o eplica e o he wise nea ly
impossible o measu e phenomena, like he Klein pa adox [
33
–
35
], Zi e bewegung [
36
–
39
],
o he p ope ies o space ime wi h a nega i e cu a u e [40–43].
Pe haps one o i s mos in e es ing applica ions de i es om he in e ac ion be ween
g aphene and an ex e nal elec ic ield. As he band s uc u e is so impo an o i s elec-
onic p ope ies, i is quickly deduced han hey can be modi ied and uned easily by adjus -
ing he ield, allowing ea s such as mo ing he Fe mi ene gy be ween bands and changing
he ype and densi y o cha ge ca ie s— he ambipola elec ic ield
e ec [44,45]—o
widening he gap be ween alence and conduc ion bands o ob ain an insula ing beha iou ,
i needed. Taking also in o accoun he high mobili y o elec ons in his ma e ial and i s
Mic omachines 2023,14, 2035. h ps://doi.o g/10.3390/mi14112035 h ps://www.mdpi.com/jou nal/mic omachines
Mic omachines 2023,14, 2035 2 o 17
nanome ic scale, g aphene is a e y p omising al e na i e ma e ial o as and inc edibly
small elec onic componen s, such as ield-e ec ansis o s [
46
]; he ballis ic anspo
would ensu e a quick esponse and educed hea ing, and he obus ness and anspa ency
o he ma e ial could ex end i s use o lexible de ices and sc eens [47–49].
A known e ec o an elec ic ield on g aphene is he appea ance o he cu a u e.
When a ans e sal ield is applied, he ma e ial bends alongside he di ec ion o he ield,
as has been bo h p edic ed and obse ed ecen ly [
50
–
53
]. While cu ed g aphene is
no a s ange iew, as i na u ally appea s in he ma e ial i sel e en when suspended
and
unpe u bed [29,54–58]
, he e ec s o he de o ma ion on he elec onic beha iou
a e no able. In insic ipples a e known o dis u b he o he wise ee elec on pa hs [
27
],
and local de o ma ions, known as nanobubbles, can be easily p oduced in he ma e ial,
allowing o a localised uning o elec ic conduc i i y and mo e exo ic phenomena, like
pseudo-magne ic ield gene a ion [
59
,
60
]. I has been ecen ly shown ha double quan um
do s can be c ea ed by hese pseudo-magne ic ields in nanobubbles and ha hei quan um
s a es can be manipula ed o c ea e a con ollable qubi [61].
Thus, ou aim in his pape abou g aphene nano lakes is o s udy he e ec s o an elec-
ic ield on a sphe ically de o med sys em, analysing he elec onic spec um ia empo al
e olu ion h ough he quan um egene a ion phenomenon— he pa ial o o al eco e y o
he ini ial s a e o a wa epacke a e a ce ain amoun o ime, known as he egene a ion
o e i al ime. An a omis ic me hod, like he Densi y Func ional Theo y (DFT), will se e
as he main ool, as i has p e iously o nano lakes wi h
sphe ical geome ies [62]
, as well
as hose wi h cylind ical and hype bolical geome ies [
63
], in he absence o an ex e nal
ield. Some o hose esul s will be used he e o compa a i e pu poses.
2. Ma e ials and Me hods
As his wo k is a con inua ion o ou p e ious esea ch abou quan um e i als in
g aphene, he me hodology, as well as he sys em, will emain he same: a hexagonal
g aphene nano lake o 10 benzenic ings pe edge, wi h zig-zag edges and hyd ogen
passi a ed; he shape will be achie ed h ough he applica ion o he sphe ical su ace
equa ion, Equa ion (1), o se e al alues o Rbe ween 30 Å and 1000 Å:
z=qR2−x2−y2. (1)
This lake will be subjec ed o a s uc u al op imisa ion calcula ed wi hin he Local
Densi y App oxima ion (LDA) using he Gaussian 16 package [
64
] un il o ces on all
a oms a e below 0.0045 Ha ee/Boh and displacemen s below 0.0018 Boh . While o he
common unc ionals, such as B3LYP o Gene alised G adien App oxima ions (GGAs),
can be used o smalle sys ems, LDA esul s o g aphene sys ems a e o a simila o
highe
quali y [65,66]
, hey equi e lowe compu a ional wo k, and his unc ional has
been success ully used o s udy ca bon nanos uc u es [
67
–
73
]. Me a-GGAs p o ide be e
accu acy han LDA bu he compu ing ime is much highe . Taking in o accoun ha some
o ou calcula ions needed 1 yea o CPU ime a he LDA le el, using hose unc ionals
was ou o he ques ion. The e o e, we ollowed he ad ice in a ecen benchma k pape on
ab ini io desc ip ions o ca bon nanoma e ials using 15 di e en unc ionals [
74
] and used
LDA. The basis se will emain 6-31G** (wi h p and d unc ions as pola isa ion aids) [
75
],
as i has p o ided good quali y esul s on all calcula ions made so a .
Highly sphe ical nanobubbles can be achie ed wi h ex e nal elec ic ields [
51
], so
a pe ec sphe ical de o ma ion could be enough o his s udy; howe e , o he sake o
expe imen al ealism and ai compa isons, wo di e en se s o geome ic es ic ions
will be applied o he nano lake (Figu e 1): (i) maximal es ic ion, o cing all ca bon
a oms o lie on he sphe ical su ace and (ii) minimal es ic ion, o cing only he 12 ca bon
a oms on he e ices o emain on he sphe ical su ace. The i s case may seem pu ely
academic, bu i is a i s app oxima ion o a possible expe imen al se up. One me hod o
s abilise g aphene o elec onic applica ions is using hexagonal bo on ni ide (h-BN) as a
Mic omachines 2023,14, 2035 3 o 17
subs a e [
76
–
78
]. A sphe ical bo on ni ide ulle ene [
79
–
81
] could be used o suppo a
sphe ical g aphene nano lake.
(a) Fixed su ace (b) Fixed e ices
Figu e 1.
Top (uppe ow) and side (lowe ow) iews o he wo bounda y condi ions o a sphe ical
nano lake wi h ini ial
R=
40
Å
. Clea di e ences be ween bo h op imised s uc u es can be seen,
especially on he edges. The a oms ixed in he second case a e highligh ed in blue. Images gene a ed
wi h Gauss iew 6 [82].
Rega ding he elec ic ield, conside ing he o iginal la nano lake was cons uc ed
in he xy plane, he ields applied will be in he di ec ion o he zaxis—and hus pe -
pendicula o he unpe u bed lake plane—wi h h ee in ensi ies (0.0050 a.u., 0.0100 a.u.,
and 0.0250 a.u.) in wo opposi e di ec ions (one in he same di ec ion as he lake de o ma-
ion and he o he in he opposi e di ec ion) each, labelled as
±
50,
±
100, and
±
250, wi h he
posi i e sign co esponding o he ield in he same di ec ion as he lake de o ma ion.
These ields could seem o ha e an as onishing in ensi y, conside ing ha a ield o 1 a omic
uni equals 5
×1011 Vm−1
, bu a he nanome ic scale such ields can be achie ed by
se ing he elec odes close enough.
3. Resul s
The s udy o he e ec o an elec ic ield has been ca ied ou om bo h mechanical
and elec onic poin s o iew, he i s h ough he cu a u e ene gy and he la e by
calcula ing he empo al e olu ion o a wa epacke and checking i quan um egene a ion
occu s. To his end, sphe ical su aces o a ious cu a u e adii ha e been used o de o m
he lake, pe o ming an op imisa ion on each combina ion o adius, es ic ion se , and
elec ic ield.
3.1. Cu a u e Ene gy
As was he case in ou p e ious wo ks, he mechanical s udy will use he o al ene gy
o each nano lake, aking he la con igu a ion as he o igin o ene gies, i.e.,
E=Ecu ed −E la , (2)
whe e
Ecu ed
is he ene gy o he cu ed lake and
E la
is he ene gy o he plana con igu-
a ion (bo h ene gies calcula ed wi h he same elec ic ield).
Figu e 2shows he cu a u e ene gy plo ed agains 1
/R2
—which is he Gaussian
cu a u e o he lake o he maximal es ic ion case— o di e en ield s eng hs. In
o de o make igu es easie o unde s and, we ha e used poin ma ke s o esul s co e-
Mic omachines 2023,14, 2035 4 o 17
sponding o maximal es ic ion and dashed lines o esul s co esponding o minimal
es ic ion. In his la e case, he lines a e me ely guides o he eye connec ing he esul s
co esponding o consecu i e alues o 1
/R2
(we ha e used he same se o alues as o
he maximal es ic ion case). While all cu es show ha , as expec ed, lakes wi h highe
cu a u es ha e highe ene gies, he ends a y depending o he ela i e di ec ion o
he ield; o ze o ields, he plo has a nea ly pa abolic shape, wi h he la case being he
lowes ene gy geome y; o posi i e ields, he plo s a e also pa abolic, bu he minimal
ene gy is eached o a non-plana s uc u e, which showcases he cu a u e induced on
he ma e ial as a ans e sal ield is applied; las ly, nega i e ields esul in an ini ial ise in
ene gy a lowe cu a u es, as he ield is ying o induce a de o ma ion opposi e o he
p e-exis ing one, an e ec coun e ed by bending as cu a u e g ows.
Figu e 2.
Cu a u e ene gy s. 1
/R2
plo s o di e en alues o ield in ensi y and di ec ion—wi h
posi i e ields in ed ones and nega i e ones in blue ones—wi h ma ke s and do ed lines o o al
and minimal es ic ion, espec i ely.
S onge ields lead o analogous beha iou s, bu he e ec s a e bigge as he ield
inc eases. The op imal s uc u e is mo e cu ed and less ene ge ic o posi i e ields, while
he ini ial ene gy inc ease appea s ea lie and is g ea e o nega i e ields. Howe e ,
a s ong enough nega i e ield can o e ide he cu a u e e ec and p oduce a maximum
in ene gy.
3.2. Elec ic Dipole Momen
Due o i s close in e ac ion wi h ex e nal ields, he beha iou o he elec ic dipole
momen has also been s udied. F om he lake shape and symme y, a null momen would
be expec ed o he la case and a non-ze o momen on he nega i e zaxis o he sphe ical
ones; his momen would be in ensi ied wi h posi i e ields, as he cha ge dis ibu ion ha
o igina ed i becomes mo e p onounced and dec eases o e en lips wi h nega i e ields.
In Figu e 3, we ha e plo ed he zcomponen o he dipole momen agains 1
/R2
o
each o he se en ield alues. As expec ed, in gene al, he dipole along he zdi ec ion
g ows in absolu e magni ude wi h an inc easing cu a u e, since a bigge cu a u e allows
o a g ea e cha ge displacemen . In he absence o an ex e nal elec ic ield (g een poin s),
he plana s uc u e (1
/R2=
0) has no global dipole momen due o he symme y o he
elec on cloud wi h espec o he g aphene plane.
Mic omachines 2023,14, 2035 5 o 17
Figu e 4shows he dependence o one pa icula case wi h he ield, showing an almos
pe ec ly linea end, as expec ed. Wi hou an elec ic ield, he dipole momen is no
exac ly ze o because cu a u e causes he cen e o cha ge o he elec on cloud (nega i e)
o no be a he exac same posi ion as he cen e o cha ge o he a omic nuclei (posi i e).
The i s is sligh ly below he second and his ac c ea es a e y small downwa ds dipole.
Wi h a posi i e elec ic ield, i.e., he ield poin ing upwa ds, a omic nuclei end o mo e
up (and wi h hem he cen e o posi i e cha ge) while he elec onic cloud ends o mo e
down (and wi h i he cen e o nega i e cha ge). This ansla es in o a la ge nega i e
dipole momen ha inc eases wi h he s eng h o he ield. The opposi e happens when
he applied elec ic ield is nega i e, i.e., when he ield poin s downwa ds, a omic nuclei
end o mo e down, while he elec onic cloud ends o mo e up. This esul s in a la ge
posi i e dipole momen ha inc eases wi h he ield s eng h.
Figu e 3.
zcomponen alue o he elec ic dipole momen ec o agains 1
/R2
o all se en ield cases.
Figu e 4.
zcomponen o he elec ic dipole agains ex e nal ield, o a nano lake wi h
R
= 150 Å.
Because o he ela i ely small a ia ion wi hin he same ield, only a single case is displayed he e,
o cla i y.
Mic omachines 2023,14, 2035 6 o 17
I is in e es ing o see he beha iou o nega i e ields in Figu e 3, whe e in his case
he dipole componen ini ially dec eases be o e g owing. Fo posi i e ields, he cha ge
shi induced by he ield and he cu a u e wo k oge he o achie e a g ea e downwa ds
dipole; o nega i e ields, hey oppose, and as cha ges can mo e longe dis ances in
hese condi ions— he cen e and edges a e a he away— he cu a u e achie es a o ally
opposi e e ec .
The di ec ion o he dipola momen sugges s a highe elec on densi y on he sphe e
inne egion, some hing al eady s udied on la ge cu ed ca bon nanos uc u es [
30
,
83
],
in which he cu a u e educes he e ec i e dis ance be ween a oms and p o okes a
ehyb idisa ion on hem, wi h a highe o e lap and a consequen ene gy dec ease. While
he e a e wo ks s udying he connec ion be ween he local cha ge dis ibu ion and g aphene
ippling [
84
], hese a e o ien ed o a mo e con inuous and ex ended sys em; alas, i is no
jus i ied o assume a di ec connec ion be ween ou small, s a ic sys em and a much bigge
and mo e dynamic phenomenon like g aphene co uga ion.
3.3. Regene a ion Times
Quan um egene a ion phenomena will be s udied using he elec onic spec um o
each nano lake as a s a ing poin . Fo ha pu pose, we de ine a wa epacke as a linea
combina ion o eigen unc ions unwi h a ying coe icien s cn:
|Ψ(0)i=
∞
∑
n=0
cn|uni. (3)
We can calcula e he s a e o he packe a e an a bi a y ime
in o de o s udy i s
ime e olu ion:
|Ψ( )i=
∞
∑
n=0
cn|unie−i
¯hEn , (4)
wi h
En
being he eigen alue co esponding o he
|uni
eigens a e. The weigh s o each
eigen unc ion will ollow a Gaussian dis ibu ion de ined by le el numbe s—posi i e in ege s
o unoccupied s a es and nega i e o occupied ones—cen ed a ound he LUMO+4 (
n0
= 5)
and a wid h
σ
o 0.7, ensu ing a wa epacke na ow enough o obse e
quan um egene a ion
:
cn=1
σ√πe−(n−n0)2
2σ2. (5)
Using le el numbe s allows us o use an analy ical app oxima ion o he egene a ion
imes, as desc ibed in [
85
]. This app oxima ion begins wi h a Taylo expansion o he
spec um E(n)a ound he cen al le el n0:
En=En0+E0
n0(n−n0) + 1
2!E00
n0(n−n0)2+1
3!E000
n0(n−n0)3+. . . . (6)
By combining Equa ions (4) and (6), we each an expanded e ol ed s a e:
|Ψ( )i=
∞
∑
n=0
cn|unie−i
¯h(En0+E0
n0(n−n0)+ 1
2! E00
n0(n−n0)2+1
3! E000
n0(n−n0)3+...) , (7)
in which each e m on he exponen ial unc ion cons i u es a empo al scale. This ex-
p ession allows o he de ini ion o se e al egene a ion imes, di ec ly connec ed o he
spec um de i a i es:
TZb =π¯h
|En0|, (8)
TCl =2π¯h
|E0
n0|, (9)
Mic omachines 2023,14, 2035 7 o 17
TRe =2π¯h
|E00
n0|/2! , and (10)
TSup =2π¯h
|E000
n0|/3! . (11)
The main imes o in e es he e will be
TCl
, he classical ime, and
TRe
, he e i al ime,
as hey will be easie o obse e and a ionalise. Being dependen on he i s and second
de i a i es, espec i ely, he classical ime gi es in o ma ion abou he absolu e alue o
he ene gy di e ences and hus he comp ession o he spec um, while he e i al ime
in o ms abou he simila i y be ween ene gy di e ences and hus abou he egula i y o
he spec um.
TZb
is ela ed o Zi e bewegung, a ela i is ic phenomenon only obse ed on pa ic-
ula wa epacke s (ca s a es), di e en om ou s. He e, i is jus a me e phase ac o . In
con as ,
TSup
, he supe - e i al ime, would co espond o a highe scale oscilla ion on he
empo al e olu ion, which is di icul o de ec due o he dec easing alue o highe -o de
de i a i es and as such will no be conside ed ei he .
The in e pola ing unc ion selec ed o calcula e he de i a i es appea ing in hese
imes is a second-deg ee polynomial, ob ained by a leas squa es i o he h ee cen al
le els o he wa epacke . By using h ee poin s— he le el numbe s and hei ene gies—an
exac exp ession o he polynomial can be ob ained:
E(n) = b−a
2(n2−n2
0) + 1
2(a+b+2an0−2bn0)n+En0, (12)
wi h
n0
being he cen al o bi al numbe ,
En0
i s ene gy, and
a
and
b
he absolu e ene gy
di e ences wi h he lowe (
n0−
1) and uppe (
n0+
1) le els, espec i ely. Bo h i s and
second de i a i es can be analy ically compu ed easily:
E0(n) = (b−a)n+1
2(a+b+2an0−2bn0)(13)
and
E00(n) = b−a. (14)
These exp essions will be use ul in la e sec ions.
As o he ac ual empo al e olu ion, i can be moni o ed h ough he au oco ela ion
unc ion, de ined as he o e lap o ini ial ( =0) and la e ( >0) s a es:
A( ) = hΨ(0)|Ψ( )i=
∞
∑
m=0
∞
∑
n=0
c∗
mcnhum|unie−i
¯hEn (15)
Wo king wi h o hono mal s a es in he wa epacke makes i possible o subs i u e
K onecke ’s del a unc ions o he o e lap in eg als and cancel all cases whe e n6=m:
A( ) =
∞
∑
m=0
∞
∑
n=0
c∗
mcne−i
¯hEn δmn =
∞
∑
n=0|cn|2e−i
¯hEn . (16)
The physically sound quan i y is he squa ed modulus o he au oco ela ion unc ion:
|A( )|2=
∞
∑
m=0
∞
∑
n=0|cm|2|cn|2e−i
¯h(En−Em) . (17)
Taking in o accoun he de ini ion o he cosine unc ion, we can ew i e his exp ession
as ollows:
|A( )|2=
∞
∑
n=0|cn|4+
∞
∑
n=0
∞
∑
m<n
2|cm|2|cn|2cos(ωmn ), (18)
Mic omachines 2023,14, 2035 8 o 17
whe e
ωmn = (Em−En)/¯h
. This o mula ela es he empo al e olu ion o he ene gy
di e ences be ween di e en le els on he packe , a he han he ene gies hemsel es. As a
sum o cosines, a comple e e i al is gua an eed (i he spec um is commensu able) a e a
long enough ime, when he leas common mul iple o all indi idual pe iods is eached.
We can ocus on ou pa icula wa epacke o u he simpli y his exp ession. While
is i ue ha he wa epacke comp ises i e le els, he coe icien dis ibu ion and hei
squa es on he gene al o mula makes he h ee cen al ones (4, 5, and 6) he main con ibu-
o s, so he gene al exp ession can be simpli ied. We can igno e any e ms in ol ing o he
le els. We can go e en u he and emo e also he
ω46
e m, as he dis ibu ion is na ow
enough o make i s con ibu ion ba ely no iceable, esul ing in he ollowing exp ession:
|A( )|2≈2|c4|2|c5|2cosa
¯h +2|c5|2|c6|2cosb
¯h , (19)
whe e we ha e emo ed he i s suma o y and p opo ionali y cons an s, as he pe iodici y
o he unc ion emains he same, and
a
and
b
a e, like be o e, he ene gy di e ences
be ween le el 5 and le els 4 and 6, espec i ely. Being a sum o cosines, and conside ing
c4
and
c6
a e iden ical due o he dis ibu ion’s symme y, i can be easily con e ed o a
p oduc o cosines:
|A( )|2≈4|c4|2|c5|2cosa+b
2¯h cos a−b
2¯h . (20)
The wo new a gumen s ound he e a e in e es ing: he i s one is he a e age o bo h main
ene gy di e ences, and he second gi es in o ma ion abou how simila hey a e, which
makes hem pe ec candida es o he o igin o bo h classical and e i al
imes, espec i ely
.
Finally, Figu e 5shows a ypical iew o
|A( )|2
plo ed agains ime, wi h a clea
pa e n o a high- equency oscilla ion modula ed by a low- equency one, whose pe iods
co espond o he alues o
TCl
and
TRe
, espec i ely; he o me is loca ed a he i s
maximum o he ime e olu ion and he la e a he i s maximum o he en eloping cu e.
This co esponds o he shape o a gene al sum (o p oduc ) o cosines, co obo a ing ou
p e ious ma hema ical explana ion.
Figu e 5. Tempo al e olu ion plo o a ixed su ace nano lake, wi h R= 150 Å in a −50 ield, as an
exempla y case. Do ed lines ep esen he analy ical alues calcula ed o bo h TCl and TRe.
Mic omachines 2023,14, 2035 9 o 17
In gene al, he obse a ion o bo h imes is an easy ask, bu i is no gua an eed: as
bo h imes app oxima e in alue, he in e e ence be ween hem inc eases, esul ing in a
la ge shi o he i s maximum, quickly de ia ing om he
TCl
analy ical alue, making
he isualisa ion o TRe ha de , i no impossible.
Summa ising, he e a e wo di e en ways o calcula e egene a ion imes. The i s
one is looking o he maxima in he modulus o he au oco ela ion unc ion by nume ically
sea ching o he i s maximum o he unc ion ( o
TCl
) o he i s maximum o i s
en eloping cu e ( o
TRe
). The second one is using he analy ical exp essions gi en in
Equa ions (9) and (10).
3.3.1. Classical Time
Nume ical and analy ical alues o classical ime a e plo ed agains 1
/R2
in Figu e 6
o he ixed su ace case and in Figu e 7 o he ixed e ices case. Taking he ixed
su ace as an ideal si ua ion, we can see ha , wi hou any ield applied, bo h nume ical and
analy ical alues o
TCl
g ow wi h 1
/R2
o low cu a u es; his end does no las , as a
hea y de ia ion be ween hem a ises as we p og ess o mo e cu ed lakes, esul ing in an
o e all mono onic g ow h o analy ical
TCl
and a seemingly pa abolic plo o nume ical
TCl
. The disc epancy be ween he wo alues is a ibu ed o he in e e ence om
TRe
.
Analy ical
TCl
, being in e sely ela ed o he i s de i a i e, shows a comp ession o he
spec um in he le els s udied. The small dis o ion on he highes posi i e ield is due,
as will become clea la e , o an impo an shi on he le els o he packe .
This base end is in ensi ied wi h posi i e ields, wi h he disc epancy happening a
lowe cu a u es and lowe alues o
TCl
wi h s onge ields, eaching a limi alue a a
ce ain 1
/R2
. Fo nega i e ields, bo h alues s a wi h a dec ease—wi h a good ag eemen
be ween hem—and seem o s abilise wi h an inc easing cu a u e. These limi alues
ba ely change wi h he ield in ensi y bu do wi h i s di ec ion.
The ixed e ices case is simila , hough he e a e sudden changes on he plo due o
he shi o a mo e op imal s uc u e as he cu a u e g ows. Howe e , he main e ec s a e
s ill obse able, so p e ious analysis and conclusions s ill apply.
Figu e 6.
Classical ime agains 1
/R2
o all ixed su ace calcula ions, wi h ma ke s o nume ical
alues and do ed lines o analy ical ones.
Mic omachines 2023,14, 2035 16 o 17
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Disclaime /Publishe ’s No e:
The s a emen s, opinions and da a con ained in all publica ions a e solely hose o he indi idual
au ho (s) and con ibu o (s) and no o MDPI and/o he edi o (s). MDPI and/o he edi o (s) disclaim esponsibili y o any inju y o
people o p ope y esul ing om any ideas, me hods, ins uc ions o p oduc s e e ed o in he con en .
