Modeling of Magnetic Films: A Scientific Perspective

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. 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/). 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 .