Ci a ion: Misiu e , D.; Holcman, V.
Modeling o Magne ic Films: A
Scien i ic Pe spec i e. Ma e ials 2024,
17, 1436. h ps://doi.o g/10.3390/
ma17061436
Academic Edi o : Anas asios
J. Tasiopoulos
Recei ed: 2 Ma ch 2024
Re ised: 15 Ma ch 2024
Accep ed: 18 Ma ch 2024
Published: 21 Ma ch 2024
Copy igh : © 2024 by he au ho s.
Licensee MDPI, Basel, Swi ze land.
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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/).
ma e ials
Re iew
Modeling o Magne ic Films: A Scien i ic Pe spec i e
Denis Misiu e * and Vladimí Holcman
Depa men o Physics, Facul y o Elec ical Enginee ing and Communica ion, B no Uni e si y o Technology,
Technicka 2848/8, 61600 B no, Czech Republic; [email p o ec ed]
*Co espondence: [email p o ec ed]
Abs ac : Magne ic hin- ilm modeling s ands as a dynamic nexus o scien i ic inqui y and echno-
logical ad ancemen , poised a he angua d o ma e ials science explo a ion. Le e aging a di e se
sui e o compu a ional me hodologies, including Mon e Ca lo simula ions and molecula dynamics,
esea che s me iculously dissec he in ica e in e play go e ning magne ism and hin- ilm g ow h
ac oss he e ogeneous subs a es. Recen s ides, no ably in mul iscale modeling and machine lea ning
pa adigms, ha e engende ed a pa adigm shi in p edic i e capabili ies, acili a ing a nuanced unde -
s anding o hin- ilm dynamics spanning dispa a e spa io empo al egimes. This in e disciplina y
syne gy, complemen ed by a an ga de expe imen al modali ies such as in si u mic oscopy, p omises
a apes y o ans o ma i e ad ancemen s in magne ic ma e ials wi h a - eaching implica ions
ac oss mul i ace ed domains including magne ic da a s o age, spin onics, and magne ic sensing
echnologies. The con luence o compu a ional modeling and expe imen al alida ion he alds a
new e a o scien i ic igo , a o ding unpa alleled insigh s in o he eal- ime dynamics o magne ic
ilms and bols e ing he ideli y o p edic i e models. As esea che s cha an ambi iously uncha ed
ajec o y, he bu geoning ealm o magne ic hin- ilm modeling bu geons wi h p omise, poised o
unlock no el pa adigms in ma e ials science and enginee ing. Th ough his in ica e nexus o heo e -
ical elucida ion and empi ical alida ion, magne ic hin- ilm modeling he alds a u u e eple e wi h
inno a ion, ca alyzing a enaissance in echnological possibili ies ac oss di e se indus ial landscapes.
Keywo ds: magne ic ilms; subs a es; i s -p inciples calcula ion; densi y unc ional heo y;
elec onic
s uc u e; molecula dynamics simula ions; in e ace phenomena; Mon e Ca lo simula ion
1. In oduc ion
In he ealm o ma e ials science and condensed ma e physics, he in ica e in e play
be ween magne ic ilms and di e se subs a es has eme ged as a cap i a ing a enue o
esea ch wi h a - eaching implica ions. The abili y o ailo magne ic p ope ies a he
nanoscale has opened up a my iad o possibili ies o echnological ad ancemen s spanning
om da a s o age and sensing o spin onics and quan um compu ing. Cen al o hese
de elopmen s is he indispensable ole o modeling echniques, which empowe esea che s
o deciphe he complex dynamics and beha io o magne ic ilms on a ied subs a es.
Modeling echniques ha e e olu ionized ou comp ehension o magne ic phenomena
by p o iding in aluable insigh s in o he unde lying physical mechanisms, in e ac ions,
and esponses. These echniques encompass a di e se a ay o me hodologies, anging
om a omis ic simula ions o con inuum app oaches, each o e ing unique ad an ages and
pe spec i es. As we del e in o he scien i ic dimensions o modeling magne ic ilms on
di e se subs a es, i becomes e iden ha a comp ehensi e explo a ion o hese echniques
is essen ial o un a eling he in ica e apes y o magne ic beha io .
The p ima y ocus o his e iew is o p o ide a comp ehensi e o e iew o he cu en
s a e-o - he-a modeling echniques employed in s udying magne ic ilms on di e se
subs a es. We aim o elucida e he unde lying p inciples, capabili ies, and limi a ions o
hese me hods while shedding ligh on he a ied pe spec i es hey b ing o he o e on .
By syn hesizing he collec i e knowledge amassed om bo h expe imen al obse a ions
Ma e ials 2024,17, 1436. h ps://doi.o g/10.3390/ma17061436 h ps://www.mdpi.com/jou nal/ma e ials
Ma e ials 2024,17, 1436 2 o 32
and heo e ical endea o s, we endea o o con ibu e o a deepe unde s anding o he
in ica e in e play be ween magne ic ilms and subs a es.
This explo a ion will be s uc u ed as ollows: Fi s , we will del e in o he ounda ional
concep s ha unde pin he magne ic in e ac ions and phenomena obse ed in hin ilms
on di e en subs a es. Subsequen ly, we will emba k on a jou ney h ough he ealm o
modeling echniques, beginning wi h a omis ic simula ions ha cap u e he undamen al
a omic-scale in e ac ions. Mo ing o wa d, con inuum models will ake cen e s age,
o e ing a b oade pe spec i e on he collec i e beha io o magne ic domains and hei
esponse o ex e nal s imuli.
As we na iga e h ough hese modeling echniques, we will highligh he unique in-
sigh s hey p o ide in o a ious aspec s o magne ic ilm–subs a e sys ems, such as su ace
e ec s, domain wall dynamics, and magne ic aniso opy. Mo eo e , we will emphasize
he impo ance o syne gy be ween expe imen al obse a ions and heo e ical p edic ions,
illus a ing how modeling en iches ou unde s anding by elucida ing complex phenomena
ha may no be eadily disce nible h ough expe imen s alone.
2. G ow h Mechanisms o Thin Films
The g ow h mechanism o hin ilms is in ica ely in luenced by a ious ac o s, such
as su ace mo phology, subs a e o ien a ion, and deposi ion p ocess pa ame e s. These ac-
o s play a c ucial ole in mi iga ing de ec s in he p oduced ilms, including decomposi ion,
luc ua ion, and laye in e di usion.
Du ing he ini ial s ages o ilm g ow h, e apo a ed pa icles o he deposi ed ma e ial
unde go abso p ion and chemical di usion p ocesses. The a e o hese pa icles is de e -
mined by su ace mo phology; hey may ei he be e lec ed o abso bed in o he subs a e
su ace. Abso p ion, in pa icula , in ol es he in e ac ion be ween he e apo a ed pa icles
and he subs a e su ace, cha ac e ized by a s icking coe icien ep esen ing he a io o
abso bed pa icles o he o al amoun deposi ed [1].
Abso p ion plays a pi o al ole in shaping he c ys allini y and mic os uc u e o
he esul ing ilms. I encompasses wo dis inc o ms: physical abso p ion and chemical
abso p ion, which di e in he s eng h o a omic in e ac ion. The in e play be ween
hese in e ac ions can be elucida ed h ough he Lenna d–Jones cu e, which desc ibes he
epulsi e and a ac i e o ces be ween a oms [1].
Abso p ion plays a pi o al ole in shaping he c ys allini y and mic os uc u e o
he esul ing ilms. I encompasses wo dis inc o ms: physical abso p ion and chemical
abso p ion, which di e in he s eng h o a omic in e ac ion. The in e play be ween hese
in e ac ions can be elucida ed h ough he Lenna d–Jones cu e (Figu e 1), which desc ibes
he epulsi e and a ac i e o ces be ween a oms [1].
Ma e ials 2024, 17, x FOR PEER REVIEW 3 o 32
Figu e 1. Lenna d–Jones cu e.
Typically cha ac e ized by wo minima, he Lenna d–Jones cu e illus a es he a -
ac i e and epulsi e o ces be ween a oms, co esponding o he minimum po en ial en-
e gy o in e ac ion. The i s minimum signi ies long- ange a ac i e in e ac ions, while
he second ep esen s epulsi e o ces a ising om elec on cloud o e lap a ound he a -
oms [1].
𝑉(𝑟) = 4𝜀[(𝜎
𝑟 )12 − (𝜎
𝑟 )6]
(1)
The combined e ec o he minima in he Lenna d–Jones cu e esul s in a po en ial
well wi h a minimum a a speci ic sepa a ion dis ance, which is c ucial o de e mining
molecula in e ac ion s abili y. The dep h o his po en ial well (ε in he Lenna d–Jones
cu e) signi ies he s eng h o a ac i e o ces, while he posi ion o he minima (σ in he
Lenna d–Jones cu e) de ines he ange o in e ac ion.
In physical abso p ion, pa icles a e a ac ed o he su ace h ough a ac i e in e -
ac ions, cha ac e ized by an abso p ion po en ial (Ep). As e apo a ed pa icles o he de-
posi ed ma e ial app oach he su ace, hey lose kine ic ene gy and o m bonds wi h su -
ace a oms, he eby educing ee su ace ene gy. On he o he hand, chemical abso p ion
in ol es he c ea ion o chemical bonds be ween he elec ons o e apo a ed pa icles and
subs a e a oms, deno ed by an abso p ion po en ial (Ec).
The p ocess o physical and chemical abso p ion can be isualized as a unc ion o
dis ance e sus po en ial. Op imal physical abso p ion occu s a a g ea e dis ance om
he su ace compa ed o chemical abso p ion. The dispa i y be ween hese wo s ages is
delinea ed by he e ec i e ene gy ba ie (Ea) (Figu e 2).
Figu e 2. Physical abso p ion as an a ac i e in e ac ion.
Figu e 1. Lenna d–Jones cu e.
Ma e ials 2024,17, 1436 3 o 32
Typically cha ac e ized by wo minima, he Lenna d–Jones cu e illus a es he a ac-
i e and epulsi e o ces be ween a oms, co esponding o he minimum po en ial ene gy
o in e ac ion. The i s minimum signi ies long- ange a ac i e in e ac ions, while he
second ep esen s epulsi e o ces a ising om elec on cloud o e lap a ound he a oms [
1
].
V( ) = 4ε[(σ
)12 −(σ
)6](1)
The combined e ec o he minima in he Lenna d–Jones cu e esul s in a po en ial
well wi h a minimum a a speci ic sepa a ion dis ance, which is c ucial o de e mining
molecula in e ac ion s abili y. The dep h o his po en ial well (
ε
in he Lenna d–Jones
cu e) signi ies he s eng h o a ac i e o ces, while he posi ion o he minima (
σ
in he
Lenna d–Jones cu e) de ines he ange o in e ac ion.
In physical abso p ion, pa icles a e a ac ed o he su ace h ough a ac i e in-
e ac ions, cha ac e ized by an abso p ion po en ial (Ep). As e apo a ed pa icles o he
deposi ed ma e ial app oach he su ace, hey lose kine ic ene gy and o m bonds wi h su -
ace a oms, he eby educing ee su ace ene gy. On he o he hand, chemical abso p ion
in ol es he c ea ion o chemical bonds be ween he elec ons o e apo a ed pa icles and
subs a e a oms, deno ed by an abso p ion po en ial (Ec).
The p ocess o physical and chemical abso p ion can be isualized as a unc ion o
dis ance e sus po en ial. Op imal physical abso p ion occu s a a g ea e dis ance om
he su ace compa ed o chemical abso p ion. The dispa i y be ween hese wo s ages is
delinea ed by he e ec i e ene gy ba ie (Ea) (Figu e 2).
Ma e ials 2024, 17, x FOR PEER REVIEW 3 o 32
Figu e 1. Lenna d–Jones cu e.
Typically cha ac e ized by wo minima, he Lenna d–Jones cu e illus a es he a -
ac i e and epulsi e o ces be ween a oms, co esponding o he minimum po en ial en-
e gy o in e ac ion. The i s minimum signi ies long- ange a ac i e in e ac ions, while
he second ep esen s epulsi e o ces a ising om elec on cloud o e lap a ound he a -
oms [1].
𝑉(𝑟) = 4𝜀[(𝜎
𝑟 )12 − (𝜎
𝑟 )6]
(1)
The combined e ec o he minima in he Lenna d–Jones cu e esul s in a po en ial
well wi h a minimum a a speci ic sepa a ion dis ance, which is c ucial o de e mining
molecula in e ac ion s abili y. The dep h o his po en ial well (ε in he Lenna d–Jones
cu e) signi ies he s eng h o a ac i e o ces, while he posi ion o he minima (σ in he
Lenna d–Jones cu e) de ines he ange o in e ac ion.
In physical abso p ion, pa icles a e a ac ed o he su ace h ough a ac i e in e -
ac ions, cha ac e ized by an abso p ion po en ial (Ep). As e apo a ed pa icles o he de-
posi ed ma e ial app oach he su ace, hey lose kine ic ene gy and o m bonds wi h su -
ace a oms, he eby educing ee su ace ene gy. On he o he hand, chemical abso p ion
in ol es he c ea ion o chemical bonds be ween he elec ons o e apo a ed pa icles and
subs a e a oms, deno ed by an abso p ion po en ial (Ec).
The p ocess o physical and chemical abso p ion can be isualized as a unc ion o
dis ance e sus po en ial. Op imal physical abso p ion occu s a a g ea e dis ance om
he su ace compa ed o chemical abso p ion. The dispa i y be ween hese wo s ages is
delinea ed by he e ec i e ene gy ba ie (Ea) (Figu e 2).
Figu e 2. Physical abso p ion as an a ac i e in e ac ion.
Figu e 2. Physical abso p ion as an a ac i e in e ac ion.
In he Volme –Webe g ow h mechanism ( e e o Figu e 3), ma e ial a oms a e ini ially
p esen as clus e s in he apo phase and subsequen ly condense on o he subs a e,
o ming a hin ilm [
2
]. This g ow h p ocess akes place a he a omic o molecula le el
and is signi ican ly impac ed by a ious subs a e pa ame e s, including empe a u e,
deposi ion a e, and he p ope ies o he deposi ed ma e ial, such as compa ibili y. The
Volme –Webe g ow h mechanism has p o en o be ad an ageous in he ab ica ion o hin
ilms o a ange o applica ions, including semiconduc o de ices like ansis o s, sola cell
uni s, and op ical ilms [2].
Ma e ials 2024, 17, x FOR PEER REVIEW 4 o 32
In he Volme –Webe g ow h mechanism ( e e o Figu e 3), ma e ial a oms a e ini-
ially p esen as clus e s in he apo phase and subsequen ly condense on o he subs a e,
o ming a hin ilm [2]. This g ow h p ocess akes place a he a omic o molecula le el
and is signi ican ly impac ed by a ious subs a e pa ame e s, including empe a u e,
deposi ion a e, and he p ope ies o he deposi ed ma e ial, such as compa ibili y. The
Volme –Webe g ow h mechanism has p o en o be ad an ageous in he ab ica ion o
hin ilms o a ange o applica ions, including semiconduc o de ices like ansis o s,
sola cell uni s, and op ical ilms [2].
Figu e 3. Volme –Webe model: (a) c ea ion o clus e s; (b) clus e s o deposi ed ma e ial.
In he an de Me we mechanism (see Figu e 4), hin ilm ma e ial is deposi ed on o
he subs a e in he o m o small pa icles gene a ed h ough nuclea ion and g ow h p o-
cesses in he apo phase [3]. These pa icles subsequen ly adhe e o he subs a e, whe e
hey u he de elop in o a con inuous ilm. The g ow h o hese pa icles is con ingen
upon ac o s such as he a e o ma e ial supply o he subs a e, subs a e empe a u e,
and he p ope ies o he me al ma e ial in ol ed [3].
Figu e 4. F ank– an de Me we model: (a) c ea ion o monolaye ; (b) 3D islands o deposi ed ma e ial.
Abso bed pa icles subsequen ly agg ega e in o clus e s o abso bed a oms, ini ia ing
he o ma ion o h ee-dimensional (3D) islands. These islands hen e ol e in o monolay-
e s o deposi ed ma e ial h ough a p ocess known as nuclea ion. P io o nuclea ion, he
g ow h o 3D islands is accompanied by he accumula ion o s ess, pa icula ly ension,
du ing he ini ial ilm o monolaye deposi ion [3].
In he con ex o laye deposi ion, he i s laye holds pa amoun impo ance. I s
cha ac e is ics a e delinea ed by c ucial pa icle size and an essen ial ene gy ba ie , o
Gibbs unc ion. These pa ame e s dic a e he ini ial s age o ilm g ow h and p o oundly
in luence subsequen laye o ma ion and ilm p ope ies (2).
∆G o al = 4
3𝜋𝑟3∆𝐺𝑣+4𝜋𝑟2𝛶
(2)
In he con ex o hin- ilm g ow h, Gibbs ee ene gy plays a c ucial ole in de e min-
ing he easibili y o he deposi ion p ocess. Gibbs ee ene gy (ΔG) is a undamen al con-
cep in he modynamics, ep esen ing he maximum amoun o wo k ha can be ob ained
om a sys em a cons an empe a u e and p essu e.
In he con ex o hin- ilm deposi ion, Gibbs ee ene gy change de e mines whe he
he p ocess is ene ge ically a o able o no . I ΔG < 0, he p ocess is spon aneous and can
p oceed wi hou he inpu o ex e nal ene gy. I ΔG > 0, he p ocess is non-spon aneous
and equi es he inpu o ene gy o occu .
Fo he g ow h o he i s laye in hin- ilm deposi ion, Gibbs ee ene gy plays a
c i ical ole in de e mining he easibili y o nuclea ion and adhesion o he deposi ed ma-
e ial o he subs a e. The dec ease in Gibbs ee ene gy associa ed wi h he o ma ion o
he i s laye indica es he s abili y o he sys em and he likelihood o success ul ilm
g ow h.
Figu e 3. Volme –Webe model: (a) c ea ion o clus e s; (b) clus e s o deposi ed ma e ial.
Ma e ials 2024,17, 1436 4 o 32
In he an de Me we mechanism (see Figu e 4), hin ilm ma e ial is deposi ed on o he
subs a e in he o m o small pa icles gene a ed h ough nuclea ion and g ow h p ocesses
in he apo phase [
3
]. These pa icles subsequen ly adhe e o he subs a e, whe e hey
u he de elop in o a con inuous ilm. The g ow h o hese pa icles is con ingen upon
ac o s such as he a e o ma e ial supply o he subs a e, subs a e empe a u e, and he
p ope ies o he me al ma e ial in ol ed [3].
Ma e ials 2024, 17, x FOR PEER REVIEW 4 o 32
In he Volme –Webe g ow h mechanism ( e e o Figu e 3), ma e ial a oms a e ini-
ially p esen as clus e s in he apo phase and subsequen ly condense on o he subs a e,
o ming a hin ilm [2]. This g ow h p ocess akes place a he a omic o molecula le el
and is signi ican ly impac ed by a ious subs a e pa ame e s, including empe a u e,
deposi ion a e, and he p ope ies o he deposi ed ma e ial, such as compa ibili y. The
Volme –Webe g ow h mechanism has p o en o be ad an ageous in he ab ica ion o
hin ilms o a ange o applica ions, including semiconduc o de ices like ansis o s,
sola cell uni s, and op ical ilms [2].
Figu e 3. Volme –Webe model: (a) c ea ion o clus e s; (b) clus e s o deposi ed ma e ial.
In he an de Me we mechanism (see Figu e 4), hin ilm ma e ial is deposi ed on o
he subs a e in he o m o small pa icles gene a ed h ough nuclea ion and g ow h p o-
cesses in he apo phase [3]. These pa icles subsequen ly adhe e o he subs a e, whe e
hey u he de elop in o a con inuous ilm. The g ow h o hese pa icles is con ingen
upon ac o s such as he a e o ma e ial supply o he subs a e, subs a e empe a u e,
and he p ope ies o he me al ma e ial in ol ed [3].
Figu e 4. F ank– an de Me we model: (a) c ea ion o monolaye ; (b) 3D islands o deposi ed ma e ial.
Abso bed pa icles subsequen ly agg ega e in o clus e s o abso bed a oms, ini ia ing
he o ma ion o h ee-dimensional (3D) islands. These islands hen e ol e in o monolay-
e s o deposi ed ma e ial h ough a p ocess known as nuclea ion. P io o nuclea ion, he
g ow h o 3D islands is accompanied by he accumula ion o s ess, pa icula ly ension,
du ing he ini ial ilm o monolaye deposi ion [3].
In he con ex o laye deposi ion, he i s laye holds pa amoun impo ance. I s
cha ac e is ics a e delinea ed by c ucial pa icle size and an essen ial ene gy ba ie , o
Gibbs unc ion. These pa ame e s dic a e he ini ial s age o ilm g ow h and p o oundly
in luence subsequen laye o ma ion and ilm p ope ies (2).
∆G o al = 4
3𝜋𝑟3∆𝐺𝑣+4𝜋𝑟2𝛶
(2)
In he con ex o hin- ilm g ow h, Gibbs ee ene gy plays a c ucial ole in de e min-
ing he easibili y o he deposi ion p ocess. Gibbs ee ene gy (ΔG) is a undamen al con-
cep in he modynamics, ep esen ing he maximum amoun o wo k ha can be ob ained
om a sys em a cons an empe a u e and p essu e.
In he con ex o hin- ilm deposi ion, Gibbs ee ene gy change de e mines whe he
he p ocess is ene ge ically a o able o no . I ΔG < 0, he p ocess is spon aneous and can
p oceed wi hou he inpu o ex e nal ene gy. I ΔG > 0, he p ocess is non-spon aneous
and equi es he inpu o ene gy o occu .
Fo he g ow h o he i s laye in hin- ilm deposi ion, Gibbs ee ene gy plays a
c i ical ole in de e mining he easibili y o nuclea ion and adhesion o he deposi ed ma-
e ial o he subs a e. The dec ease in Gibbs ee ene gy associa ed wi h he o ma ion o
he i s laye indica es he s abili y o he sys em and he likelihood o success ul ilm
g ow h.
Figu e 4. F ank– an de Me we model: (a) c ea ion o monolaye ; (b) 3D islands o deposi ed ma e ial.
Abso bed pa icles subsequen ly agg ega e in o clus e s o abso bed a oms, ini ia ing
he o ma ion o h ee-dimensional (3D) islands. These islands hen e ol e in o monolaye s
o deposi ed ma e ial h ough a p ocess known as nuclea ion. P io o nuclea ion, he
g ow h o 3D islands is accompanied by he accumula ion o s ess, pa icula ly ension,
du ing he ini ial ilm o monolaye deposi ion [3].
In he con ex o laye deposi ion, he i s laye holds pa amoun impo ance. I s
cha ac e is ics a e delinea ed by c ucial pa icle size and an essen ial ene gy ba ie , o
Gibbs unc ion. These pa ame e s dic a e he ini ial s age o ilm g ow h and p o oundly
in luence subsequen laye o ma ion and ilm p ope ies (2).
∆G o al =4
3π 3∆G +4π 2Y(2)
In he con ex o hin- ilm g ow h, Gibbs ee ene gy plays a c ucial ole in de e mining
he easibili y o he deposi ion p ocess. Gibbs ee ene gy (
∆
G) is a undamen al concep
in he modynamics, ep esen ing he maximum amoun o wo k ha can be ob ained om
a sys em a cons an empe a u e and p essu e.
In he con ex o hin- ilm deposi ion, Gibbs ee ene gy change de e mines whe he
he p ocess is ene ge ically a o able o no . I
∆
G < 0, he p ocess is spon aneous and can
p oceed wi hou he inpu o ex e nal ene gy. I
∆
G > 0, he p ocess is non-spon aneous
and equi es he inpu o ene gy o occu .
Fo he g ow h o he i s laye in hin- ilm deposi ion, Gibbs ee ene gy plays a
c i ical ole in de e mining he easibili y o nuclea ion and adhesion o he deposi ed
ma e ial o he subs a e. The dec ease in Gibbs ee ene gy associa ed wi h he o ma ion
o he i s laye indica es he s abili y o he sys em and he likelihood o success ul
ilm g ow h.
This compa ibili y c i e ion is elucida ed by Young’s equa ion [
4
], a undamen al
p inciple in he science o we ing ha desc ibes he ela ionship be ween su ace ensions
a he in e ace o a liquid–solid sys em. Young’s equa ion p o ides a simple ye elegan
ep esen a ion o he o ces in ol ed in we ing phenomena, acili a ing an unde s anding
o a ious p ac ical applica ions such as adhesion and he beha io o di e en ma e ials [
4
].
Y s <Y s +Y s ×cosθ(3)
Young’s Equa ion (3) ela es he p essu e di e ence ac oss he cu ed su ace o he
ension on he su ace.
Y s <Y s +Y s (4)
Y s=Y s +Y s (5)
The Young equa ion s ipula es ha op imal monolaye g ow h occu s when he con ac
angle app oaches 0 deg ees o ini ia es wi hin he ange o 0 o 90 deg ees [
4
]. This
phenomenon, known as epi axy, encompasses wo in ica e p ocesses: homoepi axy, whe e
Ma e ials 2024,17, 1436 5 o 32
he deposi ed laye mi o s he ma e ial composi ion o he subs a e, and he e oepi axy,
whe e he deposi ed laye ea u es a dis inc ma e ial composi ion om he subs a e [4].
Homoepi axy e e s o a ilm ha sha es he same c ys allog aphic o ien a ion and
composi ion as he subs a e, e ec i ely ex ending he subs a e i sel [
5
,
6
]. Con e sely, in
he e oepi axy, he subs a e closely ma ches he o ien a ion o he deposi ed monolaye .
When pa icles a e abso bed on o he subs a e, su ace ene gy dec eases as he ac i e
su ace a ea diminishes [
7
]. Howe e , he deposi ed monolaye ypically possesses a
dis inc chemical s uc u e om he subs a e. P e e en ial binding o abso ben pa icles
o he subs a e a he han o each o he occu s when la ice pa ame e s a e equal o o
e y simila o he deposi ed ma e ial. When la ice cons an s di e , i o en esul s in
he o ma ion o s ess wi hin he deposi ed laye , which in u n p omo es he g ow h o
island-laye s uc u es a he han a con inuous monolaye [7].
∆G∗={16πY
3(∆G +ω2)}{2−3cosθ+cos(θ)3
4}(6)
The g ow h o a monolaye induces s ess due o di e ences in la ice pa ame e s,
leading o he accumula ion o s ain po en ial. This s ain po en ial ampli ies as mo e
monolaye s s ack up. As his s ain po en ial inc eases beyond a c i ical alue and he s ain
ene gy o he deposi ed laye su passes su ace ension, subsequen laye s can o m abo e
he ini ially deposi ed one. Du ing deposi ion, a ious de ec s may a ise, consequen ly
ele a ing he o e all Gibbs ene gy. Consequen ly, he size o he ini ially deposi ed laye
dec eases, a ibu ed o a educ ion in he ene gy ba ie o ini ial abso p ion (6) [7].
Nuclea ion and he simul aneous combina ion o monolaye s culmina e in he o -
ma ion o he inal mic os uc u e o he hin ilm. The mo phology o his s uc u e is
in luenced by subs a e empe a u e, subs a e mo phology, deposi ion a e, and su ace
di usion. By manipula ing o combining hese ac o s, he esul ing ilm mo phology can
be ca ego ized in o ou subclasses: Z1, ZT, Z2, and Z3 [8].
As he ilm con inues o g ow, i begins o o m basic s uc u al zones, which a e no
uni o mly dis ibu ed ac oss i s su ace, wi h hickness a ying due o blu ed ansi ions
be ween zones. Some imes, dis inguishing be ween hese zones p o es challenging. The e
a e ou undamen al s uc u al zone models—Z1, ZT, Z3, and Z4 [
8
]—dependen on
c ucial pa ame e s such as he a io o no malized empe a u e (Ts) o su ace di usion
(Tm) and he ene gy ans e o a oms a high ene gy. The ansi ion be ween Z1 and Z4
models hea ily elies on his a io. Fo ins ance, he Z1 model p edomina es when he
Ts/Tm a io is low (<0.1 Pa) compa ed o su ace di usion. Su ace di usion mani es s as
columns app oxima ely 10–20 nm in diame e o ming on he subs a e su ace, sepa a ed
by small gaps se e al nanome e s wide. On his pa icula s uc u e, an a ay o cones is
supe imposed, wi h he cones coalescing in o domains ac oss he su ace, he size o which
inc eases wi h hickness [2].
This mode closely esembles Z1 (wi h Ts/Tm~0.1 Pa), cha ac e ized by negligible
su ace di usion. Howe e , unlike Z1, his mode lacks he p esence o gaps and domain
cones. Z1 mode o en esul s in signi ican po osi y, leading o issues such as leakage, low
elec ical conduc i i y, and inhomogenei y, which a e gene ally undesi able. Howe e ,
po osi y can be ad an ageous in ce ain echnological applica ions, such as gas leakage
de ec o s, ca aly ic eac ions, and ligh abso p ion [2].
As he a io o no malized empe a u e o su ace di usion (Ts/Tm) inc eases (Ts/Tm
> 0.3 Pa), he sys em ansi ions in o he Z2 mode. In his mode, su ace di usion becomes
inc easingly signi ican . Column o ma ion in Z2 mode is cha ac e ized by igh g ain
bounda ies be ween he columns. The diame e o hese columns inc eases wi h he no mal-
ized empe a u e and su ace di usion a io, esul ing in a mo e e ined c ys alline s uc u e.
The ansi ion om Z1 o Z2 modes is mo e p onounced a highe empe a u es [2].
In he Z3 mode (Ts/Tm > 0.5), he a io is e en la ge compa ed o he p e ious modes,
indica ing s onge su ace di usion. The su ace o Z3 mode appea s smoo h, albei wi h
sligh g oo es ac oss i s su ace. The mo phology o his mode is cha ac e ized by equiaxed

Ma e ials 2024,17, 1436 6 o 32
g ain c ys als, wi h he size o hese c ys als being p opo ional o he hickness o he
deposi ed ilm [2].
3. Modeling Techniques
3.1. Quan um Mechanical Modeling
Quan um mechanical modeling is a powe ul heo e ical amewo k used o s udy he
elec onic s uc u e and p ope ies o magne ic ilms on subs a es. I p o ides a de ailed
unde s anding o he beha io o elec ons in he sys em, allowing o he calcula ion o
ene gy le els, wa e unc ions, and elec onic in e ac ions [9].
Quan um mechanical models, oo ed in he ounda ional p inciples o physics, o e
unpa alleled accu acy in cha ac e izing he beha io o indi idual elec ons and hei
in ica e in e play [
10
]. This p ecision is pa amoun o cap u ing nuances in magne ic
in e ac ions wi hin ilms, encompassing a ibu es such as spin o ien a ions, exchange
in e ac ions, and magne ic aniso opy [
11
]. Quan um mechanical amewo ks p o ide a
mic oscopic lens o a hom he unde lying quan um s a es and elec on dis ibu ions ha
unde lie magne ic p ope ies. This unde s anding ex ends o phenomena like magne ism
swi ching, domain wall mo emen , and magne ic hys e esis, os e ing deepe comp ehen-
sion [11]. The p edic i e capabili y o quan um mechanical simula ions guides he design
and explo a ion o no el ma e ials and con igu a ions. By en isaging uncha ed magne ic
ilm p ope ies, hese models expedi e ma e ial disco e y and op imiza ion, p opelling
ad ancemen s in magne ic-based echnologies [
12
]. Quan um mechanical simula ions
adep ly encapsula e phenomena like unneling and supe posi ion, c i ical o decoding and
manipula ing magne ic beha io on he nanoscale. Such e ec s hold pa icula ele ance
in domains such as quan um compu ing and nanoscale magne ic de ices [12].
The compu a ional demands and ime cons ain s associa ed wi h quan um mechan-
ical calcula ions, pa icula ly o ex ensi e sys ems o p olonged ime ames, p esen
challenges. These limi a ions cu ail he size o sys ems amenable o accu a e model-
ing and he empo al ex en s ha can be simula ed. To accommoda e he in icacies o
eal-wo ld scena ios, quan um mechanical models o en necessi a e app oxima ions and
abs ac ions [
12
]. These expedien s can in oduce disc epancies, pa icula ly in ins ances
ea u ing obus co ela ions, in ica e geome ies, o ex eme condi ions. Magne ic ilms
o en encompass a medley o ma e ials, spanning mul iple elemen s, de ec s, and in e acial
egions. Accommoda ing hese mul i ace ed cha ac e is ics h ough quan um mechanical
modeling demands ad anced heo e ical and compu a ional app oaches. The e icacy o
quan um mechanical models hinges on pa ame e s gleaned om empi ical obse a ions o
o he sou ces [
12
]. The sensi i i y o model p edic ions o hese pa ame e s can engende
unce ain ies in he esul an ou comes. While quan um mechanical models yield p ecise
nume ical ou comes, hei in ui i e in e p e a ion o he unde lying physical mechanisms
migh no always be eadily disce nible. Ex ac ing meaning ul insigh s may necessi a e a
comp ehensi e g asp o quan um physics p inciples (Figu e 5).
Cen al o he e icacy o quan um mechanical modeling is i s p owess in explica ing
elec onic s uc u e and in e molecula dynamics wi hin ma e ials wi h unpa alleled p e-
cision (Figu e 6). The dimensionali y- es ic ed na u e o hin ilms con e s upon hem
dis inc i e elec onic ai s ha a e di e gen om hei bulk coun e pa s. This elucida ion
augmen s ou undamen al unde s anding o hin- ilm phenomena and concu en ly s im-
ula es he concep ualiza ion o inno a i e de ices endowed wi h bespoke unc ionali ies.
Mo eo e , quan um mechanical modeling u nishes a i ual c ucible o un a eling he
mechanis ic, he mal, and op ical a ibu es o hin ilms. The gamu o he mal anspo
phenomena, pi o al in nanoscale hea managemen , can be me iculously un eiled, he eby
channeling he de elopmen o he mally conduc i e ma e ials wi h ele a ed e iciency.
Fu he mo e, he op ical signa u es inhe en o hin ilms, including abso p ion, e lec ion,
and emission, can be me iculously o e old, he eby engende ing he design o ad anced
op ical coa ings and senso s wi h ailo ed ligh manipula ion capabili ies.
Ma e ials 2024,17, 1436 7 o 32
Ma e ials 2024, 17, x FOR PEER REVIEW 7 o 32
modeling demands ad anced heo e ical and compu a ional app oaches. The e icacy o
quan um mechanical models hinges on pa ame e s gleaned om empi ical obse a ions
o o he sou ces [12]. The sensi i i y o model p edic ions o hese pa ame e s can engen-
de unce ain ies in he esul an ou comes. While quan um mechanical models yield p e-
cise nume ical ou comes, hei in ui i e in e p e a ion o he unde lying physical mecha-
nisms migh no always be eadily disce nible. Ex ac ing meaning ul insigh s may neces-
si a e a comp ehensi e g asp o quan um physics p inciples (Figu e 5).
Figu e 5. Simula ed TiO2 hin- ilm g ow h.
Cen al o he e icacy o quan um mechanical modeling is i s p owess in explica ing
elec onic s uc u e and in e molecula dynamics wi hin ma e ials wi h unpa alleled p e-
cision (Figu e 6). The dimensionali y- es ic ed na u e o hin ilms con e s upon hem
dis inc i e elec onic ai s ha a e di e gen om hei bulk coun e pa s. This elucida-
ion augmen s ou undamen al unde s anding o hin- ilm phenomena and concu en ly
s imula es he concep ualiza ion o inno a i e de ices endowed wi h bespoke unc ional-
i ies. Mo eo e , quan um mechanical modeling u nishes a i ual c ucible o un a eling
he mechanis ic, he mal, and op ical a ibu es o hin ilms. The gamu o he mal
anspo phenomena, pi o al in nanoscale hea managemen , can be me iculously un-
eiled, he eby channeling he de elopmen o he mally conduc i e ma e ials wi h ele-
a ed e iciency. Fu he mo e, he op ical signa u es inhe en o hin ilms, including ab-
so p ion, e lec ion, and emission, can be me iculously o e old, he eby engende ing he
design o ad anced op ical coa ings and senso s wi h ailo ed ligh manipula ion capabil-
i ies.
Figu e 6. Simula ed ilm subs a e in e ace.
Figu e 5. Simula ed TiO2 hin- ilm g ow h.
Ma e ials 2024, 17, x FOR PEER REVIEW 7 o 32
modeling demands ad anced heo e ical and compu a ional app oaches. The e icacy o
quan um mechanical models hinges on pa ame e s gleaned om empi ical obse a ions
o o he sou ces [12]. The sensi i i y o model p edic ions o hese pa ame e s can engen-
de unce ain ies in he esul an ou comes. While quan um mechanical models yield p e-
cise nume ical ou comes, hei in ui i e in e p e a ion o he unde lying physical mecha-
nisms migh no always be eadily disce nible. Ex ac ing meaning ul insigh s may neces-
si a e a comp ehensi e g asp o quan um physics p inciples (Figu e 5).
Figu e 5. Simula ed TiO2 hin- ilm g ow h.
Cen al o he e icacy o quan um mechanical modeling is i s p owess in explica ing
elec onic s uc u e and in e molecula dynamics wi hin ma e ials wi h unpa alleled p e-
cision (Figu e 6). The dimensionali y- es ic ed na u e o hin ilms con e s upon hem
dis inc i e elec onic ai s ha a e di e gen om hei bulk coun e pa s. This elucida-
ion augmen s ou undamen al unde s anding o hin- ilm phenomena and concu en ly
s imula es he concep ualiza ion o inno a i e de ices endowed wi h bespoke unc ional-
i ies. Mo eo e , quan um mechanical modeling u nishes a i ual c ucible o un a eling
he mechanis ic, he mal, and op ical a ibu es o hin ilms. The gamu o he mal
anspo phenomena, pi o al in nanoscale hea managemen , can be me iculously un-
eiled, he eby channeling he de elopmen o he mally conduc i e ma e ials wi h ele-
a ed e iciency. Fu he mo e, he op ical signa u es inhe en o hin ilms, including ab-
so p ion, e lec ion, and emission, can be me iculously o e old, he eby engende ing he
design o ad anced op ical coa ings and senso s wi h ailo ed ligh manipula ion capabil-
i ies.
Figu e 6. Simula ed ilm subs a e in e ace.
Figu e 6. Simula ed ilm subs a e in e ace.
Quan um mechanical modeling also enables he calcula ion o a ious elec onic
p ope ies o magne ic ilms on subs a es [
13
]. These p ope ies include elec ic bands,
s uc u e, densi y o s a es, magne ic momen s, and magne ic en opy. Unde s anding
su ace changes can p o ide insigh in o he ilm’s magne ic beha io , i s esponse o
ex e nal magne ic ields, and he in luence o he subs a e on he elec onic s uc u e [
13
].
Con luence be ween expe imen al endea o s and quan um mechanical modeling can
po en ially o ches a e ans o ma i e shi s ac oss my iad domains.
Quan um mechanics me hods combined wi h mic omagne ic simula ions enable
he explo a ion and ealiza ion o no el spin onic de ice concep s, such as sky mion-
based de ices, spin-wa e de ices, and opological insula o -based de ices, wi h unique
unc ionali ies and enhanced pe o mance [14].
Addi ionally, quan um mechanical modeling can p o ide in o ma ion abou ilm–
subs a e in e aces. I allows o he ins iga ion o bonding cha ac e is ics, cha ge ans e ,
and in e acial elec onic s a es. Quan um mechanical modeling, pa icula ly DFT calcula-
ions [
15
], has been widely used in he s udy o magne ic ilms on subs a es. By accu a ely
cap u ing he elec ic s uc u e and i s p ope ies, his app oach p o ides aluable insigh
and p edic ions, guiding expe imen al in es iga ions and he design o magne ic ma e ials
and de ices [
15
]. Quan um mechanical modeling is a i al ool o unde s anding he
beha io o magne ic ilms on subs a es a he a omic and elec onic le els. I o e s a heo-
e ical ounda ion o s udying elec onic s uc u es, magne ic p ope ies, and in e acial
e ec s, con ibu ing o he ad ancemen o magne ic ilm echnology [16] (Table 1).
Elec onic s uc u e calcula ions a e compu a ional me hods used in quan um mechan-
ical modeling o p edic and analyze he beha io o a oms, molecules, and ma e ials a
he a omic and molecula le el. These calcula ions aim o sol e he Sch ödinge equa ion,
which desc ibes he beha io o quan um mechanical sys ems, pa icula ly he mo ion
o elec ons wi hin a oms and molecules. The Sch ödinge equa ion, howe e , canno be
Ma e ials 2024,17, 1436 8 o 32
sol ed analy ically o sys ems wi h mo e han one elec on, excep o a ew simple cases.
The e o e, elec onic s uc u e calcula ions ely on app oxima ions and nume ical me hods
o ind solu ions ha a e accu a e enough o be use ul o unde s anding chemical and
physical p ope ies.
Table 1. Ad an ages and disad an ages o quan um mechanics me hods.
Ad an ages Disad an ages
-E icien o la ge sys ems, hough i o e simpli ies de ails.
-Accu acy may be comp omised, especially o complex dynamics.
-Enables pa ame e s udy. -Requi es ca e ul pa ame e uning o s abili y.
-Cap u es ealis ic beha io s. -Limi ed in cap u ing ine-scale physics.
-Scales well o la ge sys ems. -In e p e a ion may be complex, equi ing alida ion.
-Compa ible wi h expe imen al da a. -Simpli ied models may no cap u e all dynamics accu a ely.
The mo i a ion behind elec onic s uc u e calcula ions lies in hei abili y o p o ide
de ailed insigh s in o he elec onic p ope ies o ma e . Some o he ad an ages o using
elec onic s uc u e calcula ions include:
Elec onic s uc u e calcula ions allow scien is s o gain a p o ound unde s anding o
chemical bonding in molecules and ma e ials. By analyzing he dis ibu ion o elec ons
and hei in e ac ions, esea che s can p edic molecula geome ies, bond s eng hs, and
eac i i y. These calcula ions can accu a ely p edic a ious molecula p ope ies such as
ene gy le els, elec onic spec a, dipole momen s, and pola izabili ies. In ma e ials science,
elec onic s uc u e calcula ions play a pi o al ole in p edic ing and designing ma e ials
wi h desi ed p ope ies. By calcula ing elec onic band s uc u es, densi y o s a es, and
o he elec onic p ope ies, esea che s can iden i y ma e ials wi h speci ic elec onic and
op ical cha ac e is ics o a ious applica ions, including elec onics, pho o ol aics, and
ca alysis. Elec onic s uc u e calcula ions p o ide insigh s in o he mechanisms o chemical
eac ions by simula ing he elec onic s uc u e o eac an s, in e media es, and ansi ion
s a es. Elec onic s uc u e calcula ions can complemen expe imen al obse a ions by
p o iding de ailed insigh s in o he unde lying elec onic s uc u e o molecules and
ma e ials. Th ough compu a ional me hods, esea che s can explo e a wide ange o
molecula and ma e ial p ope ies wi hou he need o expensi e and ime-consuming
expe imen al p ocedu es. Elec onic s uc u e calcula ions can be applied o a di e se ange
o sys ems, om simple molecules o complex ma e ials.
In he ealm o quan um mechanics, a dis inc a enue o inqui y in ol es he o -
mula ion o heo e ical amewo ks aimed a sc u inizing he cha ac e is ics o in ica e
molecula sys ems, denomina ed as quan um molecula dynamics [
17
]. This app oach
elies upon he applica ion o he Bo n–Oppenheime app oxima ion, pos ula ing he seg-
ega ion o a iables in o subs an ial nuclei and agile elec ons. The quan um molecula
dynamics me hodology, as in oduced by Ca –Pa inello, en ails he u iliza ion o unc ion
minimiza ion echniques [17].
The concep unde lying Ca –Pa inello molecula dynamics (CPMD) in ol es he
simul aneous compu a ion o elec onic and a omic subsys ems. The ajec o y o nuclei is
dic a ed by he ensemble o hei espec i e coo dina es {R
i
|i = 1,
. . .
,N
n
}. The elec onic
deg ees o eedom a e de ined by a collec ion o quan um-mechanical wa e unc ions
{Ψj|j = 1,. . .,Ne} [17].
Mi
..
Ri=−∂E
∂Ri
=Fi(7)
µ..
ψj( , ) = −ˆ
Hψj( , ) + ∑
k∧jkψk( , )(8)
In he gi en con ex , he symbol Fi ep esen s he esul an o ce exe ed on he a om.
E and ˆ
H ep esen he ene gy unc ional and he Kohn–Sham Hamil onian.
µdeno es he ic i ious elec on mass.
∧jk s ands o inde ini e Lag ange mul iplie s.
Ma e ials 2024,17, 1436 9 o 32
The selec ion o he ic i ious elec on mass is aimed a expedi ing he con e gence
o he algo i hm. In pa allel, he Lag ange mul iplie s play a pi o al ole in gene a ing
addi ional o ces ha en o ce he o hono mali y o he wa e unc ions [17] (7,8).
In he Ca –Pa inello heo y, he o al ene gy o a nanosys em is a unc ion o he
coe icien s in he expansion o he elec on wa e unc ion o e a gi en basis. When
speci ic coe icien s a e minimized, he sys em unde goes cooling and s abiliza ion. The
ini ial segmen o exp essions comp ises classical New onian mo ion equa ions o a se
o pa icles, which a e i e a i ely sol ed un il equilib ium is a ained [
18
]. Consis ency
be ween he ion and elec on subsys ems is achie ed h ough he concu en minimiza ion
o he ene gy unc ional. I is impo an o no e ha his me hod does no po ay he
genuine dynamics o he nanosys em bu a he simula es i s ic i ious e olu ion, leading
o an equilib ium s a e wi h s able ene gy o a mul i-elec on pa icle sys em [18].
The Ca –Pa inello molecula dynamics me hod is ecognized o i s abili y o accu-
a ely eplica e he p ope ies o semiconduc o and dielec ic ma e ials [
19
]. Howe e ,
o me allic sys ems in close p oximi y o he band gap, whe e nume ous s a es possess
closely spaced eigen alues, e en a mino al e a ion in o al ene gy can lead o signi ican
luc ua ions in elec on densi y. Mo eo e , he applicabili y o he Ca –Pa inello app oach
is cons ained o a limi ed numbe o a oms in a nanosys em due o he in ica e na u e o
sol ing he equa ions and he associa ed compu a ional expenses [19].
An al e na i e o Ca –Pa inello molecula dynamics is Bo n–Oppenheime molecula
dynamics (BOMD), ounded on he Bo n–Oppenheime app oxima ion [
20
]. This me hod
in ol es he seg ega ion o he nuclea and elec onic subsys em desc ip ions. The mo ion
o he nuclei is go e ned by classical mechanical equa ions, while he ene gy and o ces
ac ing on hem a e compu ed by sol ing he Sch ödinge equa ion o elec onic wa e
unc ions a each ime s ep. This app oach is ad an ageous in managing he complexi y
o me allic sys ems and ci cum en ing compu a ional challenges associa ed wi h la ge
nanosys ems [20].
Mi
..
Ri=−∇i[min
ψ1,...,ψNe
E(R1, . . . , RNn;ψ1, . . . , ψNe)](9)
In he gi en con ex , he o al ene gy (E) is subjec o minimiza ion h ough he op i-
miza ion o elec on wa e unc ions a ixed nuclea coo dina es [
21
]. Typically, his mini-
miza ion is achie ed by sol ing he Kohn–Scham equa ions o employing mul idimensional
op imiza ion algo i hms (9). The BOMD me hod, which in ol es he seg ega ion o nuclea
and elec onic subsys em desc ip ions, p o es highly accu a e and can e ec i ely desc ibe
he p ope ies o dielec ics and me als. Howe e , bo h BOMD and CPMD me hods incu
subs an ial compu a ional cos s, limi ing hei applica ion o smalle nanosys ems [21].
To add ess his limi a ion and enhance he capaci y o he sys em unde in es iga ion,
e o s ha e been di ec ed owa d he de elopmen o me hods ha combine classical
molecula modeling wi h elec onic s uc u e calcula ions [
21
]. This app oach is exem-
pli ied by embedded a om po en ials such as he Embedded A om Model (EAM) and
Modi ied Embedded A om Me hod (MEAM). The Embedded A om Me hod is de i ed
om heo e ical conside a ions o he elec on densi y unc ional and desc ibes nanosys em
beha io h ough a se o equa ions, o e ing a means o educe compu a ional complexi y
and ex end i s applicabili y o la ge sys ems [
21
]. The EAM o igina es om he heo e ical
p inciples o he elec on densi y unc ional and a icula es he beha io o a nanosys em
h ough he ollowing se o equa ions:
Mi
..
Ri=−∂E
∂Ri
=Fi, (10)
Ui=Fρ.
i+∑
J;1·=j
ϕRi−Rj.(11)
Ma e ials 2024,17, 1436 16 o 32
This compu a ional demand can limi he easibili y o s udying ex emely la ge sys ems
o conduc ing simula ions o e long pe iods. While p o iding s a is ical accu acy, Mon e
Ca lo simula ions o en in ol e simpli ica ions o he unde lying physical in e ac ions.
These app oxima ions can po en ially a ec he ideli y o esul s, pa icula ly in cases
whe e quan um mechanical e ec s play a c ucial ole. Mon e Ca lo simula ions ypically
assume ha he sys em eaches he mal equilib ium du ing he simula ion [45].
Howe e , his assump ion may no hold in scena ios in ol ing non-equilib ium
p ocesses o apid changes in sys em dynamics. Mon e Ca lo simula ions a e gene ally
classical in na u e and may no ully cap u e quan um mechanical e ec s signi ican in
ce ain magne ic sys ems. Quan um phenomena such as spin unneling o en anglemen
may no be adequa ely ep esen ed in classical Mon e Ca lo simula ions [46].
In e p e ing he esul s o Mon e Ca lo simula ions can be challenging, especially
when dealing wi h complex sys ems. Ex ac ing meaning ul physical insigh s om he
s a is ical da a gene a ed by simula ions necessi a es a p o ound unde s anding o bo h he
simula ion echnique and he magne ic phenomena unde in es iga ion [46].
Despi e hese challenges, Mon e Ca lo simula ions emain a powe ul ool in he s udy
o magne ic ilms, p o iding aluable insigh s in o hei beha io and p ope ies. I is
impe a i e o esea che s o na iga e hese complexi ies wi h ca e, le e aging he s eng hs
o Mon e Ca lo simula ions while mi iga ing hei limi a ions, o maximize he u ili y o
his compu a ional app oach in ad ancing ou unde s anding o magne ic sys ems [47].
By combining Mon e Ca lo simula ions wi h expe imen al da a and o he modeling
echniques, esea che s can e ine hei unde s anding o hin- ilm g ow h mechanisms [
48
].
Op imiza ion o he posi ion p ocesses and p edic ion o he p ope ies o he esul ing ilms
imp o es he in eg a ion o simula ion and expe imen a ion and enables mo e e icien and
a ge ed de elopmen o ilms o a wide ange o applica ions [48] (Table 2).
Table 2. Ad an ages and disad an ages o Mon e Ca lo simula ions.
Ad an ages Disad an ages
-Flexible o complex sys ems. -High compu a ional cos o la ge sys ems o
long simula ions.
-P o ides accu a e esul s wi h
p ope implemen a ion.
-Resul s a e subjec o s a is ical e o s,
equi ing mul iple uns o eliabili y.
-E icien o ce ain p oblems. -Challenges in accu a ely modeling
complex in e ac ions.
-Allows explo ing pa ame e space. -May no cap u e all eal-wo ld aspec s.
-Inco po a es ealis ic models. -Ou comes a e sensi i e o ini ial condi ions.
Mon e Ca lo simula ion and modeling o dynamic simula ion shed ligh on dynamic
beha io , he mal e ec s, and he mal phenomena. These modeling app oaches play a
c ucial ole in elucida ing he magne ic p ope ies, anspo beha io , and esponse o
magne ic ilms on subs a es, he eby acili a ing he design and op imiza ion o mag-
ne ic de ices.
3.3. Densi y Func ional Theo y Calcula ions
Densi y Func ional Theo y (DFT) calcula ions, a co ne s one o mode n compu a ional
chemis y and physics, ha e eme ged as a ans o ma i e app oach wi h a - eaching
implica ions o di e se scien i ic domains. Densi y unc ional heo y (DFT) calcula ions a e
widely used compu a ional me hods in scien i ic esea ch o in es iga ing he p ope ies
o ma e ials, including hin ilms. DFT p o ides e y aluable insigh in o he elec onic
s uc u e, chemical bonds, and physical cha ac e is ics o hese ilms [49].
DFT is based on he p inciple ha he elec onic densi y con ains all he necessa y in-
o ma ion abou he sys em. By sol ing he Sch ödinge equa ions o sel -consis ency, DFT
allows o he de e mina ion o he g ound-s a e p ope ies and equilib ium con igu a ion
o he magne ic ilms [50].

Ma e ials 2024,17, 1436 17 o 32
The co e p inciples o densi y- unc ional heo y encompass he Hohenbe g–Kohn and
Kohn–Sham heo ems. The o iginal Hohenbe g–Kohn and Kohn–Sham heo ems eadily
lend hemsel es o expansi e adap a ion beyond hei ini ial o mula ions, spanning a
b oad spec um o physical scena ios.
In DFT calcula ions, he elec ic s uc u e o he ilm subs a e sys em is desc ibed by
he elec on densi y, which is go e ned by he Kohn–Sham equa ion [
50
] (Equa ion (17))
as ollows:
ˆ
HksΨi( )="−¯h2
2m∇2+Ve ( )#Ψi( ) = εiΨi( )(17)
whe e:
Ψi( )is he wa e unc ion o he i elec on.
εiis he ene gy o he i elec on.
∇2is he Laplacian ope a o .
Ve ( )
is he e ec i e po en ial, which includes he ex e nal po en ial due o he a omic
nuclei and any addi ional ex e nal po en ial p esen in he sys em.
The Kohn–Sham equa ion is he undamen al equa ion in densi y unc ion heo y o
he beha io o elec ons in a ma e ial, including hin ilms [
50
]. This equa ion ep esen s
an e ec i e single-pa icle p oblem, whe e he elec ons mo e in an e ec i e manne ha
includes he in e ac ion wi h he a omic nuclei and he exchange-co ela ion po en ial. The
exchange-co ela ion po en ial accoun s o he e ec s o elec on–elec on in e ac ions,
which a e challenging o desc ibe exac ly and a e o en app oxima ed in p ac ical calcula-
ions. The Kohn–Sham equa ion allows us o map he in e ac ing many elec on p oblems
o a se o non-in e ac ing single elec on equa ions [50].
To sol e he Kohn–Sham equa ions, a ious nume ical echnologies can be employed,
such as blending o weigh basis se s, localiza ion o a omic o bi als, o eal-space g ids [
50
].
These echniques disc e ize he elec onic wa e unc ions and pe o m in e ac i e calcula-
ions o con e ge on o he sel -consis en solu ion [50].
The co e p inciples o densi y- unc ional heo y encompass he Hohenbe g–Kohn and
Kohn–Sham heo ems. The o iginal Hohenbe g–Kohn and Kohn–Sham heo ems eadily
lend hemsel es o expansi e adap a ion beyond hei ini ial o mula ions, spanning a
b oad spec um o physical scena ios.
The Landau–Li shi z–Gilbe calcula ions o hin ilms in ol e se e al case s eps.
Fi s , geome y op imiza ion is pe o med o de e mine he op imized a omic posi ion in
he la ice pa ame e s o he hin- ilm s uc u e. This in ol es minimizing he o al ene gy
o he sys em by i e a i ely adjus ing he a omic posi ion un il cohe ence is achie ed [
51
].
The o al ene gy is ypically calcula ed using app op ia e exchange-co ela ion unc ions,
such as he gene aliza ion g adien app oxima ion (GGA) o hyb id unc ionals. Nex , he
elec onic s uc u e calcula ion is conduc ed. Once he op imiza ion geome y is ob ained,
he elec onic s uc u e o he hin ilm is compu ed [
51
]. This s ep in ol es sol ing
he Kohn–Sham equa ions (Hohenbe g–Kohn heo em), which desc ibe he beha io o
he elec ons in he sys em. By sol ing his equa ion sel -consis en ly (Equa ion (18)),
in o ma ion abou he ene gy le els, band s uc u e, and densi y o he s a es o he ilm
can be ob ained [52].
ˆ
He=Σi(−¯h2
2m
∂2
∂ i2−ΣI
ZIe2
 i−Rj
) + ∑i<j
e2
 i− j
(18)
He e,
ˆ
He
ep esen s he elec onic Hamil onian. The i s e m ep esen s he kine ic
ene gy o elec ons, wi h
¯h2
2m∂2
∂ i2
being he kine ic ene gy ope a o . The second e m is he
po en ial ene gy due o he in e ac ion o he i- h elec on wi h a nucleus loca ed a Rj, gi en
by
ZIe2
| i−Rj|
. The hi d e m is he po en ial ene gy due o he elec on–elec on in e ac ion,
gi en by e2
| i− j| o all dis inc pai s i,j o elec ons.
Ma e ials 2024,17, 1436 18 o 32
In DFT, he elec on densi y
ρ
( ) se es as he undamen al a iable, in con as o
he many-body wa e unc ion used in adi ional quan um mechanics [
53
]. The elec on
densi y is conside ably mo e manageable han he many-body wa e unc ion since i
depends on only h ee spa ial coo dina e a iables, i espec i e o he numbe o elec ons
in he sys em. This simpli ica ion has been jus i ied by he Hohenbe g–Kohn heo em,
which es ablishes a one- o-one co espondence be ween he ex e nal po en ial and he
g ound-s a e elec on densi y, ende ing he elec on densi y a su icien and con enien
desc ip o o he quan um sys em [53].
The Hohenbe g–Kohn heo em es ablishes a one- o-one co espondence be ween he
g ound-s a e elec on densi y and he ex e nal po en ial [
54
]. Consequen ly, i he g ound-
s a e elec on densi y is known, he ex e nal po en ial is uniquely de e mined. Fu he mo e,
he physical p ope ies associa ed wi h he g ound-s a e wa e unc ion can, in p inciple,
be unambiguously de i ed om he elec on densi y [
54
]. Speci ically, he kine ic and
elec on–elec on in e ac ion ene gies o he g ound s a e can be exp essed as uni e sal
unc ionals o he elec on densi y, deno ed as E
kin
[
ρ
] and E
ee
[
ρ
], espec i ely. The e m
“uni e sal” indica es ha he unc ional o ms a e independen o he speci ic ex e nal
po en ial [54].
The heo em also p o ides a a ia ional p inciple. When we de ine he ollowing
ene gy unc ional:
E [ρ] = Ekin[ρ] + Zρ( ) ( )d +Eee[ρ](19)
o some ex e nal po en ial V( ), he unc ional sa is ies he inequali y as ollows:
E [ρ]≥E [ρ0]=E0(20)
whe e
ρ0
(E
0
) is he g ound-s a e elec on densi y (ene gy) unde he po en ial V( ), espec-
i ely. The e o e, he g ound-s a e elec on densi y can be ob ained by sea ching o he
elec on densi y ha minimizes he ene gy unc ional E [ρ][55].
Al hough he a ia ional p inciple in Equa ion (19) appea s simple, a signi ican
challenge lies in he ac ha he exac o ms o he unc ionals, E
kin
[
ρ
] and E
ee
[
ρ
], a e
unknown. To add ess his issue, Kohn and Sham in oduced he concep o “o bi als” o
app oxima e he kine ic ene gy unc ional E
kin
[
ρ
]. This inno a i e app oach has pa ed he
way o pe o ming DFT calcula ions wi h su icien accu acy o p ac ical applica ions [
55
].
The e o e, he majo i y o mode n DFT implemen a ions u ilize he Kohn–Sham
scheme. We will del e in o his scheme in mo e de ail below. Rega ding he di ec a ia-
ional app oach, known as o bi al- ee DFT, which is less accu a e compa ed o he Kohn–
Sham app oach bu o e s he ad an age o as e compu a ions, cu en esea ch e o s a e
ocused on cons uc ing accu a e kine ic ene gy unc ionals. One s a egy in ol es s i ing
o ep oduce he Kohn–Sham kine ic ene gy as accu a ely as possible [55].
The Kohn–Sham scheme in oduces an auxilia y non-in e ac ing sys em designed
o yield he same elec on densi y as ha o he in e ac ing sys em. The non-in e ac ing
sys em is desc ibed by he single-pa icle Sch ödinge equa ion, commonly e e ed o as
he Kohn–Sham equa ion, as ollows:
&−¯h2
2m
∂2
∂ i2+ e ( )'ϕi( )=εiϕi( )(21)
whe e
e ( )
is an e ec i e po en ial,
ϕi( )
is he Kohn–Sham s a e, and
εi
is he Kohn–
Sham ene gy eigen alue. The elec on densi y is gi en by he ollowing equa ion:
ρ( ) = ∑i=1|ϕi( )|2(22)
The hin ilms a e cha ac e ized by hei su aces, which can signi ican ly in luence
hei p ope ies. To accu a ely model he hin- ilm su ace, a ious subs a e ea men s
a e applied. These ea men s may in ol e he use o he acuum egion o s imula e
Ma e ials 2024,17, 1436 19 o 32
an isola ed ilm, he addi ion o he acuum s ep o mimic he semi-in ini e ilm, o he
in oduc ion o app op ia e su ace e mina ions [
56
]. In many cases, a supe cell app oach
is employed o model he hin ilm. This app oach consis s o pe iodically epea ing he
hin ilm s uc u ed in wo dimensions while ea ing he hi d dimension as ini e. I allows
o he in es iga ion o he ilm p ope ies while minimizing in e ac ion be ween adjus ed
ilms [56] (Figu e 10).
Ma e ials 2024, 17, x FOR PEER REVIEW 20 o 32
Figu e 10. Su ace opology on di e en loca ions (a,b) o ch omium oxide ilm.
DFT p o ides a obus amewo k o accu a ely calcula ing he elec onic s uc u e
o ma e ials, including magne ic ilms [57]. I can p edic p ope ies such as band s uc-
u e, densi y o s a es, and magne ic momen s, o e ing insigh s in o he unde lying phys-
ics go e ning magne ic in e ac ions [58]. DFT can accu a ely cap u e magne ic in e ac-
ions by accoun ing o he a angemen o elec ons’ spins. This enables he s udy o spin
o ien a ions, exchange in e ac ions, and magne ic aniso opy, shedding ligh on he un-
damen al mechanisms d i ing magne ic beha io in ilms [58]. DFT is e sa ile and appli-
cable o a wide ange o ma e ials, om simple me als o complex compounds. I can be
employed o explo e he magne ic p ope ies o a ious composi ions, c ys al s uc u es,
and ilm hicknesses, aiding in he design and op imiza ion o no el magne ic ilms [59].
DFT calcula ions allow o p edic i e modeling o magne ic ilms, enabling esea che s o
design ma e ials wi h speci ic magne ic p ope ies. This accele a es ma e ial disco e y
and inno a ion in ields like da a s o age, senso s, and spin onics [60]. DFT cap u es
quan um mechanical e ec s, such as elec on unneling and wa e-like beha io , which
a e i al in unde s anding and manipula ing magne ic beha io a he nanoscale. These
e ec s a e pe inen o cu ing-edge echnologies like quan um compu ing and nanomag-
ne ic de ices [60].
DFT calcula ions can be compu a ionally demanding, especially o la ge sys ems o
ex ended ime scales [61]. The need o subs an ial compu a ional esou ces limi s he size
o sys ems ha can be s udied and he du a ion o simula ions. DFT calcula ions ely on
exchange-co ela ion unc ionals, which a e app oxima ions o he complex elec on–elec-
on in e ac ions. The choice o unc ionali y can in luence he accu acy o esul s, pa ic-
ula ly in s ongly co ela ed sys ems common in magne ic ma e ials. Modeling magne ic
ilms o en equi es accu a e ea men o su ace and in e ace e ec s, which can be chal-
lenging wi hin DFT [61]. These egions may exhibi di e en elec onic s uc u es and
magne ic beha io s compa ed o he bulk, equi ing ad anced echniques. DFT calcula-
ions a e ypically pe o med a absolu e ze o empe a u es and do no inhe en ly cap u e
he mal e ec s o dynamic p ocesses. Inco po a ing empe a u e and dynamics o en ne-
cessi a es addi ional me hodologies and simula ions. In e p e ing DFT esul s equi es a
deep unde s anding o bo h quan um mechanics and ma e ial-speci ic conside a ions. Ex-
ac ing meaning ul physical insigh s om elec onic s uc u e da a can be complex, pa -
icula ly o esea che s wi hou a s ong backg ound in quan um physics [61] (Table 3).
DFT calcula ions can also be ex ended o s udy he abso p ion and di usion o a oms
o molecules on he hin- ilm su ace [62] (Figu e 11). By calcula ing abso p ion ene gies
and di usion ba ie s, aluable insigh in o he eac i i y and ca aly ic p ope ies o he
ilm can be gained. Vib a ional p ope ies o hin ilms can be analyzed using densi y unc-
ional pe u ba ion heo y (DFPT) o ini e di e ence me hods. These echniques p o ide
in o ma ion abou he pho on dispe sion, ib a ional equencies, and he mal p ope ies
o hin ilms [62].
Figu e 10. Su ace opology on di e en loca ions (a,b) o ch omium oxide ilm.
Wi hin he domain o explo ing magne ic ilms and hei in ica e cha ac e is ics, DFT
simula ions ha e eme ged as a p edominan compu a ional ins umen [
56
]. DFT p esen s
a quan um-mechanical a enue o comp ehending he elec onic a angemen and ene gy
aspec s o subs ances, ende ing i ap o deciphe ing he in ica e magne ic p ope ies
demons a ed by slende ilms. Ne e heless, akin o any modeling me hodology, DFT
compu a ions encompass a unique a ay o s eng hs and limi a ions when employed in
sc u inizing magne ic ilms [57].
DFT p o ides a obus amewo k o accu a ely calcula ing he elec onic s uc u e o
ma e ials, including magne ic ilms [
57
]. I can p edic p ope ies such as band s uc u e,
densi y o s a es, and magne ic momen s, o e ing insigh s in o he unde lying physics
go e ning magne ic in e ac ions [
58
]. DFT can accu a ely cap u e magne ic in e ac ions by
accoun ing o he a angemen o elec ons’ spins. This enables he s udy o spin o ien a-
ions, exchange in e ac ions, and magne ic aniso opy, shedding ligh on he undamen al
mechanisms d i ing magne ic beha io in ilms [
58
]. DFT is e sa ile and applicable o a
wide ange o ma e ials, om simple me als o complex compounds. I can be employed o
explo e he magne ic p ope ies o a ious composi ions, c ys al s uc u es, and ilm hick-
nesses, aiding in he design and op imiza ion o no el magne ic ilms [
59
]. DFT calcula ions
allow o p edic i e modeling o magne ic ilms, enabling esea che s o design ma e ials
wi h speci ic magne ic p ope ies. This accele a es ma e ial disco e y and inno a ion in
ields like da a s o age, senso s, and spin onics [
60
]. DFT cap u es quan um mechanical
e ec s, such as elec on unneling and wa e-like beha io , which a e i al in unde s and-
ing and manipula ing magne ic beha io a he nanoscale. These e ec s a e pe inen o
cu ing-edge echnologies like quan um compu ing and nanomagne ic de ices [60].
DFT calcula ions can be compu a ionally demanding, especially o la ge sys ems
o ex ended ime scales [
61
]. The need o subs an ial compu a ional esou ces limi s he
size o sys ems ha can be s udied and he du a ion o simula ions. DFT calcula ions ely
on exchange-co ela ion unc ionals, which a e app oxima ions o he complex elec on–
elec on in e ac ions. The choice o unc ionali y can in luence he accu acy o esul s,
pa icula ly in s ongly co ela ed sys ems common in magne ic ma e ials. Modeling mag-
ne ic ilms o en equi es accu a e ea men o su ace and in e ace e ec s, which can be
challenging wi hin DFT [
61
]. These egions may exhibi di e en elec onic s uc u es and
magne ic beha io s compa ed o he bulk, equi ing ad anced echniques. DFT calcula-
ions a e ypically pe o med a absolu e ze o empe a u es and do no inhe en ly cap u e
he mal e ec s o dynamic p ocesses. Inco po a ing empe a u e and dynamics o en
Ma e ials 2024,17, 1436 20 o 32
necessi a es addi ional me hodologies and simula ions. In e p e ing DFT esul s equi es
a deep unde s anding o bo h quan um mechanics and ma e ial-speci ic conside a ions.
Ex ac ing meaning ul physical insigh s om elec onic s uc u e da a can be complex,
pa icula ly o esea che s wi hou a s ong backg ound in quan um physics [
61
] (Table 3).
Table 3. Ad an ages and disad an ages o DFT calcula ions.
Ad an ages Disad an ages
-E icien o b oad applica ions. -Relies on app oxima ions, a ec ing accu acy.
-Applicable o a ious sys ems. -Limi ed o small- o-medium-sized sys ems.
-P edic s p ope ies accu a ely. -Requi es signi ican compu a ional esou ces o
high accu acy.
-Op imizes a omic s uc u es accu a ely. -Sensi i e o unc ional and pa ame e choices.
-In eg a es well wi h o he me hods. -Complex in e p e a ion, especially o non-expe s.
DFT calcula ions can also be ex ended o s udy he abso p ion and di usion o a oms
o molecules on he hin- ilm su ace [
62
] (Figu e 11). By calcula ing abso p ion ene gies
and di usion ba ie s, aluable insigh in o he eac i i y and ca aly ic p ope ies o he
ilm can be gained. Vib a ional p ope ies o hin ilms can be analyzed using densi y
unc ional pe u ba ion heo y (DFPT) o ini e di e ence me hods. These echniques
p o ide in o ma ion abou he pho on dispe sion, ib a ional equencies, and he mal
p ope ies o hin ilms [62].
Ma e ials 2024, 17, x FOR PEER REVIEW 21 o 32
Table 3. Ad an ages and disad an ages o DFT calcula ions.
Ad an ages
Disad an ages
-E icien o b oad applica ions.
-Relies on app oxima ions, a ec ing accu acy.
-Applicable o a ious sys ems.
-Limi ed o small- o-medium-sized sys ems.
-P edic s p ope ies accu a ely.
-Requi es signi ican compu a ional esou ces o high
accu acy.
-Op imizes a omic s uc u es accu a ely.
-Sensi i e o unc ional and pa ame e choices.
-In eg a es well wi h o he me hods.
-Complex in e p e a ion, especially o non-expe s.
Figu e 11. Deposi ed magne ic ilm wi h BaSO4 pa icles.
In a s udy ela ed o he DFT and DFPT heo e ical in es iga ions o in e acial p op-
e ies in CaVO3 hin ilms, Beck and Ede e (2020) u ilized DFPT and DFT o simula e he
impac o he pola CaVO3 in e ace on he physical p ope ies [62] (Figu e 11). The esul s
indica e ha compa ison be ween expe imen al and compu a ional esul s necessi a es
me iculous a en ion o he co esponding bounda y condi ions. In ano he s udy con-
duc ed by Ka iani and Aschaue (2022), hey used densi y unc ion heo y o simula e
oxygen acancies in he sys em S MnO3, which is g own on S TiO3 subs a es. The esul s
highligh ha su ace and in e ace e ec s ha e a subs an ial impac on he s abili y and
elec ic subs uc u e o oxygen acancies [63].
The nex s udy conduc ed by Unal e al. (2007) is ela ed o he densi y unc ional
heo y in es iga ion o he ini ial biolaye g own o Ag on NiAl hin ilms. Densi y unc-
ion heo y analysis o suppo ed Ag ilms on NiAl wi h an ideal s uc u e e eals ha he
bilaye g ow h mode is acili a ed by a quan um side e ec [64].
A e ob aining he esul s om DFT calcula ions, pos -p ocessing analysis ech-
niques a e applied o in e p e and analyze he da a. This may in ol e isualizing he
cha ge di e ence dis ibu ion, plo ing he band s uc u e, calcula ing su ace ene gies, o
in es iga ing he elec onic densi y o s a es.
3.4. Mic omagne ic Simula ions
Mic omagne ic simula ions a e a powe ul compu a ional echnique used o s udy
he magne iza ion beha io and magne ic p ope ies o hin ilms and nanos uc u es a
he mesoscopic scale [65]. These echniques p o ide insigh in o he spa ial dis ibu ion o
magne iza ion, magne os a ic in e ac ions, domain s uc u e, and analy ics o magne ic
sys ems. Magne ic s imula ions ea he magne ic ma e ial as a collec ion o magne ic mo-
men s o spins, ypically ep esen ed on a disc e e g id o mesh [65]. Each spin in e ac s
wi h i s neighbo ing spins h ough exchange in e ac ion, which go e ns he alignmen
and dynamics o he magne iza ion [66]. The magne ic model conside s a ious ene gy
con ibu ions, including exchange ene gy, aniso opy ene gy, magne os a ic ene gy, and
ex e nal magne ic ield con ibu ions. Mic omagne ic simula ion in ol es sol ing he
equa ion o mo ion o he magne ic momen s o spin o e ime. This can be achie ed
using nume ical echniques such as he Landau–Li shi z–Gilbe equa ion o i s a iance
(23) [66].
Figu e 11. Deposi ed magne ic ilm wi h BaSO4pa icles.
In a s udy ela ed o he DFT and DFPT heo e ical in es iga ions o in e acial p op-
e ies in CaVO
3
hin ilms, Beck and Ede e (2020) u ilized DFPT and DFT o simula e
he impac o he pola CaVO
3
in e ace on he physical p ope ies [
62
] (Figu e 11). The
esul s indica e ha compa ison be ween expe imen al and compu a ional esul s neces-
si a es me iculous a en ion o he co esponding bounda y condi ions. In ano he s udy
conduc ed by Ka iani and Aschaue (2022), hey used densi y unc ion heo y o simula e
oxygen acancies in he sys em S MnO
3
, which is g own on S TiO
3
subs a es. The esul s
highligh ha su ace and in e ace e ec s ha e a subs an ial impac on he s abili y and
elec ic subs uc u e o oxygen acancies [63].
The nex s udy conduc ed by Unal e al. (2007) is ela ed o he densi y unc ional
heo y in es iga ion o he ini ial biolaye g own o Ag on NiAl hin ilms. Densi y unc ion
heo y analysis o suppo ed Ag ilms on NiAl wi h an ideal s uc u e e eals ha he
bilaye g ow h mode is acili a ed by a quan um side e ec [64].
A e ob aining he esul s om DFT calcula ions, pos -p ocessing analysis echniques
a e applied o in e p e and analyze he da a. This may in ol e isualizing he cha ge di e -
ence dis ibu ion, plo ing he band s uc u e, calcula ing su ace ene gies, o in es iga ing
he elec onic densi y o s a es.
Ma e ials 2024,17, 1436 21 o 32
3.4. Mic omagne ic Simula ions
Mic omagne ic simula ions a e a powe ul compu a ional echnique used o s udy
he magne iza ion beha io and magne ic p ope ies o hin ilms and nanos uc u es a
he mesoscopic scale [
65
]. These echniques p o ide insigh in o he spa ial dis ibu ion
o magne iza ion, magne os a ic in e ac ions, domain s uc u e, and analy ics o magne ic
sys ems. Magne ic s imula ions ea he magne ic ma e ial as a collec ion o magne ic
momen s o spins, ypically ep esen ed on a disc e e g id o mesh [
65
]. Each spin in e ac s
wi h i s neighbo ing spins h ough exchange in e ac ion, which go e ns he alignmen and
dynamics o he magne iza ion [
66
]. The magne ic model conside s a ious ene gy con i-
bu ions, including exchange ene gy, aniso opy ene gy, magne os a ic ene gy, and ex e nal
magne ic ield con ibu ions. Mic omagne ic simula ion in ol es sol ing he equa ion o
mo ion o he magne ic momen s o spin o e ime. This can be achie ed using nume ical
echniques such as he Landau–Li shi z–Gilbe equa ion o i s a iance (23) [66].
dM
d =−γM×H(e )+αM×(dM
d )(23)
whe e:
Mis he magne iza ion ec o o he ma e ial.
H(e )is he e ec i e magne ic ield expe ienced by he ma e ial.
α
is he Gilbe damping pa ame e ha cha ac e izes he dissipa ion o ene gy du ing he
magne iza ion dynamics.
γ
is he gy omagne ic a io, a undamen al cons an ela ed o he magne ic p ope ies o
he ma e ial.
The Landau–Li shi z–Gilbe equa ion desc ibes he p ecession and elaxa ion o
magne iza ion in esponse o applied magne ic ields and o ques. Nume ical in eg a ion
me hods, such as he Runge–Ku a algo i hm, a e commonly used o sol e he Landau–
Li shi z–Gilbe equa ion and simula e magne iza ion dynamics [67].
Mic omagne ic simula ion equi es inpu pa ame e s ha desc ibe he ma e ial p op-
e ies and geome y o he hin ilm. These pa ame e s include exchange s i ness, con ac ,
sa u a ion magne iza ion, aniso opy cons an , ex e nal magne ic ield, and sample dimen-
sions [
67
]. Some o hese pa ame e s can be ob ained om expe imen al measu emen s,
while o he s can be es ima ed om heo e ical calcula ions o empi ical da a. Bounda y
condi ions de ine he beha io o he spin a he edges o he simula ion domain. Common
bounda y condi ions include pe iodic bounda y condi ions, which simula e an in ini e
sys em, and ixed bounda y condi ions, which ix he magne iza ion di ec ion a he edges.
The choice o bounda y condi ions depends on he speci ic condi ions being s udied o he
desi ed beha io o he magne iza ion [68].
Magne ic simula ions gene a e la ge amoun s o da a, including he special dis i-
bu ion o magne iza ion, ene gy dis ibu ions, and dynamic beha io o e ime [
68
]. Vi-
sualiza ion echniques such as colo mapping o ec o ield ep esen a ions a e used o
isualize he magne iza ion pa e ns and domain s uc u es. Analysis ools a e employed o
ex ac key pa ame e s, such as domain wall eloci y, swi ching ields, o magne ic ba ie s,
om he simula ion esul s [68].
Magne ic s imula ion has been success ully applied o a wide ange o magne ic sys-
ems, including hin ilms, nanopa icles, magne ic he e os uc u es, and pa e n magne ic
s uc u es [
69
]. They p o ide aluable insigh in o magne iza ion dynamics, domain wall
mo ion, spin wa es, and o he magne ic phenomena. By adjus ing inpu pa ame e s and
s udying di e en geome ies and ma e ials, esea che s can explo e he e ec o a iable
ac o s on he magne ic beha io o hin ilms and nanos uc u es [69].
Mic omagne ic simula ions ha e eme ged as a c ucial compu a ional ins umen o
comp ehending he in ica e phenomena displayed by magne ic ilms [
70
]. These simula-
ions o e an in e media e-scale pe spec i e on magne ic sys ems, e ealing he dynamics
and in e plays o indi idual magne ic momen s wi hin he ilm. While g an ing p o ound

Ma e ials 2024,17, 1436 22 o 32
unde s andings o magne ic cha ac e is ics and conduc , mic omagne ic simula ions also
b ing o h speci ic p os and cons [70].
The mic omagne ic me hodology is es ablished h ough a coa se-g ained app oach,
whe ein he a omic s uc u e o he sys em is a e aged ou and ep esen ed by a col-
lec ion o mic omagne ic blocks [
71
]. Each mic omagne ic block encompasses mul iple
a oms, wi h hei magne ic momen s assumed o align pa allel o each o he . Consequen ly,
he size o hese mic omagne ic blocks mus be smalle han he leng h scale associa ed
wi h cha ac e is ic magne ic inhomogenei ies in he sys em. This leng h scale is deno ed
as
l=min(δ,lex)
, whe e
δ=π√A/K
ep esen s he Bloch domain wall wid h and
lex =pA/µ0M2
is he exchange leng h [
71
]. He e A, K, and M co espond o he exchange
s i ness cons an , aniso opy densi y cons an , and spon aneous magne iza ion, espec-
i ely. I is no ewo hy ha he me hod encoun e s limi a ions nea empe a u es close o
he Cu ie poin , whe e sho -wa eleng h he mal luc ua ions become signi ican . In such
cases, a ully a omic desc ip ion becomes impe a i e; howe e , an al e na i e app oach
based on he Landau–Li shi z–Bloch equa ion is an in iguing op ion [71].
Mo eo e , a ious ypes o bounda y condi ions can be adap ed o sui he speci ic
equi emen s o he sys em unde conside a ion [72].
The o ien a ion o a omic magne ic momen s wi hin a mic omagne ic block is cha -
ac e ized by a uni ec o S
i
, deno ed as a block spin (wi h he subsc ip iindica ing he
block’s posi ion). The comple e magne ic s a e o he sys em is ully speci ied by he se o
all block spins, deno ed as {S
i
}. The magne ic ene gy o he sys em comp ises he ollowing
ou e ms: E=E
H
+E
K
+E
X
+E
D
(Equa ions (24)–(26)), whe e E
H
,E
K
,E
X
, and E
D
ep esen
he Zeeman, aniso opy, exchange, and dipola con ibu ions o ene gy, espec i ely [72].
EH=−µVa∑MH·Si(24)
He e,
µ=
4
×
10
−7N/A2
is he acuum pe meabili y, V
a
=a
3
is he mic omagne ic
block olume, Miis he spon aneous magne iza ion o block i, and H is he ex e nal
magne ic ield [73].
Aniso opy ene gy (EK) is calcula ed as ollows:
Ek=−Va∑K((ni·Si)2(25)
The aniso opy ene gy accoun s o he p e e ence o magne ic momen s o align along
hei easy axis. K
i
is he aniso opy cons an densi y o block i, and n
i
is he uni ec o
along i s easy axis [73].
Exchange ene gy (EX) is calcula ed as ollows:
Ex=−ΣJSiSj
2(26)
The exchange ene gy a o s neighbo ing magne ic momen s o be aligned. This is
desc ibed by he Heisenbe g o mula [73].
Dipola ene gy (ED) is calcula ed as ollows:
ED=−µ
4π∑JMiMj
 i− j
3(27)
The dipola ene gy co esponds o he magne os a ic in e ac ion be ween classical
dipoles [
74
]. Mic omagne ic simula ions equi e knowledge o pa ame e s such as spon a-
neous magne iza ion (M
i
), aniso opy cons an densi y (K
i
), and exchange couplings (J
ij
).
These pa ame e s a e o en ob ained om expe imen al da a o , al e na i ely, can be e alu-
a ed om i s -p inciple elec onic s uc u e calcula ions. The la e allows o a mul iscale
app oach, p o iding insigh s in o he a ia ions o pa ame e s nea in e aces. Howe e ,
Ma e ials 2024,17, 1436 23 o 32
such calcula ions can be compu a ionally expensi e, and he accu acy o i s -p inciples
me hods may ha e limi a ions o ce ain ma e ials [74].
In p ac ical applica ions, pa ame e s such as M
i
,K
i
, and J
ij
may be es ima ed om
expe imen al da a, and he in e laye exchange coupling (IEC) may be ea ed as a ee
pa ame e o app oxima ed as an a e age o bulk exchanges [74].
Mic omagne ic simula ions allow esea che s o s udy he beha io o magne ic ilms
wi h high spa ial and empo al esolu ion [
75
]. This enables he explo a ion o dynamic
p ocesses such as domain wall mo ion, magne iza ion swi ching, and spin dynamics. Mic o-
magne ic simula ions cap u e he inhe en complexi y o magne ic sys ems by inco po a ing
ac o s like c ys al aniso opy, exchange in e ac ions, and ex e nal magne ic ields [
75
]. This
ealism is essen ial o accu a ely ep esen ing he beha io o eal-wo ld magne ic ilms.
Mic omagne ic simula ions can handle ela i ely la ge sys ems, encompassing millions
o spins, which is c ucial o s udying ealis ic magne ic ilm geome ies and sizes. Re-
sea che s can sys ema ically in es iga e he impac o a ious pa ame e s, such as ilm
hickness, exchange cons an s, and ex e nal ields, on magne ic beha io [
75
]. This aids in
op imizing magne ic ilms o speci ic applica ions. Mic omagne ic simula ions p o ide
insigh s in o he dynamic e olu ion o magne iza ion o e ime, cap u ing ansien be-
ha io s ha a e c ucial o unde s anding p ocesses like magne iza ion p ecession and
elaxa ion (Figu e 12) [75].
Ma e ials 2024, 17, x FOR PEER REVIEW 24 o 32
ansien beha io s ha a e c ucial o unde s anding p ocesses like magne iza ion p e-
cession and elaxa ion (Figu e 12) [75].
Figu e 12. C ys alline ch omium oxide ilm.
Mic omagne ic simula ions can be compu a ionally demanding, pa icula ly o
la ge sys ems o when simula ing ex ended ime scales [76]. The complexi y o he simu-
la ions can equi e subs an ial compu a ional esou ces and ime. Mic omagne ic simula-
ions a e o en conduc ed a ze o empe a u e o wi h minimal conside a ion o he mal
e ec s. Inco po a ing ini e- empe a u e e ec s can be challenging and may equi e addi-
ional modeling app oaches. Mic omagne ic simula ions a e classical in na u e and do no
inhe en ly cap u e quan um mechanical e ec s ha could be ele an in ce ain magne ic
sys ems, pa icula ly a he nanoscale [76]. The accu acy o mic omagne ic simula ions is
in luenced by he nume ical me hods and app oxima ions used in he simula ion code.
The choice o disc e iza ion scheme and ime-s epping algo i hm can impac he ideli y
o he esul s. Ex ac ing meaning ul physical insigh s om mic omagne ic simula ion
da a can be challenging, especially o complex sys ems. In e p e a ion o en equi es a
deep unde s anding o bo h he simula ion echnique and he unde lying magne ic phe-
nomena [76] (Table 4).
Table 4. Ad an ages and disad an ages o mic omagne ic simula ions.
Ad an ages
Disad an ages
-E icien o la ge sys ems.
-O e simpli ica ion may lead o inaccu acies.
-Facili a es pa ame e s udies.
-Limi ed accu acy o complex dynamics.
-Cap u es ealis ic beha io s.
-Requi es ca e ul uning o nume ical s abili y.
-Scales well o la ge sys ems.
-Limi ed in cap u ing ine-scale physics.
-Compa ible wi h expe imen al da a.
-In e p e a ion may be complex, equi ing ali-
da ion.
An example o success ully applied mic omagne ic simula ion was a s udy con-
duc ed by Ruiz–Gómez e al. (2023) [77]. In he ollowing s udy, mic omagne ic simula-
ion was ex ensi ely used o simula e geome y and magne ic ex u e. The essen ial in o -
ma ion om he s udy is ha he physics in he ilm being s udied can be accu a ely cap-
u ed using mic omagne ic simula ions wi h he speci ied ma e ial pa ame e s [77]. The e
is no need o in oduce addi ional e ms o conside he in luence o de ec s. The e o e,
mic omagne ic simula ions a e su icien o ca e ully p edic ing he beha io o ul a- hin
magne ic nanos uc u es wi hou he necessi y o employing a omis ic spin dynamics
s udies, a leas on he scale o se e al nanome e s [77].
Ano he s udy is ela ed o he mic omagne ic simula ion o domains in hin ilms
wi h di e en aniso opy di ec ions, which was conduc ed by Solo e e al. (2020) [78].
The in es iga ion o magne iza ion p ocesses and domain s uc u es in he ilms was
Figu e 12. C ys alline ch omium oxide ilm.
Mic omagne ic simula ions can be compu a ionally demanding, pa icula ly o la ge
sys ems o when simula ing ex ended ime scales [
76
]. The complexi y o he simula ions
can equi e subs an ial compu a ional esou ces and ime. Mic omagne ic simula ions
a e o en conduc ed a ze o empe a u e o wi h minimal conside a ion o he mal e ec s.
Inco po a ing ini e- empe a u e e ec s can be challenging and may equi e addi ional
modeling app oaches. Mic omagne ic simula ions a e classical in na u e and do no in-
he en ly cap u e quan um mechanical e ec s ha could be ele an in ce ain magne ic
sys ems, pa icula ly a he nanoscale [
76
]. The accu acy o mic omagne ic simula ions is
in luenced by he nume ical me hods and app oxima ions used in he simula ion code. The
choice o disc e iza ion scheme and ime-s epping algo i hm can impac he ideli y o he
esul s. Ex ac ing meaning ul physical insigh s om mic omagne ic simula ion da a can
be challenging, especially o complex sys ems. In e p e a ion o en equi es a deep unde -
s anding o bo h he simula ion echnique and he unde lying magne ic phenomena [
76
]
(Table 4).
Ma e ials 2024,17, 1436 24 o 32
Table 4. Ad an ages and disad an ages o mic omagne ic simula ions.
Ad an ages Disad an ages
-E icien o la ge sys ems. -O e simpli ica ion may lead o inaccu acies.
-Facili a es pa ame e s udies. -Limi ed accu acy o complex dynamics.
-Cap u es ealis ic beha io s. -Requi es ca e ul uning o nume ical s abili y.
-Scales well o la ge sys ems. -Limi ed in cap u ing ine-scale physics.
-Compa ible wi h expe imen al da a. -In e p e a ion may be complex, equi ing alida ion.
An example o success ully applied mic omagne ic simula ion was a s udy conduc ed
by Ruiz–Gómez e al. (2023) [
77
]. In he ollowing s udy, mic omagne ic simula ion was
ex ensi ely used o simula e geome y and magne ic ex u e. The essen ial in o ma ion
om he s udy is ha he physics in he ilm being s udied can be accu a ely cap u ed
using mic omagne ic simula ions wi h he speci ied ma e ial pa ame e s [
77
]. The e is
no need o in oduce addi ional e ms o conside he in luence o de ec s. The e o e,
mic omagne ic simula ions a e su icien o ca e ully p edic ing he beha io o ul a- hin
magne ic nanos uc u es wi hou he necessi y o employing a omis ic spin dynamics
s udies, a leas on he scale o se e al nanome e s [77].
Ano he s udy is ela ed o he mic omagne ic simula ion o domains in hin ilms
wi h di e en aniso opy di ec ions, which was conduc ed by Solo e e al. (2020) [
78
]. The
in es iga ion o magne iza ion p ocesses and domain s uc u es in he ilms was ca ied ou
using mic omagne ic simula ions. The simula ions in ol e a hin- ilm model ha conside s
in-plain and pe pendicula magne ic aniso opy. The ocus o his s udy was o explo e
he magne ic mic os uc u e, pa icula ly he equilib ium magne iza ion con igu a ion and
he magne iza ion p ocesses in he hin ilms o a ious hicknesses. The mic omagne ic
simula ion esul s e ealed ha when he ilm hicknesses we e equal o sligh ly la ge han
a c ucial alue, a pe iodic magne ic mic os uc u e o med, cha ac e ized by he absence o
well-de ined domain bounda ies [78].
4. Challenges and Fu u e Di ec ions
4.1. Expe imen al Valida ion and Inco po a ion o Dynamic E ec s
Expe imen al alida ion o magne ic p edic ion o ilms in ol es conduc ing expe -
imen s o e i y he accu acy and a iabili y o heo e ical o compu a ional models in
p edic ing he magne ic p ope ies o hin ilms.
Thin- ilm samples a e p epa ed wi h con olled composi ion, hickness, and s uc u e.
Va ious deposi ion echniques a e employed o deposi hin ilms on o sui able subs a es.
Ca e ul con ol o he posi ion pa ame e ensu es ep oducibili y and minimizes ex e nal
in luence on he magne ic p ope ies.
Va ious cha ac e iza ion echniques a e employed o measu e and analyze he mag-
ne ic p ope ies o hin- ilm samples. These echniques can include me hods such as ib a -
ing sample magne ome y (VSM) o supe conduc ing quan um in e ace de ice (SQUID)
magne ome y, magne ic o ce mic oscopy (AFM), magne o-op ical measu emen s (Ke
mic oscopy), and magne ic esonance echniques (magne ic esonance o nuclea mag-
ne ic esonance). The choice o echnique depends on speci ic magne ic p ope ies being
in es iga ed, such as magne iza ion, magne ic aniso opy, coe ci i y, o magne ic domain
s uc u e. Expe imen al esul s ob ained om he magne ic cha ac e iza ion echnique
we e compa ed wi h he p edic ions made by he heo e ical o compu a ional models.
These models may in ol e undamen al magne ic heo ies, magne ic simula ions,
o ini ial calcula ions based on he densi y unc ion heo y. The compa ison in ol es
e alua ing he ma ch be ween he expe imen al da a and he p edic ed alues o a ious
magne ic pa ame e s, such as magne ic momen s, coe ci e ield, o hys e esis beha io .
The expe imen al da a in compa ison wi h he cha ac e iza ion a e analyzed o iden i y any
disc epancies o de ia ions. S a is ical analysis me hods, such as e o analysis, eg ession
analysis, o hypo hesis es ing, may be employed o assess he le el o ag eemen o
disag eemen be ween he expe imen al esul s and he p edic ed alues. Any obse ed
Ma e ials 2024,17, 1436 25 o 32
dispa i ies can p o ide insigh s in o he limi a ions and sho comings o he heo e ical
models and sugges a eas o u u e imp o emen . The expe imen al alida ion p ocess
o en in ol es an in e ac i e app oach, whe e he ob ained esul s and insigh om he
ini ial compa ison a e used o e ine he imp o ed heo e ical models and sugges a eas o
u he imp o emen . This i e a i e p ocess helps o enhance he accu acy and eliabili y
o magne ic p edic ion and ensu es a be e unde s anding o he unde lying physical
phenomena. Expe imen al alida ion o magne ic p edic ions o hin ilms is c ucial o
alida ing heo e ical models, e i ying he accessibili y o magne ic heo ies, and p o iding
a basis o u he de elopmen s in he ield.
Inco po a ion o dynamic e ec s in he modeling o hin ilms in ol es conside ing
he ime-dependen beha io and indi ec ions o he magne ic p ope ies and o he ele-
an phenomena wi hin he ilms. Dynamical e ec s can be inco po a ed by pe o ming
ime-dependen simula ions, whe e he beha io o hin ilm is s udied o e a ange o ime
in e als. This in ol es sol ing he ele an equa ions o mo ion o go e ning equa ions
using he me hod ha accoun s o ime e olu ion. Fo example, in magne ic hin ilms,
he Landau–Li shi z–Gilbe equa ion can be sol ed o cap u e he dynamic magne iza ion
beha io , including p ecision, elaxa ion, and damping e ec s. Dynamic e ec s can be
s udied by subjec ing he hin ilm o ex e nal exci a ions o pe u ba ions. This can be
achie ed by applying ime- a ying magne ic ields, empe a u e changes, o mechanical
s ain o he sys em. The esponse o he hin ilm o hese exci a ions can hen be an-
alyzed o unde s and i s dynamic beha io . Techniques such as linea esponse heo y,
loque heo y, o ime-dependen pe u ba ion heo y a e o en employed o cha ac e ize
he sys em’s dynamic esponse. Spin wa es and magnons a e collec i e exci a ions in
magne ic ma e ials ha play a signi ican ole in hei dynamic beha io . Inco po a ing
spin wa e, o magnon, and dynamics in he hin- ilm models allows o he s udy o wa e
p opaga ion, dispe sion, and in e ac ion phenomena. Techniques such as spin wa e heo y,
mic omagne ic s imula ions, o dynamic ma ix me hods can be u ilized o inco po a e
hese dynamic e ec s. Ul a as lase echniques and ime- esol ed measu emen s enabled
he s udy o ex emely as dynamic p ocesses in hin ilms. By using em osecond lase
pulses and ime- esol ing de ec ion me hods, esea che s can p o e he ul a as magne-
iza ion dynamics, spin dynamics, and elaxa ion p ocesses in hin ilms. These hings
p o ide isible insigh in o p ocesses such as magne iza ion swi ching, demagne iza ion,
and ul a as phase ansi ions in oil.
Dynamic Mon e Ca lo simula ions a e use ul o inco po a ing dynamic e ec s in he
modeling o hin ilms. Dynamic Mon e Ca lo simula ion cap u es he s ochas ic na u e
o dynamic p ocesses such as su ace di usion, a omic mo ion, o de ec o ma ion. By
simula ing hese p ocesses o e ime, Dynamic Mon e Ca lo simula ions p o ide insigh s
in o he g ow h kine ics, su ace oughness, o de ec e olu ion in he hin ilm.
Inco po a ing dynamic e ec s in he modeling o hin ilms enhances he unde s and-
ing o hei ime-dependen beha io , ansien phenomena, and s abili y. I p o ides
aluable insigh in o p ocesses such as magne iza ion dynamics, spin wa e p opaga ion,
ul a as phenomena, and g ow h kine ics. These dynamic e ec s a e c ucial o he design
and op imiza ion o hin ilm-based de ices such as magne ic memo ies, spin onic de ices,
o senso s (Table 5).
Table 5. Compa ison o simula ion me hods.
Aspec Mon e Ca lo Simula ions DFT Quan um Mechanics Mic omagne ic Simula ions
Desc ip ion
U ilizes andom sampling
o ob ain nume ical
esul s, o en applied in
s a is ical mechanics and
inance.
Compu es he elec onic
p ope ies o ma e ials by
sol ing he Sch ödinge
equa ion by app oxima ing
he elec on densi y.
Desc ibes he beha io o
pa icles a he a omic and
suba omic le els using
ma hema ical o mula ions.
Models he beha io o magne ic
ma e ials a mic oscopic scales,
simula ing magne ic s uc u es
and dynamics.
Ad ancing
Fea u es
E icien o la ge sys ems
wi h complex in e ac ions.
Applicable o la ge sys ems;
used in ma e ials science
and chemis y.
Facili a es accu a e
p edic ions o molecula
s uc u es and eac ions.
Valuable o s udying magne ic
ma e ials and de ices in
nano echnology.
Ma e ials 2024,17, 1436 32 o 32
77.
F an iu, M.A. Mic omagne ic Simula ions o Magne ic Thin Films. Ph.D. Thesis, Uni e si y o G oningen, G oningen, The
Ne he lands, 2019; pp. 1–46.
78.
Ruiz-Gómez, S.; Pé ez, L.; Masca aque, A.; San os, B.; El Gabaly, F.; Schmid, A.K.; de la Figue a, J. S acking in luence on he
in-plane magne ic aniso opy in a 2D magne ic sys em. Nanoscale 2023,15, 8313–8319. [C ossRe ] [PubMed]
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 .