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University of Southern California Dissertations and Theses
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Molecular dynamics simulations of nanostructures
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Molecular dynamics simulations of nanostructures
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! ! MOLECULAR)DYNAMICS)SIMULATIONS) OF)NANOSTRUCTURES) ! ! by! ! ! Zaoshi!Yuan! ! ! ! ! A!Dissertation!Presented!to!the ! FACULTY!OF!THE!USC!GRADUATE!SCHOOL! UNIVERSITY!OF!SOUTHERN!CALIFORNIA! In!Partial!Fulfillment!of!the! Requirements!for!the!Degree! DOCTOR!OF!PHILOSOPHY! (MATERIALS!SCIENCE)! ! ! May!2013! ! ! Copyright!2013!! ! Zaoshi!Yuan! ! ! ii! Acknowledgements) First,!I!would!like!to!gratefully!acknowledge!the!supervision!of!my!advisor,! Dr.!Priya!Vashishta,!who!agreed!to!take!me!on!as!a!graduate!student!and!funded!my! Ph.D.!for!the!past!years.!I!am!also!thankful!for!his!warm!encouragement,!thoughtful! guidance!and!excellent!example!he!provided!as!a!successful!and!talented!physicist! and!professor.!! I!feel!tremendously!lucky!to!have!had!the!opportunity!to!work!with!Dr.! Aiichiro!Nakano.!Dr.!Nakano!has!been!a!great!mentor!and!father!figure!to!me.!He! gave!me!his!full!sup port!during!my!time!as!a!graduate!student,!has!worked!side!by! side!with!me!at!the!work!of!my!dissertation,!and!encouraged!me!to!immerse!myself! in!something!I!was!passionate!about.!Not!only!he!instilled!in!me!a!love!for!atomistic! simulation!but!taught!me!th e!true!essence!of!education!as!well.!I!have!never!met!a! professor!more!generous!with!his!time!and!experience.!Without!him!my!thesis!work! would!not!have!been!possible.! ! I!am!grateful!to!Dr.!Rajiv!Kalia!for!serving!on!my!committee.!Dr.!Kalia!has! been!a!great!source!of!encouragement!during!my!time!at!USC.! ! The!members!of!Collaboratory!of!Advanced!Computing!and!Simulation!(CACS)! group!have!contributed!immensely!to!my!personal!and!professional!time.!The!group! has!been!a!great!source!of!friendships!as!well!as!goo d!advice!and!collaboration.!I! appreciate!the!work!done!by!past!and!present!CACS!group!members,!Dr.!Hsiu YPin! Chen!and!Dr.!Weiqiang!Wang,!Dr.!Weiwei!Mou!and!Adarsh!Shekar,!whose!work! ! iii! were!contagious!and!motivational!for!me.!I!would!like!to!acknowledge!honora ry! group!member!Dr.!KenYichi!Nomura.!I!fully!appreciate!his!enthusiasm,!intensity,!his! willingness!to!help!me!on!tough!Fortran!bugs!and!amazing!ability!to!see!through!the! veil! of! graphical! simulation! results.! Other! group! members! that! I! have! had! the! pleasure!to!work!with!or!alongside!are!Amit!Choubey,!Ying!Li,!Manaschai!Kunaseth,! Pankaj!Rajak,!Dr.!Hikmet!Dursun,!Dr.!Richard!Clark,!Dr.!Richard!Seymour,!Dr.!Peng! Liu!and!Dr.!Anne!Hemeryck.! ! In!regards!to!the!study!on!semiconductor!nanowires,!I!thank!Dr.!Daniel! Dapkus,!Dr.!Chongwu!Zhou!and!Dr.!Steve!Cronin! as!well!as!their!group!members!for! the!continuous!effort!in!growing,!testing!and!improving!nanowires.! Their!work!at! Center!of!Energy!Nanoscience!(CEN)!at!USC!served!as!the!underlining!incentive!of! my! dissertation.! Dr.! Dapkus! gave! me! advice! on! nanowire! growth! and! testing.! Maoqing!Yao!and!Chunyung!Chi!shared!and!discussed!with!me!detailed!information! on!nanowire!growth!processes,!defects!and!experimental!work!at!present.! ! I! gratefully! acknowledge! the! funding! sourc es! that! made! my! Ph.D.! work! possible.!I!was!honored!to!be!a!Mork!Fellow!for!the!first!two!years!of!my!graduate! life!and!I!was!funded!by!the!U.S.!Department!of!Energy!(DOE)!for!the!last!three!and! half!years.!! I!would!like!to!thank!all!my!wonderful!friends!an d!colleagues.!I!can!think!of! no!finer!individual!than!Amit!Choubey,!who!has!been!an!exceptional!friend!to!me! both!in!and!out!of!the!lab.!Mr.!and!Ms.!Lammerts!have!been!one!of!my!strongest! supporters! and! I! am! very! grateful! for! their! encouragement! and! advice s.! Dr.! ! iv! Quanzheng!Li!from!Harvard!has!inspired!me!with!his!infectious!enthusiasm!and! amazing!outlook!on!life.!My!true!friend!Yi!Tan!had!made!my!graduate!years!so!rich! and!enjoyable.!! Finally,!I!would!like!to!thank!my!mother!Lihua!Cheng!and!grandmother!Guige ! Cheng,!for!their!love!and!support.!This!dissertation!is!dedicated!to!them.! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! !!!!! Zaoshi!Yuan!! ! ! ! ! ! ! University!of!Southern!California! ! !! ! ! ! ! ! ! !!!!!!! Fall!2012! ! ! ! ! v! Table)of)Contents ) Acknowledgements!...................................................................................................................................!ii! List!of!Figures!.............................................................................................................................................!ix! List!of!Tables!............................................................................................................................................!xvi! Abstract!.....................................................................................................................................................!xvii! CHAPTER!1!INTRODUCTION!................................................................................................................!1! 1.1!Molecular!dynamics!simulation!of!nanostructures ! ...............................................!1! 1.2! Three! fundamental! problems! in! nanostructures! addressed! by! this! dissertation!....................................................................................................................................!6! 1.2.1!Grain!boundary!segregation!in!nanocrystalline!material !................!6! 1.2.2!Stacking!defects!during!nanowire!growth !...........................................!17! 1.3!Dissertation!overview!.....................................................................................................!22! CHAPTER!2!MOLECULAR!DYNAMICS!SIMULATION!METHOD!...........................................!25! 2.1!Molecular!dynamics!simulation!..................................................................................!26! 2.1.1!Newton’s!equation!of!motion!....................................................................!26! 2.1.2!Numerical!integrator! .....................................................................................!27! 2.1.3!Microcanonical!and!canonical!ensembles!............................................!30! •!Microcanonical!(NVE)!ensemble!.......................................................!30! •!Canonical!(NVT)!ensemble! ...................................................................!32! 2.2!Physical!properties! ...........................................................................................................!36! 2.2.1!ThermoYmechanical!properties!................................................................!37! ! vi! 2.2.2!Structural!properties! .....................................................................................!38! 2.2.3!Dynamic!properties!.......................................................................................!40! 2.2.4!Slip!vector!field!................................................................................................!41! 2.3!Simulation!algorithms! .....................................................................................................!41! 2.3.1!LinkedYlistYcell!method!................................................................................!42! 2.3.2!Boundary!conditions!.....................................................................................!44! 2.3.3!Parallelization!..................................................................................................!48! 2.3.4!Performance! ......................................................................................................!51! 2.4!Energy!minimization!.......................................................................................................!54! 2.4.1!The!steepest!descent!method !....................................................................!54! 2.4.2!Conjugate!gradient!relaxation!...................................................................!55! 2.4.3!Global!minimization!......................................................................................!57! CHAPTER!3!INTERATOMIC!INTERACTIONS! ...............................................................................!59! 3.1!Interatomic!interactions!and!Lennard YJones!potential!....................................!59! 3.2!ManyYbody!potentials!.....................................................................................................!62! 3.2.1!StillingerYWeber!potential!..........................................................................!62! 3.2.2!Effective!force!field!for!ZnO ! ........................................................................!64! 3.2.3!Effective!force!field!for!GaAs !......................................................................!69! 3.3!Reactive!force!field!...........................................................................................................!73! 3.3.1!Bond!order!and!bond!energy ! .....................................................................!74! 3.3.2!Lone!pair!energy!.............................................................................................!76! 3.3.3!Overcoordination!and!undercoordination!energy!...........................!77! ! vii! 3.3.4!Valence!angle!energies!.................................................................................!78! 3.3.5!Torsion!angle!energies!.................................................................................!79! 3.3.6!Hydrogen!bond!energy!.................................................................................!81! 3.3.7!Van!der!Waals!and!Coulomb!energie s!...................................................!81! 3.3.8!Validation!results!............................................................................................!82! CHAPTER!4!TRANSFORMATION!OF!GRAIN!BOUNDARIES!IN!NANOCRYSTALLINE! NICKEL!NEAR!PERCOLATION!THRESHOLD!.................................................................!84! 4.1!Background!and!motivation!.........................................................................................!85! 4.1.1!Sulfur!segregation!in!nickel!nanocrystalline!system ! .......................!85! 4.1.2!Percolation!and!physical!properties !......................................................!88! 4.2!Structural!and!thermodynamic!properties!of!Ni YS!system!.............................!90! 4.3!Interaction!range!of!sulfur!i n!nickel!crystal!and!percolation!analysis ! .......!94! 4.4!Thresholds!in!binary!Lennard YJones!systems!......................................................!99! CHAPTER! 5! CORE/SHELL! STRUCTURE! AND! NOVEL! THERMO YMECHANICAL! BEHAVIORS!IN!NANOWIRES!............................................................................................!104! 5.1!Background!and!motivation!......................................................................................!106! 5.1.1!ZnO!.....................................................................................................................!106! 5.1.2!Slip!vector!for!the!chara cterization!of!plastic!deformation!......!111! 5.2!Surface!relaxation!..........................................................................................................!115! 5.3!Nanowire!core/shell!structure!and!system!preparation !..............................!118! 5.4!Tensile!test!........................................................................................................................!121! 5.5!Core/shell!structure!and!novel!structural!transformation ! ..........................!125! ! viii! 5.6!Size!and!temperature!dependen t!brittleYtoYductile!transition!..................!128! 5.7!ThermoYmechanical!phase!diagram!......................................................................!130! CHAPTER!6!STACKINGYFAULT!GENERATION!IN!ZINCBLENDE!NANOWIRES !.........!132! 6.1!Background!and!motivation!......................................................................................!134! 6.1.1!Crystal!structure!of!GaAs !..........................................................................!134! 6.1.2!Stacking!fault!in!GaAs!NWs ! ......................................................................!137! 6.2!Surface!relaxation!..........................................................................................................!140! 6.3!Nanowire!core/shell!structure!and!system!preparation !..............................!141! 6.4!Adatom!energy!map!......................................................................................................!145! 6.5!Adlayer!energy!................................................................................................................!148! 6.6!Nucleation!growth!model!...........................................................................................!149! CHAPTER!7!CONCLUSION!................................................................................................................!153! REFERENCES! ..........................................................................................................................................!156! !! ! ) ! ix! List)of)Figures ) Figure!1.1!! (a)!Schematic!of!nanostructures!–!0D!nanoparticles,!1D!NWs,!and!2D! multilayers;! (Huang! and! Van! Swygenhoven! 2009)! (b)! variation! of! normalized! Young’s! modulus! (E)! as! a! function! of! Cu! nanoplate! thickness!based!on!molecular!statics!and!ab!initio!calculation!(Zhou! and!Huang!2004).! ..........................................................................................................!3! Figure!1.2!! Core!and!mantle!model,!showing!relative!fractions!of!grain!boundary! and! grain! interior! regions! in! the! (a)! microcrystalline! and! (b)! nanocrystalline!regimes.!(Meyers,!Mishra!et!al.!2006)!(c)!The!effect!of! grain!size!on!calculated!volume!fractions!of!intercrys tal!regions!and! triple! junctions,! assuming! a! grain Yboundary! thickness! of! 1! nm(Palumbo,!Thorpe!et!al.!1990). ! .........................................................................!7! Figure!1.3!! Scanning!electron!microscopy!images!from!fracture!surfaces!of!Ni! tensile! specimens! show! the! transition! from! ductile! tearing,! mixed! transgranular!to!brittle!intergranular!fracture!with!increased!grainY boundary!S!concentrations:!(a)!2.79!at.%!S,!(b)!11.59!at.%!S,!and!(c)! 15.86!at.%!S!(Heuer,!Okamoto!et!al.!2002). !......................................................!8! Figure!1.4!! Influence!of!grainYboundary!S!concentration!on!the!transition!from! ductile! to! brittle! fracture! as! measured! by! the! percentage! of! intergranular!fracture!(Heuer,!Okamoto!et!al.!2002). !..................................!9! Figure!1.5!! Critical!S!concentration!as!a!function!of!amorphous!fraction!(Heuer,! Okamoto!et!al.!2002).!...............................................................................................!10! Figure!1.6!! Representative!stress!vs.!strain!curves!for!(a)!fcc!Au!and!(b)!bcc!Mo! nanopillars;!(c)!a!logYlog!plot!of!flow!stress!as!a!function!of!initial! diameter!representing!the!scaling!laws!for!Mo!and!Au.!The!slope!of! strengthening!in!Au!is!nearly!2X!higher!than!that!for!Mo. !(Greer!and! De!Hosson!2011).!.......................................................................................................!14! Figure!1.7!! (a)!Measured!thermal!conductivity!of!different!diameter!Si!nanowires.! The! number! beside! each! curve! denotes! the! corresponding! wire! diameter! (Li,! Wu! et! al.! 2003);! (b)! melting! temperature! of! In! nanowires!as!a!function!of!wire!diameter.!The!solid!line!is!calculated! theoretically! while! the! symbols! denote! the! corresponding! experimental!values!(Qi!2005).!...........................................................................!15! ! x! Figure!1.8!! SEM!images!of!silicon!pillars!compressed!with!nanoindenter.!(a)!pillar! of!400nm!(b)!pillar!of!310!nm!and!(c)!critical!engineering!stress.!(The! pillars! that! showed! cracking! are! encircled;! all! the! other!pillars! deformed!purely!plastic.!The!triangle!correspond!to!experiments!with! an!MTS!and!the!diamonds!correspond!to!experiment!in!situ.)! (Östlund,! RzepiejewskaYMalyska!et!al.!2009)!....................................................................!16! Figure!1.9!! Wurtzite!(WZ)!segment!and!twin!plane!in!III YV!nanowires:!(a)!TEM! image!along!<110>!axis!of!VLS!InAs!nanowires!showing!the!presence! of!WZ!phase;!(b)!SEM!image!(tilt!angle!30˚)!revealing!the!complex! shape! of! VLS! InAs! nanowire s! and! associated! threeYdimensional! atomic!model;!(c)!TEM!image!of!zoomed!in!area!around!rotational! twin!planes!in!(b);!(Caroff,!Dick!et!al.!2008)!(d)!TEM!image!of!a!GaAs! nanowire! grown! using! selectiveYarea! growth! method! showing! a! transition!from!zincblende!(ZB)!to!WZ;!(Tomioka,!Kobayashi!et!al.! 2009)!(e)!TEM!images!of!GaAs!nanowire!with!twin!segment.!(Yoshida,! Ikejiri!et!al.!2009)! .......................................................................................................!20! Figure!1.10!! Effect! of! twin! and! wurtzite! segment! on! electrical! and! optical! properties! of! nanowire:! (a)! InAs! nanowire! resistivity! against! nanowire!diameter!and!defect!densities! (Thelander,!Caroff!et!al.!2011)! ;!(b)!transmission!of!electrons!through!silicon!stacking!faults!with! different! separations! (Stiles! and! Hamann! 1988);! (c)! Photoluminescence! spectrum! of! the! nanowire! with! and! without! planer!defects!with!different!excitation!intensities!(Bao,!Bell!et!al.! 2008).!..............................................................................................................................!21! Figure!2.1.! !Cell! decomposition! of! a! system! for! linked! list! algorithm.! Atom! i! resides!in!cell!8.!..........................................................................................................!42! Figure!2.2!!! Data!structure!for!linked!list!algorithm,!where!“E”!denotes!empty ! .....!44! Figure!2.3!! (a)!A!simulation!supercell!with !16!atoms!replicated!periodically!in!2D! space.!(b)!Shifting!supercell!boundaries!produces!a!different!set!of! atoms!but!does!not!alter!the!overall!periodic!arrangement !...................!47! Figure!2.4!! Schematic!view!of!assigning!process!ID!for!subsystems. !.........................!49! Figure!2.5!! An!illustration!of!a!projection!view!of!spatial!decomposition!of!3 YD! system!implemented!in!parallel!MD!program.!Atoms!attributes!in!the! extendedYdomain!are!cached!from!26!neighboring!nodes. !.....................!50! Figure!2.6!! Results! of! the! isogranular! (2,044,416! atoms! per! processor)! benchmark! results! of! finiteYrange! parallel! MD! algorithm! for! silica! ! xi! using! IBM! BlueGene/L! (open! symbols)! and! BlueGene/P! (solid! symbols)!computers.!The!wallYclock!time!(circles)!and!inter Yprocessor! communication!time!(squares)!per!MD!step!are!shown .!.........................!53! Figure!2.7! Execution! time! of! ReaxFF! MD! per! time! step! as! a! function! of! the! number!of!processors!P!of!BlueGene/L!(squares)!and!BlueGene/P! (circles),! where! the! number! of! atoms! is!N! =! 16,128P)(Nomura,! Seymour!et!al.!2009).!...............................................................................................!54! Figure!3.1!! LennarYJones!potential!(from!http://www.wikipedia.org/)!..................!61! Figure!3.2!! Diamond!cubic!structure.!The!angle!between!any!two!bonds!involving! the!same!atom!is!θ jik =109.47°!i.e.,!cosθ t =−1/3.!........................................!64! Figure!3.3!! Energy! per! particle! as! a! function! of! the! volume! per! particle! for! hexagonal,!zincYblende!and!rockYsalt!ZnO.!.....................................................!68! Figure!3.4!! Energy! vs.! volume! relation! for! zinc Yblende,! wurtzite! and! rockYsalt! structures!of!GaAs.! .....................................................................................................!71! Figure!3.5!! Energy! per! particle! as! a! function! of! the! volume! per! particle! for! hexagonal,! zinc! blende! and! rock Ysalt! GaAs.! José! P.! Rino) et) al.! (Vashishta,!Kalia!et!al.!1997) !................................................................................!73! Figure!4.1!! Generalized!stacking!fault!energy!(GSFE)!of!Ni!∑5(012)!GB!without!S! (A),!with!a!monolayer!of!segregated!S!(B),!and!with!amorphous!sulfide! GB! phases! (C).! Also! shown! are! shear! stress! calculated! from! the! derivative!of!the!generalized!stacking!fault!energy!with!respect!to! displacement!along!the![100]!direction!(Chen,!Shi!et!al.). !.......................!87! Figure!4.2!! (a)!Electrical!conductivity!σ c !of!polymer!based!composites!of!carbon! nanotubes!as!a!function!of!the!filler!volume!fraction! f!(Bryning,!Islam! et! al.! 2005);! (b)! electrical! conductivity!σ c !of! polymer! based! composites!of!grapheme/polystyrene!as!a!function!of!the!filler!volume! fraction!f!(Stankovich,!Dikin!et!al.!2006);!(c)!variation!of!the!relative! dielectric!constant!κ!of!Ni/BaTiO 3!+!PVDF![poly(vinylidene!fluoride)]! threeYphase! composites! with! the! volume! fraction!f!of! Ni! particles! (Dang,!Shen!et!al.!2002). !.........................................................................................!89! Figure!4.3!! (a)!NiYNi!partial!pair!distribution!function!gNiYNi(r)!of!NiYS!system!at! different!S!concentration!cS;!(b)!full!width!at!half!maximum!of!the!first! peak!of!gNiYNi(r)!as!a!function!of! cS;!(c)!height!of!the!second!peak!of! gNiY ! xii! Ni(r)!as!a!function!of!cS;!(d)!coordination!number!NNiYNi(r)!of!NiYS! system!at!different!S!concentration! cS;!.............................................................!91! Figure!4.4!! (a)!NiYNiYNi!bond!angle!distribution!(BAD)!at!S!concentrations! cS!=!8,! 14,! and! 18%.! (b)! Comparison! of! the! BAD! cS!=! 18%! with! that! of! amorphous!Ni.!.............................................................................................................!93! Figure!4.5.!! Equilibrium!volume!as!a!function!of!S!concentration. !...............................!94! Figure!4.6!! Schematic! of! the! calculation! procedure! for! the! lattice! distortion Y mediated!SYS!interaction!energy.!........................................................................!96! Figure!4.7!! SYS!interaction!energy!in!Ni!fcc!crystal!as!a!function!S YS!distance.!......!97! Figure!4.8!! The!largest!cluster!size!as!a!function!of!the!occupation!probability!for! fcc! lattice! considering! the! 1 st ,! 2 nd ,! and! 4 th ! nearest! neighbor! connectivity.!.................................................................................................................!99! Figure!4.9!! (a)!Interaction!energy!vs.!distance!between!two!B!impurity!atoms!in!A! crystal!in!binary!LennardYJones!systems!with!different!atomic!size! ratios!σAA/σBB.!(b)!Interaction!range!for!different!atomic!size!ratio.!(c)! Amorphization!threshold!in!a!binary!L YJ!system!with!different!atomic! size!ratios!σAA/σBB.!.................................................................................................!101! Figure! 4.10! Amorphization! threshold! for! binary! Lennard YJones! system! and! percolation!threshold.! ...........................................................................................!103! Figure!5.1!! ZnO!structure:!(a)!the!wurtzite!unit!cell!(b)!the!wurtzite!structure! model;!(from!http://www.wikipedia.org/).!...............................................!107! Figure!5.2!! Total!energy!vs.!volume!(both!per!ZnO!formula!unit)!for!the!three! phases:!ZB!(squares),!WZ!(diamonds),!and!RS!(circles).!The!zero!of! energy!is!the!sum!of!the!total!energy!of!an!isolated!Zn!and!isolated !O! atom.!(Weast!and!Astle!1993). !..........................................................................!108! Figure!5.3!! (a)!Young’s!modulus!as!a!function!of!NW!diameter!(Wen,!Sader!et!al.! 2008);!(b)!fracture!strain!as!a!function!of!N W!diameter!(Desai!and! Haque!2007).!............................................................................................................!110! Figure!5.4!! (a)! A! perfect! simpleYcubic! crystal! (reference! system);! (b)! displacement!of!halfYcrystals!along!cut!plane!A!by!l attice!vector!b! results!in!two!surface!steps!but!does!not!alter!the!atomic!structure! inside!the!crystal;!(c)!the!same!procedure!limited!to!a!part!of!cut!plane! A!introduces!as!an!edge!dislocation;!(d)!an!edge!dislocation!created!by! inserting!a!halfYplane!of!atoms;!(e)!a!screw!dislocation!in!which!the! ! xiii! slip!vector!is!parallel!to!the!dislocation!line;!the!slipped!area!of!the!cut! plane!is!shown!in!dark!gray!and!the!un Yslipped!area!is!shown!in!light! gray;!the!dislocation!line!is!marked!by!the!solid!line;!(f)!a!cu rved! dislocation!line!with!an!edge!orientation!at!one!end!(on!the!left)!and!a! screw!orientation!at!the!other!end!(on!the!right)!(Bulatov!and!Cai! 2006).!...........................................................................................................................!113! Figure!5.5!! A! bond! that! connects! a! pair! of! atoms! α !and!β !in! the! reference! configuration! (drawn! in! a! dashed! line).! The! displacement! s α !can! indicate!(a)!slip!if!the!slip!plane!cuts!the!bond!between!atoms! α!and! β,!or!(b)!bond!breaking!by!crack. !...................................................................!115! Figure!5.6!! Structural!relaxation!at!the!nonpolar!(1010)!surface!as!a!function!of! the!distance!(in!atomic!layers)!from!the!surface.!(a)!Tilt!angle!(in! degrees)!as!defined!in!the!inset.!(b)!The!change!in!Zn YO!bond!length! (in!%)!from!the!bulk!value.!MD!results!(blue!lines)!are!compared!with! QM!results(Meyer!and!Marx!2003)!based!on!the!PBE!approximation! (black!dot!lines).!......................................................................................................!117! Figure!5.7!! Schematic!of!simulated!ZnO!nanowire!with!diameter! D.!(b)!and!(c)! show!the!incipient!coreYshell!structure!of!ZnO!nanowire!with! D!=!2.5! nm!at!temperature!5!K.!(b)!Top!view!of!color Ycoded!slipYvector!field.! (c)!Top!view!of!bond!network,!where!the!bonds!are!colored!in!black! (shortest!bond),!blue!(2 nd !shortest),!and!green!(3 rd !shortest).!Normal! or!elongated!bonds!are!colored!in!red. ! ..........................................................!120! Figure!5.8!! Tensile!tressYstrain!curves!of!ZnO!NW!of!diameter!D!=!2.5,!5,!and!10! nm,!at!temperature!100!K. !..................................................................................!122! Figure!5.9!! Temperature!dependence!of!the!tensile!stress Ystrain!curve!of!ZnO!NW! with!diameter!D!=!2.5!nm!(a),!5!nm!(b),!and!10!nm!(c) !.........................!124! Figure!5.10!! Intrinsic!coreYshell!structure!of!the!NW!under!tension!for!diameter!5! nm!at!temperature!1000!K.!(a)!Top!view!of!slip Yvector!field.!(b)!Side! view!of!slipYvector!field!of!the!slab! shown!in!dashed!black!lines!in!(a).! (c)!Atomic!structure!of!the!same!NW!as!shown!in!(a).!Magenta!dashed! line!delineates!BCTY!and!HXYtransformed!zones.!(d)!Atomic!structure! of!the!same!NW!slab!as!in!(b).!Magenta!dashed!lines!delineate!the!BCT! and!HX!zones.! ............................................................................................................!127! Figure!5.11!! Snapshot!of!slipYvector!field!of!the!5!nm!NW!right!before!failure!at! temperature! 100! K! (a)! and! 1000! K! (b).! Corresponding! atomic! positions!before!(c)!and!after!(d)!failure!at!100K!(d),!and!those!before! ! xiv! (e)!and!after!(f)!failure!at!1000!K.!Slip Yvector!field!in!a!15!nm Ythick! slab!at!the!center!of!the!NW!is!shown. ! ...........................................................!130! Figure!5.12!! SizeYtemperature! phase! diagram! for! the! mechanical! response! of! [0001]Yoriented!ZnO!NW,!where!the!temperature!is!scaled!by!the! melting! temperature.! The! calculated! melting! temperture,! 2050K,! agrees! well! with! the! experimental! value,! 2248K ! The! circles! and! crosses!denote!response!without!and!with!structural!transformation,! respectively.! The! squares! denote! ductile! failure,! otherwise,! the! fracture!is!brittle.!....................................................................................................!131! Figure!6.1!! Crystal!structure!of!zincblende!lattice!at!(100)!direction !.....................!135! Figure!6.2!! Crystal!structure!of!ZB!(left)!and!WZ!(right)!respectively! (McMahon! and!Nelmes!2005;!Dick,!Caroff!et!al.!2010). !................................................!136! Figure!6.3!! 2D! projection! in! the! 110 !viewing! direction! of! possible! stacking! sequences!during!nanowire!growth.!(a)!shows!zincblende!stacking,! whereas!in!(b)!and!(c),!the!first!fault!plane!on!top!of!a!ZB!region!is! indicated!in!red.!If!the!stacking!sequence!continues!with!layer!A!and!C! as!in!(b),!a!twin!segment!is!formed!and!the!stacking!seque nce!is! mirrored!around!the!C!layer!under!the!fault!plane.!On!the!other!hand,! another!fault!plane!formed!on!top!of!the!red Ymarked!B!layer!will! result!in!WZ!phase!(c)!(Johansson,!Karlsson!et!al.!2008). ! .....................!138! Figure!6.4!! Two! possible! atomic! structures! of! twins! in! GaAs:! (a)! ortho! twin! projected!along![110]!direction;!and!(b)!para!twin!projected!along! [110]!direction!(Lee,!Lee!et!al.!1990).!(c)!Ortho!twin!results!in!a! switch! from! ABCABC! to! a! CBACBA! stacking.! (d)! An! ortho! twin! crystallite! can! be! created! by! 60˚! rotation! of! segments(Bolinsson,! Ouattara!et!al.!2009).!.............................................................................................!139! Figure!6.5!! GaYAs!bond!length!as!a!function!of!the!distance!(in!atomic!layers)!from! the!(110)!surface!of!GaAs!ZB!crystal. !.............................................................!141! Figure!6.6!! (a)! Schematic! of! simulated! GaAs! nanow ire.! (b)! Top! view! of! bond! network!in!NW,!where!the!bond!length!is!color Ycoded.!(c)!Top!view!of! colorYcoded!slipYvector!field.!.............................................................................!143! Figure!6.7!! Top!view!of!the!slip! vector!field!(a)!and!color Ycoded!atomic!bonds!(b)! in!the!Aa,!Bb,!and!Cc!bilayers!of!a![111] Yoriented!GaAs!NW.!...............!145! Figure!6.8!! (a)!Schematic!of!the!calculation!of!a dYatom!energy!map.!(b)!and!(c)!Ga! adatom! energy! along! the! blue! and! red! lines! marked! in! (a),! ! xv! respectively.!(d)!The!adatom!energy!map!of!the!entire!NW!top!surface. !.........................................................................................................................................!147! Figure!6.9!! (a)!Schematic!of!a!GaAs!bilayer!island!(magenta)!nucleated!at!a!corner! of!the!top!surface!of!a!NW!(cyan).!(b)!ZB!(blue)!and!fault!(red)!inland! energies!per!atom!as!a!function!of!the!total!number!of!adatoms.!The! asymptotic!limit!of!the!ZB!and !fault!energies!are!indicated!by!blue!and! red!arrows,!respectively.!.....................................................................................!149! Figure!6.10!! (a)!Gibbs!freeYenergy!changes!of!ZB!(blue)!and!fault!(red)!islands!of!a! GaYAs!bilayer!adsorbed!on!the!GaAs!NW!as!a!function!of!the!island! diameter!at!temperature!T!=!1030!K!(solid!lines)!and!1050!K!(dashed! lines).!The!peak!position!(i.e.!the!critical!nucleus!size)!of!each!curve!is! indicated!by!an!arrow.!!(b)!Critical!nucleus!sizes!o f!ZB!(blue)!and!fault! (red)! islands! as! a! function! of! the! temperature.! The! yellow! area! indicates!(DW,!T)!pairs,!for!which!NWs!are!grown!free!of!SF.!The! vapor!pressures!are!pGa!=!2.7 ×10Y6!bar!and!pAs!=!5 ×10Y4!bar.!.......!152! ! ! ) ! xvi! List)of)Tables ) Table!1.1!! Experimentally!Measured!UltraYHigh!Strengths.!............................................!3! Table!3.1!! TwoY!and!threeYbody!parameters!in!the!interaction!potential!for!ZnO. ! José!P.!Rino)et)al.!(Vashishta,!Kalia!et!al.!1997) !............................................!66! Table!3.2!! Experimental!and!calculated!values!from!MD!simulations!of!the!elastic! constants!for!zinc!oxide.!José!P.!Rino)et)al.!(Vashishta,!Kalia!et!al.! 1997)!...............................................................................................................................!67! Table!3.3!! Elastic!constants,!C11,!C12,!and!C!44,!and!bulk!modulus,!B,!of!crystalline! and!amorphous!GaAs!from!old!data!set.! José!P.!Rino)et)al.!(Vashishta,! Kalia!et!al.!1997)! .........................................................................................................!69! Table!3.4!! TwoY!and!threeYbody!parameters!of!the!interaction!potential!for!GaAs! used!in!this!dissertation.!.........................................................................................!72! Table!3.5!! Elastic!constants!of!nickel!fcc!crystal. !...............................................................!83! Table!3.6!! Comparison!of!ReaxFF!and!DFT!calculations!for!calculations!for!the! binding!energy!of!a!sulfur!impurity!at!various!locations!in!nickel!fcc! crystal.! .............................................................................................................................!83! Table!6.1!! GaYAs!bond!length!in!the!top!three!atomic!layers!of!the!(110)!surface! of!GaAs!ZB!crystal.!..................................................................................................!141! ! ! ) ! xvii! Abstract) This! dissertation! is! focused! on!multimillionYatom! molecular! dynamics! (MD)! simulations!of!nanoscale!materials.!In!the!past!decade,!nanoscale!materials!have! made! significant! commercial! impacts,! which! will! potentially! lead! to! the! next! industrial!revolution.!The!interest!lies!in!the!novel!and!promising!features!nanoscale! materials!exhibit!due!to!their!confined!sizes.!However,!not!all!novel!behaviors!are! understood! or! controllable.! Many! uncontrollable! parameters,!e.g.! defects! and! dangling!bonds,!are!known!to!hinder!the!performance!of!nanodevices.!Solutions!to! these!problems!rely!on!our!understandin g!of!fundamental!elements!in!nanoscience:! isolated!individual!nanostructures!and!their!assemblies.! ! In! this! dissertation,! we! will! address! atomistic! foundations! of! several! problems! of! technological! importance! in! nanoscience.! Specifically,! three! basic! problems! are! discussed:! (1)! embrittlement! of! nanocrystalline! metal;! (2)! novel! thermoYmechanical!behaviors!of!nanowires!(NWs);!and!(3)!planar!defect!generation! in!NWs.!With!a!scalable!algorithm!implemented!on!massively!parallel!computing! platforms!and!various!data!mining!methods,!MD!simulations!can!provide!valuable! insights!into!these!problems.! ! An!essential!role!of!sulfur!segregation Yinduced!amorphization!of!crystalline! nickel!was!recently!discovered!experimentally,!but!the!atomistic!mechanism!of!the! amorphization!remains!unexplained.!Our!MD!simulations!reveal!that!the!large!steric! size!of!sulfur!impurity!causes!strong!sulfur Ysulfur!interaction!mediated!by!lattice! ! xviii! distortion,! which! leads! to! amorphization! near! the! percolation! threshold! at! the! sulfurYsulfur!network!in!nickel!crystal.!The!generality!of!the!mechanism!due!to!the! percolation!of!an!impurity!network!is!further!confirmed!by!a!model!binary!system. ! In! our! study! of! novel! behaviors! of! semiconductor! NWs,! MD! simulations! construct!a!rich!sizeYtemperature!‘phase!diagram’!for!the!mechanical!response!of!a! zincYoxide!NW!under!tension.!For!smaller!diameters!and!higher!temperatures,!novel! transitions!are!found!from!brittle!cleavage!to!structural!transformation Ymediated! brittle!cleavage!to!ductile!failure.!Atomistic!mecha nisms!of!the!unique!nano YthermoY mechanical! behavior! are! elucidated! as! a! consequence! of! surface Ystructural! relaxation,! which! in! particular! predicts! spontaneous! formation! of! a! core/shell! structure! under! tension.! The! phase! diagram! resolves! controversies! betwe en! previous!experiments!and!theory,!and!the!predicted!‘intrinsic’!core/shell!structure! may!find!novel!device!applications. ! Generation!of!stacking!faults!(SFs)!during!the!growth!of!NWs!is!a!major! concern!for!the!efficiency!of!NW Ybased!devices!such!as!solar! cells.!MD!simulation!of! a! [111]Yoriented! gallium! arsenide! NW! reveals! an! atomistic! mechanism! of! SF! generation.!Spatial!distribution!of!the!adatom!energy!on!the!(111)B!top!surface! exhibits!a!novel!core/shell!structure!due!to!the!contraction!of!atomic!bonds!a t!the! sidewall!surfaces,!where!SFs!are!preferentially!nucleated!in!the!shell.!A!nucleation! growth!model!incorporating!the!core/shell!mechanism!suggests!a!size!and!growth Y condition!controlled!approach!for!SF Yfree!growth!of!NWs.! ! 1! CHAPTER)1 ) ) ) ) ) ) ) )))) INTRODUCTION) 1.1 Molecular)dynamics)simulation)of)nanostructures )) Nanoscience!and!nanotechnology!have!been!growing!explosively!in!the!last!decade.! They!are!revolutionizing!the!ways!in!which!materials!and!products!are!created!and! the!range!and!nature!of!functionalities!that!can!be!accessed.!In!the!year!2009,!a! quarter! of! a! trillion! dollars! were! generated! in! the! field! of! nanotechnology! worldwide,! of! which! 91!billion! dollars! were! in! U.S.! products! that! incorporate! nanoscale!components!(Roco,!Mirkin!et!al.!2011 )(Roco,!Mirkin!et!al.!2011 )(Roco,! Mirkin! et! al.! 2011).!The! potential! will! assuredly! increase! in! the! future! and! is! believed!to!lead!us!to!the!next!industrial!revolution! (Roco,!Williams!et!al.!1999 ).!! ! The! fundamental! entities! in! nanoscience! and! nanotechnology! are! nanostructures.! A! structure! becomes! nanoscale! when! at! least! one! of! its! three! dimensions!is!below!100!nm !(Huang!and!Van!Swygenhoven!2009 ).!It!could!be!in!t he! form!of!nanoparticles!or!nanograins!(0D),!nanowires !(NWs)!or!nanotubes!(1D),!and! nanoplates!or!multilayers!(2D)!(Gleiter!2000)!according!to!their!dimensionality,!as! schematically!shown!in!Figure!1.1!(a).!Those!structures!sit!on!a!magical!point!on!the! dimensional!scale:!a!length!scale!in!extent!between!isolated!atoms/molecules!and! microstructures.!Due!to!confinements!of!electron!distribut ion!in!small!dimensions,!a! ! 2! nanoscale!object!often!displays!physical!functionalities!substantially!different!from! either!individual!atoms/molecules!or!bulk!materials.!Their!electronic!and!magnetic! properties!are!often!distinguished!by!quantum!mechanical!beh avior,!while!their! mechanical!and!thermal!properties!can!be!understood!within!the!framework!of! classical!statistical!mechanics.!In!terms!of!mechanical!properties,!small!dimensions,! on!one!hand,!minimize!or!even!eliminate!the!presence!of!defects.!On!the!oth er!hand,! nanostructures!entail!large!surface!or!interface!areas.!The!absence!of!defects!makes! nanostructure!materials!stronger,!while!the!presence!of!surfaces!and!interfaces!may! bring! unexpected! effects! to! mechanical! strength.!Figure!1.1!(b)!shows! Young’s! modulus!of!a!nanostructure!as!a!function!of!dimensions,!which!was!calculated!using! a!combination!of!molecular!statics!and! ab)initio) calculation!(Zhou!and!Huang!2004).! It! was! shown! that! the! strength! of! the!copper! (Cu)!metal! nanostructure! could! increase!or!decrease!depending!on!both!sizes!and!surfaces!states! (Zhou!and!Huang! 2004).!Similar!distinguishable!behaviors!are!also!revealed!in! experiments.!In!Table! 1.1,! some! of! the! ultraYstrength! material! systems! and! phenomena! observed! in! nanocrystalline!structures!in!recent!experimental!studies!are!listed.! ! ! 3! !!!!!!! !!! ! Figure!1.1!(a)!Schematic!of!nanostructures!–!0D!nanoparticles,!1D!NWs,!and!2D! multilayers;! (Huang! and! Van! Swygenhoven! 2009 )! (b)! variation! of! normalized! Young’s!modulus!(E)!as!a!function!of!Cu!nanoplate!thickness!based!on!molecu lar! statics!and!ab!initio!calculation! (Zhou!and!Huang!2004).! ! Table!1.1!Experimentally!Measured!UltraYHigh!Strengths.! Material) Number)of)Layers) or)Diameter)(nm) ) Measured) Strength)(GPa)) Ideal)Strength) ~)E/10)(GPa)) Reference) CNT) SW! 30! 100! Falvo,!1997!! CNT) MW! 97Y110! 100! Peng,!2008!! ZnOPNW) 30! 7! 14! Wen,!2008!! SiPNW) 100Y200! 12! 17! Hoffmann,!2006!! AuPNW) 16.5! 7.3! 8! Wu,!2005!! AuPNW) 40! 5.6! 8! Wu,!2006!! AuPNP ) 300! 0.8! 8! Greer,!2006!! CNT:!carbon!nanotubes;!NP:!nanopillars;!E:!Young’s!modulus;!SW:!singleYwall;!MW:!multiYwall.!The! measured!data!were!from!tension,!compression!or!bending!experiments. )! ! 4! Nanoscale! materials! lie! on! the! borderline! of! the! smallest! human Ymade! devices!and!the!largest!molecules!of!living!systems.!Due!to!the!development!of! sophisticated!experimental!techniques!and!practical!tools,!e.g.!scanning!tunneling! microscopy!(STM)!(Binnig,!Rohrer!et!al.!1982 ),!atomic!force!microscopy!(AFM) ! (Binnig,!Quate!et!al.!1986 ),!scanning!probe!microscopy!(SPM) Ybased!techniques! (Sarid!1994)!and!lithography!techniques!(Xia!and!Whitesides!1998)!(Vieu,!Carcenac! et!al.!2000)!etc.,!researchers!are!able!to!understand,!characterize!and!manipulate! nanoscale!materials!and!processes.!However,!the!investigative!tools!and!level!of! understanding! of! basic! nanoscale! phenomena! are! still! rudimentary.! Many! fundamental!problems!remain!unveiled!due!to!the!limitation!of!experiments,!such!as! the!mysterious!competition!between!dislocation!nucleation!versus!multiplication!in! nanocrystalline! systems! (Weissmüller! and! Markmann! 2005),! dislocation! mechanisms!in!nanograins!and!their!contribution!to!the!plasticity !(Meyers,!Mishra! et!al.!2006;!Li!2007),!and!effects!of!single! defect!or!surface.!Some!key!elements!in!the! study! of! nanostructures! are! extremely! difficult! to! obtain,! such! as! surface! reconstruction,! atomic! structures! and! electronic! levels! at! interfaces,! the! stored! excess!energy,!the!free!volume,!and!the!internal!stress!distribution!and!strain Ytime! behavior!(Van!Swygenhoven!and!Weertman!2006 ).!! Atomistic!simulation!is!an!ideal!study!tool!for!nanoscale!objects!since!they! are!only!constructed!from!limited!number!of!atoms! or!molecules.!Computational! simulation!is!based!on!the!fields!of!mathematical!modeling!and!computer Ybased! simulation!(RafiiYTabar!2000),!which!allow!for!computation!and!prediction!of!the! ! 5! underlying! dynamics! of! nanostructures! and! processes.! For! smaller! dimensions,! electronic! calculations! are! feasible,! which! can! provide! full! details! of! atomic/electronic!behavior.!For!simple!atoms!and!molecules,!ab)initio!quantum! mechanical! calculation! methods!(Rowley,! Yang! et! al.! 2001)! have! successfully! produced!accurate!intermolecular!potential!functions.!While!ab)initio!calculations! may!be!satisfactory!for!simple!molecules,!even!with!today’s! most!sophisticated! computational!platforms!and!quantum!mechanical!techniques ,!system!sizes!that!can! be!studied!by!quantum!mechanical!methods!are!limited!to!a!few!hundred!atoms .! Consequently,!the!use!of!empirical!interatomic!and!intermolecular!potentials!in! simulations!is!necessary!to!study!nanostructures!consisting!of!several!millions!of! atoms!to!even!billions!of!atoms. ! The!capability!to!treat!nanostructures!with!sufficient!accuracy!using!classical! atomistic! simulations! is! invaluable! in!both! fundamental! science! and! applied! technologies.!It!can!provide!insight!into!the!formation,!evolution,!and!properties!of! nanostructures! and! mechanisms! of! nanoprocesses.! The! precision! of! calculation! depends!on!the!accuracy!of!the!interatomic!and!intermolecular !force!field.!These! simulations! can! enhance! our! understanding! of! atomic! and! molecular! scale! structures,!energetics,!dynamics,!and!mechanisms!underlying!various!physical!and! chemical!processes!that!unfolded!in!isolated!nanostructures!and!their!assemblies.!In! synergy!with!experiments,!atomistic!simulations!can!provide!useful!guidance !and! suggest!possible!parameters.!! ! 6! 1.2 Three)fundamental)problems)in)nanostructures)addressed)by)this ) dissertation) This!dissertation!focuses!on!three!fundamental!problems!in!nanostruc tures:!(1)! grain! boundary! embrittlement! of! nanocrystalline! metal;! (2)! thermo Ymechanical! behaviors!of!NWs;!and!(3)!planar!defect!generation!in!semiconductor!NWs.! ! 1.2.1 Grain)boundary)segregation)in)nanocrystalline)material ) The!unusual!mechanical!behavior!of!nano crystalline!materials!has!been!extensively! studied! for! many! years!(Meyers,! Mishra! et! al.! 2006 )!(Dang,! Shen! et! al.! 2002 ;! Yamakov,!Wolf!et!al.!2002;!Farkas,!Van!Petegem!et!al.!2005;!Farkas,!Mohanty!et!al.! 2007;!Greer,!Jang!et!al.!2009 ).!Greatly!increased!ductility!or!dramatically!enhanced! strength!and!hardness!are!observed.!Those!novel!behaviors!are!thought!to!arise! from!the!intricate!interplay!among!dislocation,!grain!boundary!(GB)!and!free!surface! all!within!a!relatively!small!volume.!However,!most!of!the!studies!to!date!have!been! focused!on!pure!nanocrystalline!systems!without!impuritie s.!It!is!known!that!the! alternation!of!chemical!composition!of!grain!boundaries!by!the!atomic!segregation! of!a!small!amount!of!impurities!could!cause!intergranular!brittleness!of!normally! ductile! nanocrystalline! materials!thereby!limiting! the! mechanical! properties! of! materials!(Koch!2007)!(Rice!and!Wang!1989)!(Messmer!and!Briant!1982).!For! nanocrystalline!materials,!the!grain!boundary!vs.!grain!interior!volume!ratio!could! be!a!degree!of!magnitude! bigger!than!microcrystalline!systems,!as!shown!in !Figure! 1.2.!Therefore,! the! influence! from! impurity! atoms! at! grain! boundaries! is! also! ! 7! magnified! in! nanocrystalline! systems.!Up! to! date,! neither! the! mechanism! of! segregation!of!individual!atoms!nor!grain!boundary!network!with!impurity!atoms! segregation!is!well!understood.! ! ! Figure!1.2!Core!and!mantle!model,!showing!relative!fractions!of!grain!boundary!and! grain!interior!regions!in!the!(a)!microcrystalline!and!(b)!nanocrystalline!regimes.! (Meyers,! Mishra! et! al.! 2006 )!(c)! The! effect! of! grain! size! on! calculated! volume! fractions!of!intercrystal!regions!and!triple!junctions,!assum ing!a!grainYboundary! thickness!of!1!nm(Palumbo,!Thorpe!et!al.!1990 ).!! A!prime!example!of!such!GB!mechanochemistry! (Gilman!1996)!is!sulfur!(S)! segregationYinduced!embrittlement!of!nickel!(Ni)! (Rogers!1968;!Heuer,!Okamoto!et! al.!2002).!Fundamental!understanding!and!control!of!embrittlement!mechanisms!in! the!NiYS!system!are!important!not!only!scientifi cally!but!also!technologically,! e.g.,!for! the!development!of!nextYgeneration!nuclear!reactors!(Allen,!Sridharan!et!al.!2008 ).! Despite!decades!of!intense!experimental! (Rogers!1968;!Heuer,!Okamoto!et!al.!2002 )! ! 8! and!theoretical!(Rice!and!Wang!1989;!Schweinfest,!Paxton!et!al.!2004 ;!Yamaguchi,! Shiga! et! al.! 2005)!efforts,! SYinduced! fracture! phenomena!remained! a! mystery.! Experiments! by! Heuer!et) al.!(Heuer,! Okamoto! et! al.! 2002 )!have! provided! key! insights!into!the!problem.!They!found!the!existen ce!of!a!critical!S!concentration!at! the! Ni! GB! that! causes! a! transition! from! ductile,! mixed! transgranular! and! intergranular!fracture!to!brittle,!intergranular!fracture.!As!shown!in! Figure!1.3!(a),! scanning!electron!micrographs!from!fracture!surfaces!of!Ni!tensile!specimen!with! 2.79!at.%!S!appear!to!be!ductile!tearing,!whereas!with!increased!S!concentration!of! 15.86!at.%!S,!fracture!surfaces!in!Figure!1.3!(c)!appear!to!be!brittle!cleavage.!The! transition! from! transgranular! to! brittle! fracture! has! also! been! quantified! by! percentage! of! intergranular! fracture! using! the! data! set! from! the! Ni! tensile! specimens!as!shown!in!Figure!1.4.! ! ! Figure!1.3!Scanning!electron!microscopy!images!from!fracture!surfaces!of!Ni!tensile! specimens!show!the!transition!from!ductile!tearing,!mixed!transgranular!to!brittle! intergranular!fracture!with!increased!grain Yboundary!S!concentrations:!(a)!2.79! at.%!S,!(b)!11.59!at.%!S,!and!(c)!15.86!at.%!S !(Heuer,!Okamoto!et!al.!2002 ).! ! ! 9! ) Figure!1.4!Influence! of! grainYboundary! S! concentration! on! the! transition! from! ductile!to!brittle!fracture!as!measured!by!the!percentage!of! intergranular!fracture! (Heuer,!Okamoto!et!al.!2002 ).! Another! important! factor! reported! by! Heuer!et) al.)is! that!the! critical! S! concentration! for! GB! embrittlement! was! found! to! coincide! with! that! for! the! amorphization!of!Ni!single!crystal!under!ion!implantation! (Heuer,!Okamoto!et!al.! 2002).! The! amorphous! fraction! is! plotted! as! a! function! of!S! concentration! determined!in!Figure!1.5.!The!figure!indicates!that!the!transition!from!crystalline!to! amorphous!structure!occurs!over!a!sharp! range!of!S!concentration.!! ! 10! ! Figure!1.5!Critical!S!concentration!as!a!function!of!amorphous!fraction !(Heuer,! Okamoto!et!al.!2002).! Yamaguchi! et)al.)(Yamaguchi,!Shiga!et!al.!2005))performed!firstYprinciples! quantumYmechanical!(QM)!calculations!to!study!the!decohesion!of!Ni!GB!with!1 Y2! monolayers!of!segregated!S!atoms,!and!proposed!S Yinduced!reduction!of!GB!tensile! strength!as!an!embrittlement!mechanism.!Although!this!study!provi des!information! about!change!in!local!chemical!bonding!environment!due!to!S!segregation,!it!does! not! explain! the! experimentally! found! relation! between! embrittlement! and! amorphization.! To! solve! this! long Yunsolved! mystery,! Chen!et) al.! employed! a! multiscale!simulation!approach!that!involves!48!million Yatom!MD!simulations!based! on!reactive!force!fields!(ReaxFF )!(Nielson,!van!Duin!et!al.!2005 ;!Nomura,!Kalia!et!al.! 2008),!which!are!trained!and!validated!by!first Yprinciples!QM!calculations!based!on! the!density!functional!theory!(DFT)! (Hohenberg!and!Kohn!1964;!Payne,!Teter!et!al.! 1992),!to!study!the!effect!of!S!segregation!on!the!fracture!of!nanocrystalline!Ni.!The! serials!of!simulations!reveal!a!direct!link!between!S Yinduced!GB!amorphization!and! ! 11! embrittlement:! an! orderYofYmagnitude! reduction! of! GB! shear! strength! due! to! amorphization! was! found,! which,! combined! with! tensile Ystrength! reduction,! provides! an! easy! cleavage! path! at! the! crack! tip.! This! amorphization Yinduced! mechanism!also!explains!an!experimen tally!observed!crossover!from!transgranular! to!intergranular!fracture!as!well!as!suppression!of!plastic!activities! (Heuer,!Okamoto! et!al.!2002).! Now! the! relation! between! S! segregationYinduced! amorphization! and! embrittlement!in!Ni!has!thus!been!clarified,! the!only!remaining!question!is,!why!and! how!the!sharp!amorphization!transition!occurs!within!such!a!narrow!concentration! range!at!the!critical!S!concentration!of!~!14.2!%.!In!this!dissertation,!we!extend!our! previous! study! and! use! ReaxFF! MD! simulations! to!investigate! the! change! of! structural!properties!of!crystalline!Ni!as!a!function!of!S!composition.!The!simulation! results!reveal!that!the!large!steric!size!of!S!impurity!causes!strong!S YS!interaction! mediated!by!the!distortion!of!Ni!crystalline!lattice!up!t o!the!next!nearestYneighbor! lattice!sites!and!that!amorphization!occurs!at!the!percolation!threshold!of!the!S YS! network!with!the!next!nearest Yneighbor!connectivity.!Furthermore,!we!confirm!the! generality!of!the!amorphization!mechanism!due!to!the!percolatio n!of!an!impurity! network! for! a! model! binary! material! described! by! Lennard YJones! interatomic! potential.!Our!work!in!this!dissertation!thus!provides!an!atomistic!mechanism!of! impurityYinduced!amorphization.! ! ! 12! 1.2.2) Novel)thermoPmechanical)behaviors)of)nanowir es)) Since!the!discovery!of!carbon!nanotubes !(Iijima!1991),!there!has!been!great!interest! in! the! study! of! one Ydimensional! nanoscale! materials:!NWs,! nanopillars! (NPs),! nanobelts! (NBs),! nanofibers,!etc.! NWs! and! NPs! are! distinct! from! their! bulk! counterparts!due!to!the!high!surface YtoYvolume!ratio!and!the!reduced!number!of! defects.!At!the!nanoscale!level,!surface Yrelated!properties!are!more!signif icant,!and! can! even! be! dominant! in! determining! the! properties! and! characteristics! of! nanostructures.! It! is! already! recognized! that! the! properties! of! nanostructures! depend!sensitively!on!surface!composition!and!surface!states/defects !(Pan!and!Zhu! 2009).! For! example,! Lu!et) al.! reported! that! thin! zinc! oxide! (ZnO) ! NWs! have! significantly! higher! electrical! conductivity! due! to! enrichment! of! surface! states ! (Chang,!Chien!et!al.!2007 ).!Lieber!et)al.!reported!that!surface!modification!of!silicon! NWs!could!lead!to!remarkable!enhancement!of!carrier!mobility !(Cui,!Zhong!et!al.! 2003).!Lewis!et)al.!showed!the!fabrication!of!high Yperformance!airYstable!silicon!NW! field!effect!transistors!(FETs)!by!modifying!the!surfaces!of!silicon!NWs!with!Si YCH3! functionality!(Haick,!Hurley!et!al.!2006 ).!It!has!been!also!pointed!out!that!surface! scattering!plays!a!dominant!role!in!the!transport!properties!of!small Ydiameter!NWs! (<!100nm),!whereas!transport!in!intermediate!diameter!(120! –!180!nm)!NWs!is! dominated! by! interface! scattering! (Motayed,! Vaudin! et! al.! 2007 ).! These! nanostructures! provide! ideal! systems! for! studying! transport! processes! of! one Y dimensionally!confined!objects!and!related!fundamental!phenomena,!and!they!are! also! very! important! for! developing! new! generation! nanodevices! with! high! ! 13! performance.!! A!number!of!research!groups!have!carried!out!mechanical!measurements!on! individual!NWs!using!different!approaches,!predominantly!using!either!AFM! (Wong,! Sheehan! et! al.! 1997)! (Song,! Wang! et! al.! 2005 )! or! an! inYhouseYbuilt! system! incorporated!within!transmission!electron!microscopy!(TEM)! (Poncharal,!Wang!et! al.!1999)!(Ruoff,!Tse!et!al.!1993 )!or!scanning!electron!microscopy!(SEM)! (Yu,!Lourie! et! al.! 2000)! (Greer! and! Nix! 2006;! Hoffmann,! Utke! et! al.! 2006 )! (Östlund,! RzepiejewskaYMalyska!et!al.!2009).!Figure!1.6!shows!one!example!of!size Ydependent! mechanical!properties! that!was! tested! by! Greer!et)al.)on! metal! NWs! with! bcc! (molybdenum)! and! fcc! (gold)! structures ! (Greer! and! De! Hosson! 2011 ).! This! experiment!identified!two!fundamental!types!of!mechanical!behaviors!of!metal!NWs ! (Brinckmann,!Kim!et!al.!2008 ).!Thermal!responses!are!also!intensively!studied!( (Li,! Wu!et!al.!2003)!(Yu,!Shi!et!al.!2005)!(Wu,!Yan!et!al.!2002)!(Qi!2005)!(Mingo!2003)).! For! example,!as! shown! in!Figure! 1.7! (a)! and! (b)! respectively,! experimentally! measured!thermal!conductivity!of!different!diameter!Si!NWs !(Li,!Wu!et!al.!2003)!and! melting!temperature!change!as!a!function!of!indium!wire!diameter! (Qi!2005)!all! show!a!strong!size!dependency.!The!reported!experimental!results!are!consistent! with! computational! findings! by! molecular! dynamics! (MD)! and! dislocation! simulations!(Deshpande,!Needleman!et!al.!2005 )!(Nicola,!Van!der!Giessen!et!al.! 2003)!(Espinosa,!Panico!et!al.!2006 )!(Devincre,!Kubin!et!al.!2001 )!(Chang,!Cai!et!al.! 2002).!! ! 14! !!!!!!!!!!!!!!! ! Figure!1.6!Representative!stress!vs.!strain!curves!for!(a)!fcc !Au!and!(b)!bcc!Mo! nanopillars;! (c)! a! logYlog! plot! of! flow! stress! as! a! function! of! initial! diameter! representing!the!scaling!laws!for!Mo!and!Au.!The!slope!of!strengthening!in!Au!is! nearly!2X!higher!than!that!for!Mo. !(Greer!and!De!Hosson!2011 ).! ! 15! ! Figure!1.7!(a)!Measured!thermal!conductivity!of!different!diameter!Si!nanowires.! The!number!beside!each!curve!denotes!t he!corresponding!wire!diameter!(Li,!Wu!et! al.!2003);!(b)!melting!temperature!of!In!nanowires!as!a!function!of!wire!diameter.! The! solid! line! is! calculated! theoretically! while! the! symbols! denote! the! corresponding!experimental!values!(Qi!2005).! Another!notable!phenomenon!is!recent!discovery! sizeYdependent!brittleYtoY ductile!(BTD)!transition!semiconductor!NPs!(Östlund,!RzepiejewskaYMalyska!et!al.! 2009)!(Östlund,! Howie! et! al.! 2011 )!(Kang! and! Cai! 2010).!Figure!1.8!shows! an! example! of! the!BTD! transition! in!nominally! brittle! materials! below! a! critical! ! 16! diameter.!The!sharp!transition!can!be!observed!between!310!to!400!nm,!where! NPs! with!diameters!greater!than!that!tend!to!fracture.! The!current!studies!on!the!sizeY dependent! mechanical! behaviors! are! focused! on! dislocation! slip! and! defect! nucleation! within! a! confined! volume!(Li! 2007):! dislocation! activation! energy! dependence! on!volume! change;! crossYslip! planes!in! confined! volume;!and!the! dislocation!nucleation,!disappearance,! repellence! or! attraction!to! free! surfaces.! None! of!those! theories!take! into! consideration!the!volume! associated! with! the! surface!structure!change.!They!also!failed!to!explain!the!presence!of!BTD!transition! in!a!dislocation!free!crystal .!! ! Figure!1.8!SEM!images!of!silicon!pillars!compressed!with!nanoindenter.!(a)!pillar!of! 400nm!(b)!pillar!of!310!nm!and!(c)!critical!engineering!stress.!(The!pillars!that! showed!cracking!are!encircled;!all!the!other!pillars!deformed!purely!plastic.!The! triangle!correspond!to!experiments!with!an!MTS!and!the!diamonds!c orrespond!to! experiment!in!situ.)!(Östlund,!RzepiejewskaYMalyska!et!al.!2009)! Understanding!such!nanoYthermoYmechanical!properties!is!critical!for!the! reliability,!manufacturability,!and!optimization!of!a!wide!ra nge!of!devices!consisting! ! 17! of!NWs!(Keten,!Xu!et!al.!2010 ).!A!prototypical!example!is!ZnO.!Although!many!efforts! on!nanomechanical!properties!have!been!made!(Wang!2004;!Elias,!LevyYClement!et! al.!2010),!the!size!and!temperature!dependence!of!the!failure!behavior!of!ZnO!NWs ! remains!unsolved.!In!this!dissertation,!a!complete!sizeYtemperature!‘phase!diagram’! for!the!mechanical!response!of!a!ZnO!NW !is!constructed!based!on!a!series!of!MD! simulation! of! [0001]Yoriented! ZnO! NW! under! different! condition .! The! phase! diagram!can! serve! as! a! bridge! to! connect !the! controversy! between! theoretical! studies!(Dai,!Cheong!et!al.!2010 )!and!observations!in!experiment!(Agrawal,!Peng!et! al.!2009).!An!intrinsic!core/shell!structure!in!semiconductor!NW!is!revealed.!The! atomistic!mechanisms!of!the!unique!nano YthermoYmechanical!behavior!and!their! relations!with!surface!will!be!unveiled.!!! 1.2.2 Stacking)defects)during)nanowire)growth ) In!order!to!build!high!functional!nanodevices,!a!great!control!of!the!crystal!structure! is!critical.!NW!synthesis!method!can!be!divided!into!2!categories !(Sun,!Kargar!et!al.! 2011):!catalyst!growth!and!catalyst Yfree!growth.!In!catalyst!growth,!metals,!such!as! Au,!Pt,!Ni,!Ag,!Al,!as!well!as!alloys!such!as!Au/Pd,!are!used!as!sinks!fo r!atomic! precipitation,!resulting!in!supersaturation!that!drives! NW!growth!(Paek,!Nishiwaki! et!al.!2009).!The!most!commonly!used!techniques!include!metal Yorganic!chemical! vapor!deposition!(MOCVD)!(Pan!and!Zhu!2009)!(!!!!INVALID!CITATION!!!!)!(Bjork,! Ohlsson!et!al.!2002),!molecular!beam!epitaxy!(MBE) !(Morales!and!Lieber!1998)! (Persson,!Larsson!et!al.!2004 ),!chemical!beam!epitaxy!(CBE) !(Messing,!Hillerich!et!al.! ! 18! 2009)!(Dick,!Deppert!et!al.!2005 )!and!chemical!vapor!deposition!(CVD) !(Green! 2003).!Interactions!between!metal!catalyst!and!semiconductor!materials!have!been! thoroughly!studied.!It!was!discovered!that!adatom!surface!diffusion!rates!play!a! significant!role!in!determining!NW!growth!rate!and!morphology.!The!edge!of!the! NW!top!surface!is!important!in!dictating!the!grown!crystalline!phase !(Glas,!Harmand! et!al.!2007).!Though!semiYempirical!explanations!based!on!surface!energies!have! been!attempted,!atomistic!mechanisms!underlying!the!edge Ymediated!stacking!fault! (SF)!nucleation!is!still!elusive!(Glas,!Harmand!et!al.!2007 ).!) Although!catalysts!in!vaporYliquidYsolid!(VLS)!growth!show!some!desirable! features!(e.g.,!it!can!provide!preferential!growth!along!a!single!axis!even!on!lattice! mismatched!substrates),!they!have!presented!serious!limitations.!One!example!is! the!metal!atom!diffusion,!which!provides!deep!impurity!levels!in! NWs.!Therefore,! catalystYfree!growth!methods,!like!selectiveYarea!MOVPE!method(Colombo,!Heiß≤!et! al.!2009;!Dong,!Tian!et!al.!2009 ),!selfYcatalyzed!VLS!through!groupYIII!elements!(Xu,! Guo!et!al.!2009)!or!SiYassisted!growth!(Tian,!Kempa!et!al.!2009 ),!are!on!demand.! CatalystsYfree!growth!is!not!a!single YmechanismYgoverned!process.!It!is!controlled! by!nucleation!and!growth!at!the!same!time!and!condition.!Factors!such!as!growth! temperature!(Bjork,!Ohlsson!et!al.!2002 ),!partial!pressure!and!(Xu,!Guo!et!al.!2009 )! substrate! orientation!(Van! Weert,! Wunnicke! et! al.! 2006 )! are! all! important! in! determining!the!NW!crystallinity,!orientation!and!morphology.! ! Although! NWs! typically! exhibit! few! crystal! defects,! the! III YV! compound! materials!(for!instance,!GaAs,!GaP,!InP!and!InAs)!based! NWs!often!show!randomly! ! 19! distributed!rotational!twin!planes!and!stacking!fa ults!in!both!catalyst!and!catalyst Y free! synthesis.!Figure!1.9!shows! some! examples! of! wurtzite! (WZ)! segment! in! zincblende!(ZB)!structure!IIIYV!NWs!and!the!presence!of! twin!planes!during!VLS!and! selectiveYarea!growth.!Despite!the!tremendous!efforts!on!controlling!the!stacking! fault! and! twin! planes,! the! minimization! of! those! defects! remain s! technically! challenging.!! The!presence!of!WZ!phase!in!ZB!structure!and!twin!segmen ts!tend!to!affect! both!optical!and!nanoelectronic!properties!of!NWs.!As!shown!in!Figure!1.10!(a),! mixture! of! the! crystal! phases! can! exhibit! up! to! 2! orders! of! magnitud e! higher! resistivity!than!singleYphase!NWs!(Thelander,!Caroff!et!al.!2011 ).!Another!research! (see!Figure!1.10!(b))!calculated!electron!scattering!at!a!single!SF!and!a!group!of!SF! interfaces!and!showed!that!they!could!degrade!electronic!properties !(Stiles!and! Hamann!1990)!(Stiles!and!Hamann!1988).!In!terms!of!optical! properties!(see!Figure! 1.10! (c)),! Wagner! et) al.) developed! a! technique! to! perform! TEM! and! microphotoluminescence!on!the!same!NW!over!a!large!range!of!optical!power!and! observed!a!shift!of!photoluminescence!on!the!rotatio nally!twinned!ZB/WZ!NW!due! to!the!change!of!band!alignment !(Bao,!Bell!et!al.!2008 ).!To!eliminate/control!the! defect,!fundamental!understanding!of!the!formation,!structure!and!energetics!of!the! planer!defects!and!growth!kinetic!process!is!required.!! ! ! 20! ! Figure!1.9!Wurtzite!(WZ)!segment!and!twin!plane!in!III YV!nanowires:!(a)!TEM!image! along!<110>!axis!of!VLS!InAs!nanowires!showing!the!presence!of!WZ!phase;!(b)!SEM! image! (tilt! angle! 30˚)! revealing! the! complex! shape! of! VLS! InAs! nanowires! and! associated! threeYdimensional! atomic! model;! (c)! TEM! image! of! zoomed! in! area! around!rotational!twin!planes!in!(b);! (Caroff,!Dick!et!al.!2008 )!(d)!TEM!image!of!a! GaAs!nanowire!grown!using!selective Yarea!growth!method!showing!a!transition! from!zincblende!(ZB)!to!WZ;!(Tomioka,!Kobayashi!et!al.!2009 )!(e)!TEM!images!of! GaAs!nanowire!with!twin!segment.! (Yoshida,!Ikejiri!et!al.!2009)! ! ! 21! ! ! Figure!1.10!Effect!of!twin!and!wurtzite!segment!on!electrical!and!optical!properties! of!nanowire:!(a)!InAs!nanowire!resistivity!against!nanowire!diameter!and!defect! densities!(Thelander,!Caroff!et!al.!2011 )!;!(b)!transmission!of!electrons!through! silicon!stacking!faults!with!different!separations !(Stiles!and!Hamann!1988);!(c)! Photoluminescence!spectrum!of!the!nanowire!with!and!without!planer!defects!with! different!excitation!intensities!(Bao,!Bell!et!al.!2008 ).!! In!this!dissertation,!we!chose!NWs!composed!of! GaAs!as!an!example!to! explore!the!underlying!mechanisms!of!stacking!defect!generation!during! NW!growth.! GaAs!NWs!have!traditionally!been!grown!by!the!VLS!technique. (Persson,!Larsson!et! al.!2004)!Recently,!an!alternative!growth!method!called!selective!area!metal!organic! vaporYphase!epitaxy!(SAYMOPVE)!has!attracted!much!attention!due!to!its!ability!to! form!atomically!sharp!interfaces!to!fabricate!axial!heterostructures!such!as!tandem! NW!solar!cells.(Tomioka,!Ikejiri!et!al.!2011 )!The!catalystYfree!SAYMOPVE!method! ! 22! also!avoids!metal!impurities!that!degrade!VLS Ygrown!NWs!(Tomioka,!Ikejiri!et!al.! 2011).!In!this!dissertation,!a!series!of!MD!simulations!are!performed.!The!simulation! results!reveal!the!role!of!intrinsic!core/shell!structure!for!adatom!energetics!on!the! (111)B!top!surface!of!a![111] Yoriented!GaAs!NW,!in!which!SFs!are!preferentially! nucleated!in!the!shell.!We!have!also!developed!a!nucleation!growth!model,!which! suggests!optimal!growth!temperatures!and!pressures!for!minimizing!SFs!as!well!as! a!novel!sizeYcontrol!growth!method!for!eliminating!them. ! 1.3 ) Dissertation)overview) ) The!central!contribution!of!this!dissertation!is!the!investigation!of!fundamental! mechanisms!of!nanoscale!material!behaviors.!The!discussion!covers!three!different! behaviors! of! three! different! materials:! transformation! of! grain! boundaries! in! nanocrystalline! nickel! (CHAPTER! 4);! novel! thermomechanical! behaviors! in!ZnO! NWs! (CHAPTER! 5);! and! stackingYfault! generation! in!GaAs!NWs! (CHAPTER! 6).! Parallel! molecular! dynamics! simulation! methods,! including! molecular! dynamics! simulation! algorithms! and! calculation! of! physica l! properties,! are! presented! in ! CHAPTER!2.!The!interatomic!interactions!used!in!MD!simulations!in!the!dissertation! along!with!their!validations!are!presented!in !CHAPTER!3.!! ! The!organization!of!this!dissertation!is!illustrated!in!the!following!diagram,! which!serves!as!a!connection!map!between!chapters!and!interatomic!poten tials!that! are!used,!in!order!to!guide!the!reading.! ! ! 23! ! ! • Chapter)Descriptions! Chapter)2!introduces!concepts,!elements,!theories!and!algorithms!in!molecular! dynamics! simulation,! from! the! physics! foundation! to! computational! methodologies.!It!also!covers!how!t he!basic!physical!quantities!used!in!later!in! the! dissertation! are! calculated,! including! thermo Ymechanical! properties,! structural!quantities,!dynamic!properties,!and!slip!vector!field.! ! ! 24! Chapter)3)continues!the!introduction,!presenting!the!most!important!in gredient! in!MD!–!interatomic!interactions.!The!Chapter!presents!the!formula,!parameter! set!and!validation!of!3!types!of!potentials:!pair!potential,!many Ybody!potential! consisting!2!and!3!body!interactions!and!reactive!force!field,!which!is!built!on!the! concept!of!chemical!bonds.!The!use!of!the!potentials!in!the!dissertation!is!shown! in!the!figure!above.! ! Chapter)4)applies!reactive!force!field!to!the!simulation!of!nanocrystalline!nickel! to! study! the! SYinduced! amorphization! at! grain! boundary.! It! proposes! a! hypothesis!using!percolation!analysis,!which!explains!the!sharp!transition!at! grain!boundary!as!a!percolation!threshold.!The!hypothesis!is!later!validated! using!a!simple!model!system!based!on!a!binary !LennardYJones!potential.!! Chapter) 5) develops! a! sizeYtemperature! ‘phase! diagram’! for! the! mechanical! response!of!zincYoxide!NWs!using!a!many!body!force!field.! The!phase!diagram! resolves! controversies! between! previous! experiments! and! theory,! and! the! predicted!‘intrinsic’!core/shell!structure!may!find!novel!device!applications. ! Chapter)6!extends!the!discovery!of!‘intrinsic’!core/shell!structure!in!wurtzite! semiconductor!NW!to!zincYblende!NW!–!GaAs.!It!is!revealed!in!this!chapter!that! the!intrinsic!core/shell!structure!might!serve!as!the!root!cause!of!stacking!fault! generation!during!selective!area!growth.!! ! Chapter)7!concludes!the!dissertation.! ) ) ! 25! CHAPTER)2 ) ) ) ) ) ) ) ) ) ) MOLECULAR)DYNAMICS)SIMULATION )METHOD) The! main! focus! of! this! chapter! is! the! molecular! dynamics! (MD)! simulat ion! methodology.!In!MD,!materials!are!viewed!as!a!collection!of!discrete!atoms.!The! atoms!interact!by!exerting!forces!on!each!other,!and!they!follow!Newton’s!equation! of! motion.! Based! on! the !interaction! model,! a! simulation! computes! the! atoms’! trajectories! numerically!(Bulatov! and! Cai! 2006).! Thus! a! molecular! simulation! necessarily!contains!the!following!ingredients! (Frenkel!and!Smit!2001):! • The!model!that!describes!the!integration!between!atoms.!This!is!usually! called!the!interatomic!potential:!V({r i }),!where!{r i }!represent!the!position!of! all!atoms.!! • Numerical!integrator!that!solves!the!atoms’!equation!of!motion.! ! • Extraction!of!useful!data!from!the!raw!atomic!trajectory!information.! ! ! Here,!an!overview!of!statistical!ensembles!will!be!first!presented,!together! with! integration! algorithms! used! in! molecular! d ynamics.! In! section! 2.2,! the! calculation! of! thermoYmechanical,! structural,! and! dynamic! properties! will! be! presented.! In! section! 2.3,! parallel! implementation! used! in! simulations! of! multimillion!atoms!will!be!discussed,!followed!by!energy Yminimization!methods! employed!in!this!thesis.!The!interatomic!potentials!used!in!this!dissertation!will!be! introduced!in!CHAPTER!3.!! ! 26! 2.1 Molecular)dynamics)simulation) ) 2.1.1 Newton’s)equation) of)motion) ) Molecular!dynamics!(MD)!is!a!simulation!approach!where!the!time!evolution!of!a!set! of!interacting!atoms!is!followed!by!numerically!solving!their!equation!of!motion.!I n! MD,!we!deal!with!classical!dynamics,!following!the!three!laws!of!Newton! (Landau,! Lifshitz!et!al.!1976):!! First)law:)Every!object!in!a!state!of!uniform!motion!tends!to!remain!in!that! state!of!motion!unless!an!external!force!is!applied!to!it.! ! Second)law:) An!object’s!mass!m,)its!accelerations! a,!and!the!applied!force! F! are!related!by!! ! ! m i r i = F i ,! ! ! ! ! ! ! !!!!!!!!!! (Eq.!2.1)! for!each!i!in!a!system!constituted!by! N!atoms.!Here,! € m i !is!the!atomic!mass,! r the!acceleration,!and! F i !is!the!total!force!acting!on!atom!i.!In!this!thesis,!we! consider!atomic!forces!that!are!derived!from! the!total!potential!energy!U,!! ! F i =− ∂U( r 1 ,..., r N ) ∂ r i .! ! ! ! ! ! !!!!!!!! !(Eq.!2.2)! Third)law:) For!every!action!there!is!an!equal!and!opposite!reaction.!! ! ! The!total!energy!is!conserved!if!the!atoms!follow!the!Newton’s!equation!of! motion!in!a!conservative!force!field!( i.e.,!when!force!can!be!written!in!terms!of! spatial!derivative!of!potential!field),!while!the!kinetic!energy!and!potential!energy! ! 27! can!themselves!can!change!values!dynamically.!Newton’s!equation!of!motion!can! also!be!written!in!the!Hamiltonian!form! (Allen,!Tildesley!et!al.!1989 ).! ! d r dt = ∂H( r, p) ∂ p !,! ! ! ! ! ! ! !!!!!!!! (Eq.!2.3)! ! d p dt =− ∂H( r, p) ∂ r .! ! ! ! ! ! ! !!!!!!!! (Eq.!2.4)! Hence,!! ! dH dt =− ∂H( r, p) ∂ r . d r dt + ∂H( r, p) ∂ p . d p dt = 0! ! ! ! !!!!!!!! (Eq.!2.5)! 2.1.2 Numerical)integrator) A!dynamical!simulation!computes!atomic!positions!as!a!function!of!time!( i.e.!their! trajectories)!given!their!initial!positions!and!velocities.!Because!Newton’s!equation! of!motion!is!second!order!in!time,!an!initial!condition!needs!to!specify!both!positions! and!velocities!of!all! the!atoms.!! ! To!solve!the!equation!of!motion!on!a!computer,!we!first!need!to!discretize!the! time.!Namely,!we!determine! r(t)!at!a!series!of!time!instances! t n .!Usually,!the!time! axis!is!discretized!uniformly:!t n = n⋅Δt,!where!Δt!is!called!the!time)step.) The!task!of! a!simulation!algorithm!is!to!find! r(t n )!for!i!=!1,2,3,!…,!given! r(0)!and! v(0)!(Allen,! Tildesley!et!al.!1989).! ! The!Verlet!algorithm!begins!by!approximating! d 2 r(t)/dt 2 !as! ! 28! ! d 2 r(t) dt 2 ≈ r(t+Δt)−2 r(t)+ r(t−Δt) Δt 2 .! ! ! ! ! !!!!!!!!! (Eq.!2.6)! Thus,! ! r(t+Δt)−2 r(t)+ r(t−Δt) Δt 2 =− 1 m dU( r(t)) d r ,!! ! ! ! !!!!!!!! !(Eq.!2.7)! ! r(t+Δt)=2 r(t)− r(t−Δt)−Δt⋅ 1 m dU( r(t)) d r .!! ! ! !!!!!!! !!(Eq.!2.8)! ! Therefore,!we!can!find! r(t+Δt)!if!we!know! r(t)!and! r(t−Δt).!In!other!words,! if!we!know! r(0)!and! r(Δt),!we!can!solve!for!all! r(t n )!for!n)=2,3,4,….! ! While!this!is!sufficient!for!computing!the!entire!trajectory!of!the!atoms,!it! does!not!specify!the!velocities!of!the!atoms,!which!are!often! needed!as!a!function!of! time!as!for!computing!various!physical!quantities.!F or!example,!the!velocities!are! needed!to!compute!the!kinetic!energy!of!the!system.!One!way!to!compute!velocity!is! through!the!approximation,!! ! v(t)≡ d r dt ≈ r(t+Δt)− r(t−Δt) 2Δt ! ! ! ! ! !!!!!!!!! (Eq.!2.9)! which!amounts!to!the!Verlet!algorithm.!According!to!this!algorithm,!we!only!know! v(t) !after! we! have! computed! r n (t+Δt) ! during! the! simulation.! This! is! one! disadvantage!of!the!original!Verlet!algorithm.! ! ! In!our!MD!simulations,!the!velocity!Verlet!algorithm!is!used,!because!it!is! timeYreversible! and! symplectic,! in! addition! to! resolving! the! aforementioned! problem! of! the! original! Verlet! algorithm.! The! velocityYVerlet! algorithm! is! a! ! 29! modification!of!the!Verlet!algorithm!where!the!position!of!atom!i!at!time! t t Δ + is! obtained!using!Taylor!series!expansion.!The!expansion!around! t!gives! ! r(t+Δt)= r(t)+Δt v(t)+ 1 2 Δt 2 a(t)+ 1 6 Δt 3 b(t)+O(Δt 4 )! ! !!!!!! !!(Eq.!2.10)! ! r(t−Δt)= r(t)−Δt v(t)+ 1 2 Δt 2 a(t)− 1 6 Δt 3 b(t)+O(Δt 4 )! !!!!!!! ! !!!!!! !!(Eq.!2.11)) Adding!the!two!equations,!one!gets ! ! r(t+Δt)= 2 r(t)− r(t−Δt)+Δt 2 a(t)+O(Δt 4 ),! ! ! !!!!! !!!(Eq.!2.12)! and!the!velocity!is!obtained!by!subtracting! the!two!equations:!! ! v i t ( ) = r i t+Δt ( ) − r i t−Δt ( ) 2Δt +O Δt 3 ( ) .!! ! ! ! !!!!! !!!(Eq.!2.13)! Therefore,!the!discretization!error!in!the!velocity!is!higher!than!that!in!the!position .! In!simulations!where!velocity!plays!an!essential!role,!the!drawbacks!of!the!Verlet! algorithm!can!be!eliminated!by!the!velocity YVerlet!algorithm.!In!this!algorithm,!the! velocity!at!half!the!time!step!is!given!by! ! v(t+ 1 2 Δt)= v(t)+ 1 2 a(t)Δt .! ! ! ! ! !!!! (Eq.!2.14)! The!position!at!time! t t Δ + !is! ! r(t+Δt)= r(t)+Δt v(t)+ 1 2 a(t)Δt 2 .! ! ! ! ! !!!!!! (Eq.!2.15)! From! the! updated! positions,! the! corresponding! forces! and! accelerations!are! computed.!The!velocity!at!time! t t Δ + !is!then!given!by! ! v(t+Δt)= v(t+ 1 2 Δt)+ 1 2 a(t+Δt)Δt,!! ! ! !!!! !!!!! (Eq.!2.16)! ! 30! where! a(t+Δt)!are!the!accelerations!of!atoms!after!the!position!update.! ! 2.1.3 Microcanonical)and)canonical)ensembles ) • Microcanonical)(NVE))ensemble) ) Consider!a!MD!simulation!of!a!system!of! N!atoms,!interacting!with!each!other! through!the!potential!function!U({ r i }),!subjected!to!periodic!boundary!conditions! on! a! simulation! box! spanned! by! three! vectors,! c 1 , c 2 , c 3 .! For! applications! in! this! thesis,!the!repeat!vectors!remain!fixed!during!the! simulation;!hence!the!simulation! volume!V =( c 1 × c 2 )⋅ c 3 !remains!constant.!In!one!case,!the!simulation!corresponds!to! a! closed! system,! for! which! the! total! energy! also! remains! constant! along! the! trajectory,!i.e.,!H({ r i , p i })=E ,!where! !! ! H = p i 2 2m i i=1 N ∑ +U({ r i }). !! ! ! ! ! !!!!!!!! (Eq.!2.17) ! ! This!simulation!samples!the!microcanonical,!or!( NVE)!ensemble,!because!the! number!of!atoms!N,!volume!V!and!total!energy!E!are!all!conserved.!In!the!NVE! ensemble,!Hamilton’s!equations!of!motion!are! ! r i = ∂H ∂ p i , p i =− ∂H ∂ r i .!! ! ! ! ! !!!!!! !(Eq.!2.18)! ! The! dynamical! trajectory! of! a! closed! Hamiltonian! system! samples! the! microcanonical!ensemble!corresponds!to!the!following!density!distribution!in!the! phase!space,! ! 31! ! f NVE ({ r i , p i })= 1 Ω(E,δE) 0 " # $ % $ !!!!!! H({ r i , p i })∈[E−δE,E] otherwise ! !!!!!! ! !!!!!!!! (Eq.!2.19)! where!δE!is!a!small!parameter!and!Ω(E,δE)!is!the!normalization!constant,! ! ! Ω(E,δE)= d N rd N p E−δE≤H({ r i , p i })≤E ∫ ,!! ! ! ! !!!!!!! (Eq.!2.20)! which!is!the!volume!of!the!phase!space!covered!by!the!microcanonical!ensemble.!As! the! system! approaches! equilibrium,! its! entropy!S! reaches! a! maximum.! This! maximum!value!of!entropy! S!is!a!function!of!N,!V!and!E,! ! S(N,V,E)=k B ln Ω(N,V,E),!! ! ! ! ! !!!!!!!! (Eq.!2.21)! ! Ω(N,V,E)= Ω(E,δE) N!h 3N ,! ! ! ! ! ! !!!!!!!! (Eq.!2.22)! in!case!of!a!monatomic!system,!where! k B =1.381×10 −23 J⋅K −1 =8.617×10 −5 eV ⋅K −1 !is! Boltzmann’s!constant,!h=6.626×10 −34 J⋅s=4.136×10 −15 eV ⋅s!is!Plank’s!constant.!In! Eq.!2.22,! Ω!is!the!number!of!distinguishable!microscopic!states!in!the!phase!space! volume Ω.!The!factor!N!!accounts!for!the!fact!that!one!cannot!distinguish!these! particles,!i.e.,!exchanging!the!position!of!two!identical!particles!does!not!lead!to!a! new!microscopic!state.! ! In!the!limit!of!large!N,!lnΩ(E,δE)!is!insensitive!to!the!choice!of! δE.!Hence! we!can!ignore!the!dependence!of!S!on!δE.!In!fact,!it!does!not!make!a!difference!in! the!value!of!S!in!this!limit,!even!if!we!redefine!Ω!to!be!the!volume!of!all!the!phase! space!that!satisfies!H({ r i , p i })≤E ,!i.e.,! ! 32! ! S(N,V,E)=k B ln 1 N!h 3N d N rd N p H({ r i , p i }≤E) ∫ # $ % % & ' ( ( !! ! ! !!!!!!! (Eq.!2.23)! In!this!case,!the!dependence!on! δE!is!completely!removed.!! • Canonical)(NVT))ensemble) ) Imagine!a!system!with!a!fixed!number!of!particles! N,!and!a!constant!volume!V.!The! system!can!exchange!heat!with!its!environment!in!such!a!way!that!its!entropy! S! remains!constant.!However,!in!practice,!it!is!difficult!to!keep!the!entropy!of!a!system! at!a!constant.!Instead,!it!is!easier!to!control!the!temperature!of!the!system.!The! corresponding!equation!of!state!is!expressed!in!terms!of!the!temperature! T.!This!can! be!done!by!using!the!Helmholtz!free!energy! A.!! ! A!=!E−TS.! ! ! ! ! ! ! ! !!!!!!! (Eq.!2.24)! The!transformation!from!E(N,V,S)!to!A(N,V,T)!in!thermodynamics!corresponds!to! the!construction!of!the!canonical!( NVT)!ensemble!from!the!microcanonical!( NVE)! ensemble.!! ! To!construct!the!canonical!( NVT)!ensemble,!we!consider!a!system!and!a! reservoir!that!together!constitute!a!closed!system.!The!total!energy!of!the!system! (E)!and!the!thermostat!( ˆ E)!together!is!E 0 =E+ ˆ E,!which!remains!constant.!The! joint!distribution!in!the!phase!space!of!the!system!and!the!thermostat!at!equilibrium! is!! ! f({ r i , p i },{ ˆ r i , ˆ p i })= const. 0 ! " # H({ r i , p i })+ ˆ E∈[E 0 −δE,E 0 ] otherwise ! ! !!!!!!! (Eq.!2.25)! ! 33! To!find!the!distribution!of!the!system!itself,! all!the!thermostat!variables!need!to!be! integrated,!! ! f({ r i , p i })∝ d N ∫ r i d N p i δ(E 0 −H({ r i , p i })− ˆ E) ∝ ˆ Ω( ˆ N, ˆ V,E 0 −H({ r i , p i })) ∝exp 1 k B S( ˆ N, ˆ V,E 0 −H({ r i , p i })) % & ' ( ) * ! ! ! !!!!! (Eq.!2.26)! where! ˆ Ω!is!the!volume!occupied!by!thermostat!variables!in!the!phase!space!that! satisfies!the!constraint! ˆ E=E 0 −H({ r i , p i }).!Notation!with! !is!used!to!represent!all! the!quantities!for!thermostat.! ! Because!the!thermostat!is!much!larger!than!the!system,!putting!it!in!contact! with! the! system! will! not! change! its! temperature.!The! equation!of! state!S!for! thermostat!!can!be!Taylor!expanded!with!respect!to! ˆ Earound!E 0 !and!only!keep!the! first!two!terms,!! ! ˆ S( ˆ N, ˆ V, ˆ E)= ˆ S( ˆ N, ˆ V,E 0 )− H({ r i , p i }) T !! ! ! ! !!!!!!!! (Eq.!2.27)! Plug!this!expression!into!Eq.! 2.26,!we!have!! ! f({ r i , p i })∝exp H({ r i , p i }) k B T " # $ % & ' !! ! ! ! ! !!!!!!!! (Eq.!2.28)! The!proportional!constant!can!be!determined!by!the!normalization!condition,! ! ! d N ∫ r i d N p i f({ r i , p i })=1! ! ! ! ! ! !!!!!!!! (Eq.!2.29)! The!final!expression!for!the!canonical!(NVT)!ensemble!is! ! ˆ a ! 34! ! f NVT ({ r i , p i })= 1 Z(N,V,T) exp − H( r i , p i ) k B T , ! ! ! ! !!!!!!!! (Eq.!2.30)! ! Z(N,V,T)= d N ∫ r i d N p i exp − H( r i , p i ) k B T # $ % & ' ( . ! ! ! ! !!!!!!!! (Eq.!2.31)! This!is!Bolzmann’s!distribution!which!characterized!by!single!parameter!T.! ! ! Nosè!(Nosé!1984)!(Nosé!1984)!proposed!an!extended!Lagrangian!where!we! can!derive!a!set!of!equations!of!motion,!which!will! also!sample!the!canonical!(NVT)! ensemble.!The!basic!idea!is!to!introduce!a!new!(thermostat)!variabl e!s!that!can! exchange!heat!with!the!system!by!scaling!its!velocities.!The!extended!Lagrangian!is! ! ! L= m i 2 s 2 r i 2 −U({ r i })+ Q 2 s 2 −gk B T obj lns i ∑ ! ! ! ! !!!!!!!! (Eq.!2.32)! where!Q!is!the!“thermal!mass”!associated!to! s,!g!=!dN+1!with!d!being!the!number!of! dimensions!(d!=!3!for!3!dimensions),!!and! T obj !is!the!temperature!we!wish!to!apply! to!the!system.!! ! Nosè!stated!that!it!is!conceptually!more!appropriate!to!interpret )s!as!a!time! scaling!variable,!i.e.,!! ! Δt ' =Δt / s! ! ! ! ! ! ! ! !!!!!!! (Eq.!2.33)! where!Δt ' !is!the!real!time!step!and!Δt!is!the!scaled!time!step.! ! ! From! the! Nosè! Lagrangian! in! Eq.! 2.32,! we! can! obtain ! the! conjugate! momentum! p i !and!p s !for! r !and!s:!! ! p i ≡ ∂L ∂ r =m i s 2 r ! ! ! ! ! ! !!!!!!!!!!!!!!!!!!!!! (Eq.!2.34)! ! 35! ! p s ≡ ∂L ∂ s =Q s!! ! ! ! ! ! ! !!!!!!!! (Eq.!2.35)! ! The!Nosè!Hamiltonian!is! ! ! H = p i ⋅ r i ∑ + p s s−L! ! !!!!= p i 2 2m i s 2 i=1 N ∑ +U r i { } ( ) + p s 2 2Q +gk B T ln(s),! ! ! !!!!! ! !!!!!!! (Eq.!2.36)! which!includes!an!additional!degree!of!freedom!corresponding!to!a!heat!bath!and! keeps! the! temperature! of! the! system! constant. !The! last!two! terms! of! Eq.!2.36! represent!the!effect!of!the!heat!bath.!Here!s!is!a!“coordinate”!for!the!heat!bath ,!ps!is! the!corresponding!“momentum”.!The!variables!s!and!Q!rescale!time!and!kinetic! energy!of!the!system!to!satisfy!the!constraint!of!canonical!ensemble.!! ! Hoover!has!modified!Nosè’s!scheme!(Hoover!1985)!and!Martyna!et)al.!have! further! extended! it!to! describe! linked! heat! baths!(Martyna,! Klein! et! al.! 1992 )! (Martyna,!Tobias!et!al.!1994)!(Martyna,!Tuckerman!et!al.!1996 ).!The!Hamiltonian!for! this!extended!NoséYHoover!model!with!M !linked!heat!baths!is! ! H = p i 2 2m i i=1 N ∑ +U r i { } ( ) + p s i 2 2Q i i=1 M ∑ +3Nk B T obj s 1 + k B i=2 M ∑ Ts i ,! ! !!!!!!! (Eq.!2.37)! which!remains!valid!even!for!small!and!stiff!systems.! The!dynamics!derived!from!the! extended!Hamiltonian!is!expressed!as ! ! r i = p i m i ,! ! ! ! ! ! ! ! !!!!!! (Eq.!2.38)! ! 36! ! p i =−∇ r i U r i { } ( ) − p i p s 1 Q 1 ,! ! ! ! ! ! !!!!!! (Eq.!2.39)! !! i s i Q p s i = ,! ! ! ! ! ! ! ! !!!!!! (Eq.!2.40)! ! p s 1 = p i 2 m i i=1 N ∑ −3Nk B T obj # $ % & ' ( − p s 1 p s 2 Q 2 ,! ! ! ! ! !!!!!! (Eq.!2.41)! ! p s j = p s j−1 2 Q j−1 −k B T obj " # $ $ % & ' ' − p s j p s j+1 Q j+1 ,!! ! ! ! ! !!!!!! (Eq.!2.42)! ! p s M = p s M−1 2 Q M−1 −k B T obj .! ! ! ! ! ! ! !!!!!! (Eq.!2.43)! 2.2 Physical)properties)) An! MD! simulation! generates! phase Yspace! trajectories! from! which! the! physical! properties!of!the!system!can!be!calculated,!so!that!direct!comparison!can!be!made! with!experiments!as!well!as!more!accurate! calculations.! ! Based!on!the!ergodic!hypothesis,!physical!properties!in!an!MD!simulation!can! be!calculated!from! ! a= lim τ→∞ 1 τ a(t)dt 0 τ ∫ = a( r, p)ρ( r, p)d r d p ∫ .! ! ! ! !!!!!! (Eq.!2.44)! a represents!the!time!average!of!a!physical!quantity ,!which,!according!to!the!ergodic! hypothesis,!is!equivalent!to!an!ensemble!average .!In!Eq.!(2.44),! r !and! p!are!the! atomic! positions! and! momenta,! respectively,! and!ρ( r, p) !is! the! probability! distribution!function.!! ! 37! 2.2.1 ThermoPmechanical)properties) PhaseYspace!configurations!are!used!in!an!MD!simulation!to!calculate!the!kinetic! energy,!temperature,!stress!tensor,!heat!capacity,! etc.!The!average!kinetic!energy! K! is!related!to!the!temperature!of!the!system, ! ! € K = 3 2 Nk B T =lim τ→∞ 1 2 m i v i t ( ) [ ] 2 dt i ∑ 0 τ ∫ ,! ! ! ! !!!! !!(Eq.!2.45)! where!N!is!the!number!of!atoms,! T!is!the!temperature,!kB)is!the!Boltzmann!constant,! mi!and! € v i )are!the!mass!and!velocity!of!atom! i,!respectively.! !! The!stress!tensor,!σ,!can!be!obtained!from!the!Virial!theorem!as ! ! € σ αβ = 1 V m i v i α v i β i ∑ + 1 2 r ij α f ij β j ∑ i ∑ ,!! ! ! ! !!!!!! (Eq.!2.46)! where! a)and)b!are!Cartesian!coordinate!indices !{x,y,z},! and! € r ij α )and)) € f ij α !are! the! distance!vector!and!force!between!atoms!i!and!j.!One!can!calculate!pressure! P!as! ! € P = 1 3 tr(σ αβ ).!! ! ! ! ! ! ! !!!!!! (Eq.!2.47)! ! The!fluctuations!of!kinetic!energy!in!the!NVE!ensemble!is!related!to!the! specific!heat!of!the!system!via ! ! € δK 2 = K − K ( ) 2 = 3 2 Nk B 2 T 2 1− 3Nk B 2C V $ % & ' ( ) ,! ! ! ! !!!!!!! (Eq.!2.48)! where! V C !is!the!constantYvolume!heat!capacity.!! ! 38! 2.2.2 Structural)properties) Calculation!of!various!correlation!functions!can!provide!information!on!the!spatial! distribution!of!atoms!in!a!system !(Vashishta!and!Rahman!1979).!One!such!function! is!the!pair!distribution!function,! g,!defined!as! ) g αβ ( r 1 , r 2 )= V 2 N α N β δ r 1 − r i ( ) δ r 2 − r j ( ) j∈ β { } i≠j ∑ i∈ α { } ∑ ,! ! ! !!!!!! (Eq.!2.49)! where!V !is!the!volume,! α N !and! β N !are!the!number!of!atoms!of!species!α !and!β ,! respectively,!and!<>!is!the!ensemble!average.!In!a!uniform!system,!the!pairY distribution!function!only!depends!on! r = r 1 − r i :! ! g αβ ( r)= V N α N β δ r− r ij ( ) j∈ β { } i≠j ∑ i∈ α { } ∑ .!! ! ! ! !!!!!! (Eq.!2.50)! The! expression! can! be! simplified! for! an! isotropic! system,! where! the! pair Y distribution!function!depends!only!on! r:! ! g αβ ( r)= V 4πr 2 N α N β δ r− r ij ( ) j∈ β { } i≠j ∑ i∈ α { } ∑ .! ! ! ! !!!!!! (Eq.!2.51)! The! total! pairYdistribution! function! is! a! weighted! sum! over! the! partial! pair! distributionYfunctions.! ! g( r)= c α c β g αβ ( r) α,β ∑ ,! ! ! ! ! ! ! !!!!!!! (Eq.!2.52)! ! 39! where! ca,b!!are!the!concentrations!of!species,! N N c , , / β α β α = .! The! atomic! coordination!numbers!are!obtained!from, ! ! N αβ ( r)= 4πN β V g αβ ( ! r ) ! r 2 d ! r 0 r ∫ .! ! ! ! ! !!!!!! !(Eq.!2.53)! ! The!static!structure!factor, € S αβ (q),!is!the!Fourier!transform!of! ) (r g αβ ! ! S αβ ( q)=δ αβ + c α c β N V g αβ ( r)e iqr d r ∫ .! ! ! ! !!!!!!!! (Eq.!2.54)! If!the!system!is!isotopic,! the!static!structure!factor!becomes, ! ! € S αβ (q) =δ αβ + 4π c α c β N V g αβ (r)−1 [ ] sin(qr) qr r 2 dr ∫ .! ! !!!!!!!! (Eq.!2.55)! The!neutron!scattering!static!structure!factor, € S N (q),!can!be!obtained!from ! ! € S N (q) = b α b β α,β ∑ c α c β S αβ (q)−δ αβ + c α c β [ ] b α c α α ∑ ' ( ) * + , 2 ,! ! ! !!!!!!!! (Eq.!2.56)! where! α b !and! β b !are!the!coherent!neutron!scattering!lengths!of!species!α !and!β ,! respectively.! ! BondYangle!distribution!is!a!three Ybody!correlation!that!provides!additional! information!about!the!spatial!distribution!of!atoms.!The!nearest!neighbor!distance! rc! obtained!from!the!first!peak!of! € g(r)!is!used!to!determine!whether!a!bond!between!a! pair!of!atoms!is!formed.!If!rij)<)r c)and)rjk)<)r c,!the!angle!of!atomic!triplet!iMjMk!is! ! 40! calculated! from! € cosθ ijk = r ij ⋅ r jk r ij ⋅ r jk !and! binned! to! construct! a! histogram.! The! bond Y angle!distribution!can!be!compared!w ith!NMR!measurements.! 2.2.3 Dynamic)properties) The!cross!correlations!between!two!quantities! A!and!B!can!be!expressed!as:! ! € G AB = A- A B- B A 2 - A 2 ( ) B 2 - B 2 ( ) .! ! ! ! ! !!!!!!!! (Eq.!2.57)! The!correlation!is!dynamic!when!both )A!and!B!vary!with!time.!If!A!and!B)are!the! same!quantities,!the!function!is!called!an!auto Ycorrelation!function,! ! G auto r,t ( ) = ΔA r+ r o ,t+t o ( ) ( ) ΔA r o ,t o ( ) ( ) r o ,t o ΔA r o ,t o ( ) ( ) 2 r 0 ,t o ,! ! ! !!!!!!!! (Eq.!2.58)! where! € ΔA!is!the!fluctuation!of! A!around!the!mean!value.! ! ! The!diffusion!constant,!Dα,!of!species!α!can!be!computed!from!the!velocity! autoYcorrelation!function,!! ! € v auto,α t ( ) = v t + t o ( )⋅v t o ( ) α,t o v t o ( ) ( ) 2 α,t o ,! ! ! ! ! !!!!!!!! (Eq.!2.59)! as! ! € D α = k B T m α v t ( ) v t o ( ) α v t o ( ) ( ) 2 α 0 ∞ ∫ dt.! ! ! ! ! ! !!!!!!!! (Eq.!2.60)! One!can!also!calculate!the!diffusion!constant!from!the!mean!square!displacement, ! ! 41! ! α α 2 t t t t D ) ( ) ( 6 1 lim 0 r r − = ∞ → .! ! ! ! ! ! !!!!!!! (Eq.!2.61)! 2.2.4 Slip)vector)fiel d) To! analyze! structural! changes! during! deformation! and! failure! of! materials,! we! employ!a!slipYvector!approach!(Zimmerman,!Kelchner!et!al.!2001 ).!The!slip!vector!of! atom! € α!is!defined!as! ! s α =− 1 n s ( r αβ − R αβ ) β(≠α) n ∑ ,! ! ! ! ! ! !!!!!!! (Eq.!2.62)! where!n!is!the!number!of!nearest Yneighbor!atoms!β!to!atom! € α,!ns!is!the!number!of! slipped!neighbors,!and! ! r αβ = r α − r β ,!! ! ! ! ! ! ! !!!!!!!! (Eq.!2.63) ! R αβ = r α0 − r β0 ,! ! ! ! ! ! ! !!!!!!!! (Eq.!2.64)!! with! r α !and! r α0 !being! the! current! and! reference! positions! of! the! atom!α,! respectively.!The!reference!configuration!is!a!uniformly!elongated!initial!material! configuration!with!the!corresponding!strain!without!defect.! The!atomic!slip!vector!is! designed!to!identify!atoms!that!have!been!displaced!relative!to!their!reference! neighbors,!and!contains!information!about!the!slip!plane!and!Burgers!vector.! ! 2.3 Simulation)algorithms)) In!this!section,!implementation!of !MD!simulation!algorithms!in!single!machine!is! first!presented,!including!linkedYlist!cell!method!and!applied!boundary!conditions.! For!largeYscale!system!containing!millions!to!billions!of!atoms,!it!is!impractical!to! ! 42! simulate!any!physical!process!within!a!reasonable!time!frame!on!a!single!processor.! Parallel!computing!divides!the!computational!task!and!executes!the!subtasks!on! multiple! interconnected! computing! resources! concurrently.! It! capitalizes! on! the! aggregated! computing! power! to! greatly! reduce! the! computing! time.! This! often! requires!an!algorithm!with!high!parallel!efficiency,!and!a!hardware!infrastructure! that!supports!distributed!computing!and!fast!inter Yunit!communication.!! 2.3.1 LinkedPlistPcell)method) Interatomic!potentials!energies!usually!involve!interactions!of!two Ybody,!threeYbody,! fourYbody!or!even!higher!order!terms.!Let!us!consider!the!simplest!case!of!a!pair! potential,!as!an!example.!The!brute!force!implementation!of!MD!simulation!to!a! system!of!N!atoms!will!have!the!computational!complexity!of!O( N 2 )!due!to!the! doubly!nested!loops!for!calculating!forces!between!all!pairs!of!atoms.! ! ! Figure!2.1.!Cell!decomposition!of!a!system!for!linked!list!algorithm.!Atom! i!resides!in! cell!8.) ! 43! Creating!a!linkedYcell!list!is!one!way!to!improve!the!scalability!of!the!MD! algorithm.!For!a!shortYrange!potential!of!a!cutoff!length! rc,!only!interactions!with! neighbor!atoms!within!distance!rc!are!considered!for!each!atom.!Potential!form!is! then!shifted!to!eliminate!the!discontinuity!at!the!cutoff.!The!whole!system!is!divided! into!cells!with!edges! α c r ,!which!are!slightly!larger!than!the!cutoff!length: ! ! α α α c c L L r / = ,!! ! ! ! ! ! ! !!!!!! !!(Eq.!2.65)! ! ! " c c r L L / α α = ,! ! ! ! ! ! ! !!!!!!!! (Eq.!2.66)! where! α! denotes! Cartesian! coordinates! index!x,!y,!z,! and! α L !represents! system! length.!Each!cell!is!assigned!a!scalar!ID! c!in!the!range!of!0!and! NC!Y1!(NC!is!the!total! number!of!cells!in!the!system).!Atoms!in!each!cell!will!only!interact!with!atoms!in! the!same!cell!and!other!26!neighboring!cells!(as!shown!in! Figure!2.1!for!a!2D! schematic!view).!! ! To!construct!a!list!of!atoms!in!each!cell,!we!use!the!linked Ylist!algorithm,! which!is!implemented!using!the!linked Ylist!data!structure!as!shown!in!Figure!2.2! (Allen,!Tildesley!et!al.!1989).!In!this!data!structure,!two!arrays!are!created.!The!first! array!is!the!head!array!as!denoted!“head”!in! Figure!2.2!with!size!equal!to!NC.!The! head! array! contains!NC!elements,! which! correspond! to! cell! scalar! ID,! and! each! element!stores!the!first!atom!in!each!cell.!The!second!array!is!the!linked Ylist!array!as! denoted!“lscl”!in!Figure!2.2!with!size!equal!to!the!number!of!atoms!in!the!system.! The!“lscl”!array!holds!the!atom!index!to!which!the!atom!points.!For!example,!in!c ell!3,! the!first!stored!atom!is!atom!7,!then!it!is!linked!to!lscl[7],!where!we!can!find!it!is! ! 44! linked!to!atom!3.!Atom!3!then!points!to!lscl[3],!which!is!atom!1.!Since!atom1!is!the! last!atom!in!cell!3,!atom!1!points!to!empty.!Then!the!search!of!the!list!stop ped.! Implementation! of! this! algorithm! reduces! the! computation! complexity! to! O( N),! which!makes!MD!simulations!of!large!systems!computationally!feasible. ! ! Figure!2.2!!Data!structure!for!linked!list!algorithm,! where!“E”!denotes!empty) 2.3.2 Boundary)conditions) ) To!appreciate!the!importance!of!boundary!conditions!for!atomistic!simulations,!let! us!consider!Avogadro’s!number,!N A =6.022×10 23 ,!which!is!the!number!of!molecules! in!one!mole!of!any!substance.!In! comparison,!a!typical!simulation!on!a!desktop! workstation!can!only!handle!10 3 !to!10 6 !atoms.!Even!in!billionYatom!simulations!on! ! 45! massively!parallel!computers,!the!total!number!of!atoms!is!still!very!small!compared! with! Avogadro’s! number.! Accordingly,! thermo dynamic! behaviors! of! materials! cannot!be!described!adequately!in!MD!simulations,!unless!influence!of!the!massive! number! of! surrounding! atoms! is! accounted! for! through! appropriate! boundary! conditions.!! ! There! are! three! widely! used! boundary! conditions:! free! b oundary,! fixed! boundary,!and!flexible!boundary!conditions.!For!free Ysurface!boundary!condition,! the!system!is!surrounded!by!vacuum,!and!there!is!no!constraint!on!the!motion!of! any!atom.!This!is!used!widely!in!simulations!of! nanoYsystems,!but!is!not!applicable! for!bulk!materials!since!it!ignores!the!effects!of!atoms!outside!the!simulation!volume! and!introduces!artificial!surfaces.!One!way!to!reduce!the!surface!artifact!in!bulk! simulations!is!to!fix!atoms!on!the!periphery!of!the!simulation!volume!at! equilibrium! positions.!Such!a!fixed!boundary!condition!is!relatively!simple!to!implement!but!has! its!own!artifacts,!especially!in!a!dynamic!system!when!all!atoms!should!be!able!to! move.!Flexible!boundary!conditions!allow!atoms!in!the!boundary!layer!to!adjust! their!positions!in!response!to!the!motion!of!inner!atoms!( (Sinclair,!Gehlen!et!al.! 1978)).! The! goal! is! to! reduce! the! surf ace! effect! as! much! as! possible.! Periodic! Boundary! Condition! (PBC)! is! a! special! case! among! many! types! of! boundary! conditions.!PBC!can!be!applied!along!one,!two!or!three!directions!of!the!simulation! cell.!! The!idea!of!BornYvon!Karman!(Born!1912)!PBC!is!to!embed!the!simulation! volume!or!simulation!cell!into!an!infinite,!periodic!array!of!replicas!or!images! (Cai,! ! 46! Bulatob!et!al.!2003).!This!is!illustrated!in! Figure!2.3!(Bulatov!and!Cai!2006)!for!a!2Y dimensional!simulation.!The!atoms!in!the!replicas!are!assumed!to!behave!exactly!in! the! same! way! as! the! atoms! in! the! original! or! primary! simulation! cell! (Allen,! Tildesley! et! al.! 1989).! Because! the! primary! and! image! cells! are! identical,! it! is! irrelevant!which!one!of!them!is!regarded!as!primary!and!which!ones!are!the!images.! ! A!remarkable!property!of!PBC!is!that!no!point!in!spac e!is!treated!any!more! specially!than!others.!The!presence!of!a!surface!breaks!the!translational!invariance,! and!at!a!first!glance,!even!when!PBC!is!applied!one!might!think!that!the!border!of!the! simulation!supercell!(solid!line!in !Figure!2.3)!creates!artificial!interfaces!that!break! the!translational!invariance.!But!this!is!not!the!case.!The!boundary!of!the!primary! cell!can!be!shifted!arbitrarily,!such! as!in!Figure!2.3.!Such!a!shift!has!no!effect!on!the! dynamics!of!any!atom,!even!though!each!periodic!replica!now!apparently!contains!a! different!arrangement!of!atoms.!In!other!words,!tr anslational!invariance!of!space!is! fully!preserved.!Hence,!there!are!no!boundaries!and!no!artificial!surface!effects! when!PBC!is!used.!In!particular,!PBC!is!a!standard!boundary!condition!for!first! principles! electronic! structure! calculations! that! rely! on! t he! planeYwave! basis! functions!requiring!translational!invariance!of!space. ! ! 47! ! Figure!2.3!(a)!A!simulation!supercell!with!16!atoms!replicated!periodically!in!2D! space.!(b)!Shifting!supercell!boundaries!produces!a!different!set!of!atoms!but!does! not!alter!the!overall!periodic!arrangement ! In! an! atomistic! simulation! with! only! short Yranged! interactions,! implementation!of!PBC!is!straightforward.!We!just!need!to!enforce!the!minimum! image!convention,!which!states!that!the!relative!displacement!vector!between!atoms! i)and!j!is!taken!to!be!the!shortest!of!all!vectors!that!connect!atom! i!to!all!periodic! replicas!of!atom!j.!Since!the!energy!and!forces!depend!only!on!the!relative!positions! between! the! atoms,! this! convention! avoids! any! ambiguity! provided! that! the! potential!cutYoff!distance!is!sufficiently!small!so!that!no!more!than!one!replica!of! atom!j!falls!within!the!cutYoff!radius!of!atom! i.!! In! fact,! PBC! can! bring! in! its! own! artifacts.! One! example! is! when! the! simulation!cell!contains!a!crystal!defect!and!is!subjected!to!PBC.!Such!a!simulation! does!not!exactly!reproduce!the!behavior!of!a!single!defect!in!a n!infinite!crystal!due! ! 48! to! the! longYrange! stress! field! around! the! defect.! Instead,! it! will! be! more! representative!of!an!infinite!periodic!array!of!defects.!Unless!the!simulation!domain! is!sufficiently!large,!the!interaction!between!primary!and!image!defects !may!change! the!simulation!results.!This!is!an!important!issue!given!the!wide!applicability!and! robustness!of!PBC!in!atomistic!simulations!of!materials.!Some!techniques!have!been! developed!to!eliminate!or!mitigate!the!artifacts!introduced!by!PBC!when!model ing! defects!in!crystals!(Cai,!Bulatob!et!al.!2003 ).! 2.3.3 Parallelization ) For!largeYscale!simulations!containing!millions!to!billions!of!atoms,!it!is!not ! possible!to!simulate!any!physical!process!within!a!reasonable!time!frame!on!a!single! processor.!Parallel!computing!can!divide!the!computational!task!and!executes!the! subtasks!on!multiple!interconnected!computing!resources!concurrently.!In!an!MD! simulation,!we!can!use!appropriate!decomposition!of!atoms!onto!processors.!Each! processor!has!knowledge!of!the!complete!system!configuration!but!only!compute! physical!properties!of!a!certain!part!of!the!system.!Since!force!calculations!in!MD!are! only!based!on!nearby!atoms!within!an!interaction!range,!it!is!possible!to!effectively! pass!the!information!between!processors!and!carry!the!computation.! ! The!parallel!decomposition!scheme!we!used!is!called!spatial!decomposition!! (Nakano,!Kalia!et!al.!1994 )!(Nakano,!Kalia!et!al.!1999 ).!In!the!spatial!decomposition! technique,!divideYandYconquer!strategy!is!used!to!partition!the!simulated!system! into!subdomains!with!equal!volume,!which!are!mapped!to!processors!in!a!parallel! ! 49! computer.! Communication! between! nodes! is! carried! out! by! using! t he! message! passing!interface!(MPI)!library.!Suppose!that!there!are! P!processors!available.!We! logically!arrange!them!as! z y x N N N P × × = ,!where! z y x N N N , , !represents!number!of! processors!along!the!x,!y,!z!directions.!Each!processor!is!given!an!ID!numbered!from! 0!to!PY1,!as!shown!in!Figure!2.4.!Atom!i!at!position!ri!=!(rix,!riy,!riz)!will!be!assigned!to! processor!p!as! ! z z y z y x p N p N N p p + + = ,! ! ! ! ! ! !!!!!!! (Eq.!2.67)! ! p α = r iα N α /L α ,! ! ! ! ! ! ! !!!!!!! (Eq.!2.68)! ! Figure!2.4!Schematic!view!of!assigning!process!ID!for!subsystems. ! where)α!denotes!the!Cartesian!direction,!x,!y,!z,!Lα!represent!the!lengths!of!the!MD! box,!and!pα!indicates!the!vector!index!in!the! x,!y!and!z!directions.!Each!processor!p! possesses!atoms!positions,!velocities!and!species!within!the!spatial!sub Ydomain! assigned!to!it.!In!order!to!perform!local!potential!and!forces!calculations!for!the! Nx! Ny! Nz! Nz! Nx! Ny! NxY1! NyY1! NzY1! ! ! ! 50! atoms!located!at!the!boundaries!of!each!sub Ydomain,!the!domain!of!each!processor! is!extended!by!a!cutoff YlengthYthick!layer!as!shown!in !Figure!2.5.!Atoms!information! at!the!extended!domain!is!cached!from!neighboring!processors.! The!caching!process! is!done!through!interYprocessor!communication.!! ! ! Figure!2.5!An! illustration! of! a! projection! view! of! spatial! decomposition! of! 3 YD! system!implemented!in!parallel!MD!program.!Atoms!att ributes!in!the!extendedY domain!are!cached!from!26!neighboring!nodes .) ! This!interYprocessor!communication!involves!sending!and!receiving!data!from!the! 26! neighboring! processors.! In! the! process! of! caching,! each! processor! sends! attributes!of!the!skinYlayer!atoms!to!left,!right,!top,!bottom,!back!and!front!neighbors! consecutively.!Also,!migration!of!atoms!and!atoms!attributes!to!proper!neighbor! p) ! 51! processors!are!done!when!atoms!move!out!of!the!domain!of!a!processor!after!the! position!updates.!! 2.3.4 Performance)) Parallel!computing!efficiency!is!closely!related!to!the!computing!hardware.!P arallel! computing! hardware! can! be! divided! into! three! types:! shared Ymemory! system,! distributedYmemory! system! and! hybrid! distributed Yshared! memory! system.! A! sharedYmemory!system!refers!to!a!system!where!the!same!memory!is!accessed!by! multiple!processors.!This!computer!architecture!increases!the !data!communication! speed.!OpenMP!is!a!common!parallel!computing!library!that!can!be!used!in!a!share Y memory!system.!On!the!other!hand,!i n!a!distributedYmemory!system,!each!processor! accesses! its! own! local! memory,! sharin g! data! through! interYprocessor! communication.!Common!parallel!programming!libraries !include!Message!Passing! Interface!(MPI).!The!last!type!is!the !hybrid!distributedYshared!memory!system,! which!is!a!combination!of!the!above!two!architectures.!For!example,!t he!20,925Y processor!Linux!cluster!at!the! High!Performance!Computing!Center!(HPCC) !and!the! 4,096Yprocessor!Linux!clusters!at!the!Collaboratory!for!Advanced!Computing!and! Simulations!(CACS)!at!the!University!of!Southern!California!use! this!architecture.! ! For!scalability!analysis,!consider!a!parallel!program!running!on! P!processors! to!solve!a!problem!of!size! W)in)T(W,P)!execution!time.!The!speed!of!the!program!is! given!by!S(W,P)!=!W/T(W,P).!Speedup!of!the!program!Sp)is!therefore! ) 1 , ( ) , ( W S p W S S p = ,! ! 52! and!the!efficiency!E!can!be!defined!as! p S E p = .!In!a!constant!problem Ysize!(or!strong)! scaling!test,!the!efficiency!is ) , ( ) 1 , ( P W T P W T E • = ,!whereas!in!an!isogranular!(or!weak)! scaling!test,! ) , ( ) 1 , ( P w P T P w T E • • = ,!where!w!is!the!problem!size!p er!processor.!The! following!figures!are!the!weak!scalability!tests!done!with!finite!range!potential! parallel!MD!algorithm!(Figure!2.6)!and!parallel!reactive!force Yfield!(ReaxFF)!MD! algorithm!(Figure!2.7).!Figure!2.6!(Nomura,!Seymour!et!al.!2009 )!shows!benchmark! results!using!IBM!BlueGene/L!and!BlueGene/P!computers!for!the!performance!of! our!parallel!MD!algorithm!for!silica.!The!isogranular!test!(2,044,416!atoms!per! processor)!shows!the!efficiency!The!weakYscaling!parallel!efficiency!is!0.975!on! 131,072!BlueGene/P!processors.!The!measured!weak Yscaling!parallel!efficiency!on! 212,992! BlueGene/L! processors! is! 0.985! based! on! the! speedup! over! 4,096! processors.! ! ! 53! ! Figure!2.6!Results!of!the!isogranular!(2,044,416!atoms!per!processor)!benchmark! results!of!finiteYrange!parallel!MD!algorithm!for!silica!using!IBM!BlueGene/L!(open! symbols)!and!BlueGene/P!(solid!symbols)!computers.!The!wall Yclock!time!(circles)! and!interYprocessor!communication!time!(squares)!per!MD!step!are!shown .) ! Figure!2.7!shows!benchmark!results!using!our!parallel!ReaxFF!MD!algorithm! for!energetic!materials!(RDX)!(Nomura,!Seymour!et!al.!2009 ).!The!grain!size!is! chosen!to!be!16,128!on!the!BlueGene/P!and!BlueGene/L!processors.!The!efficiency! for!BlueGene/L!is!0.996,!which!is! close!to!the!ideal!efficiency,!1. ! ! ! ! 54! ! Figure!2.7!Execution!time!of!ReaxFF!MD!per!time!step!as!a!function!of!the!number!of! processors!P!of!BlueGene/L!(squares)!and!BlueGene/P!(circles),!where!the!number! of!atoms!is!N!=!16,128P)(Nomura,!Seymour!et!al.!2009 ).!! 2.4 Energy)minimization) 2.4.1 The)steepest)descent)method) ) Steepest!descent!is!a!simple,!although!not!very!efficient,!iterative!algorithm!for! finding!a!(local)!minimum!of! U(R)!starting!from!an!arbitrary!initial!configuration! R,! where!R!=!(x 1 ,y 1 ,z 1 ,x 2 ,y 2 ,z 2 ,...,x N ,y N ,z N ) T .!At!every!iteration,!the!force!vector! F!is! computed!and!R!is!displaced!by!a!small!step!along! F.!The!iterations!continue!until! |F|! becomes! smaller! than! a! prescribed! tolerance! ε.! A! simple! steepest! descent! algorithm!is!given!below,!which!requires!the!step!si ze!∆!to!be!specified!as!an!input. ! Algorithm)2.1) ! 55! ) 1.) F :=−∂V(R) /∂R! ) 2.)If)| F|)<) ε,!exit.!! ) 3.)R:=R+F∆.!Go)to)1.) ) The!idea!of!the!steepest!descent!algorithm!is!that,!as!long!as!|F|!is!non Yzero,!V(R)!can! be!further!reduced!by!displacing)R!in!the!direction!of!F.!This!algorithm!is!equivalent! to!a!numerical!integration!of!an!over Ydamped!equation!of!motion,! ! f i −γ v i = 0,! ! ! ! ! ! ! !!!!!!!! ! !!!!!!! (Eq.!2.69)! where! v i !is!the!velocity!of!atom! i,!and!γ!is!the!friction!coefficient.!After!a!sufficiently! long!time,!the!particles!will!eventually!arrive!at!a!structure!corresponding!to!a!local! energy!minimum!for!which!all!forces!vanish. !The!steepest!descent!algorithm!is! numerically!inefficient,!often!requiring!many!steps!to!converge!to!a!minimum.! ! 2.4.2 Conjugate)gradient)relaxation) ) Closely!related!to!the!steepest!descent!method!is!the!conjugate!gradient!relaxation! (CGR)!algorithm.!CGR!relies!on!exactly!the!same!information!as!steep est!descent,!i.e.! atomic!forces,!but!uses!it!in!a!more!efficient!way.!CGR!goes!through!a!series!of! search!directions.!The!(local)!minimum!energy!point!along!each!search!direction!is! reached!before!CGR!proceeds!to!the!next!search!direction.!The!search!seque nce!is! constructed! in! such! a! way! that! subsequent! search! directions! “avoid”! ( i.e.! are! conjugate! to)! all! previously! searched! directions.! This! is! the! key! to! the! greater! efficiency!of!the!CGR!algorithm!compared!to!the!steepest!descent!method. ! ! 56! ! The!CGR!algorithm!works!best!in!an!idealized!situation!when!the!potential! energy!is!a!quadratic!function.!The!conjugate!condition!means!that!any!two!search! directions!d (n) !and!d (m) (for!different!m!and!n)!must!satisfy!! ! d (n)T ⋅G ⋅d (m) = 0 , ! ! ! ! ! ! ! !!!!!!!! (Eq.!2.70)! where!G!is!a!real,!symmetric,!positive Ydefinite!matrix.!Ensuring!that!the!current! search! direction! is! conjugate! to! all! previous! search! directions! seems! to! be! a! daunting!task,!especially!if!the!relaxation!requires!hundreds!of!search!directions.! The!key!to!the!success!of!CGR!is!to!ensure!that!the!current!search!direction! d (n) !is! conjugate!to!the!previous!one!d (n−1) ! in!a!simple!iterative!way! (Seiler!and!Seiler!1989).! The!algorithm!can!be!described!as!follows. ! Algorithm)2.1) )1.)Initialize)iteration)n:=1 ) 2.Compute) F :=−∂V(R) /∂R.)If)| F|)<) ε,!exit.! )3.)If)n=1,) d (n) :=F (n) ;!otherwise,!! ))))) γ (n) :=[F (n) ⋅F (n) ]/[F (n−1) ⋅F (n−1) ],!d (n) :=F (n) +γ (n) d (n−1) ;!! )4.Find)an)energy)minimum)along)direction) d (n) ! )5.)R :=R+x 0 d.! )6.)n:=n+1.)Goto)2.) ) ! 57! 2.4.3 Global)minimization) ) Neither! CGR! nor! the! steepest! descent! algorithms!guarantees! to! find! the! global! minimum!of!an!energy!function.!Depending!on!the!choice!of!the!initial!point,!both! minimization!algorithms!discussed!so!far!can!converge!to!one!of!the!local!minima.!If! the!global!minimum!is!the!target,!a!brute Yforce!approach!is!to!run!minimization! many!times!starting!from!randomly!selected!initial!configurations.!Then,!the!lowest! local!minimum!found!from!a!series!of!relaxations!provides!an!upper!bound!on!the! global!minimum.!Obviously,!such!an!approach!is!very!inefficient. ! ! Some!of! the! more! efficient! methods! for! finding! the! global! minimum! of! functions!of!many!variables!are!motivated!by!processes!occurring!in!Nature.!For! example,! genetic! algorithms! invoke! mutation! and! natural! selection! to! evolve! a! collection!of!systems!in!search!fo r!the!fittest!(having!the!lowest!energy)! (Holland! 1992)!(Sivanandam! and! Deepa! 2007)!(Haupt! and! Haupt! 2004).! Another! useful! method! is! simulated! annealing! (SA),! which! mimics! the! thermal! annealing! of! a! material,!as!its!name!suggests! (Kirkpatrick!and!Vecchi!1983)!(Kirkpatrick!1984).!SA! requires!methods!to!simulate!atomistic!systems!at!non Yzero!temperatures.!Both! Monte! Carlo! (MC)! and! MD! methods! can! be! used! for! this! purpose.! At! low! temperatures,!the!system!can!become!trapped!in!the!neighborhood!of!a!local!energy! minimum!that!depends!on!the!choice!of!the!initial!configuration.!To!lessen!this! unwanted!dependence,!SA!simulations!starts!at!a!high!temperature,!making!it!easier! for!the!system!to!escape!from!the!traps!of!local!minima!and!explore!a!larger!volume! in!the!configuration!space.!The!SA!search!then!slowly!reduces!the!temperature!to! ! 58! zero.!At!the!end!of!this!procedure,!the!system!converges!to!a!local!minimum!that!is! likely!to!be!less!depen dent!on!the!initial!configuration,!and!is!more!likely!to!be!the! global!minimum.!! ! ! ! ! ) ! 59! CHAPTER)3 ) ) ) ) ) ) ) ))))) INTERATOMIC)INTERACTIONS) As!mentioned!in!CHAPTER!2,!materials!properties!are!derived!from!the!interaction! between!their!constituent!atoms.!The!basic!interactions!make!the!atoms!assemble!in! a! particular! crystalline! structure! and! define! how! the !atoms! are! arranged! on! a! surface!or!around!a!vacancy.!In!MD!simulations,!interatomic!interaction!is!described! by!an!interatomic!potential,!which!is!essential!to!study!the!collective!behavior!of! atoms!that!constituted!a!material!and!to!understand!the!behavi or!of!the!material.! ! In!this!chapter,!four!different!interatomic!interactions!are!discussed,!from! the!most!basic!pair!interactions!—!LennardYJones!potential!—!to!potentials!that! include!pair!and!3Ybody!interactions!to!a!reactive!force!field!based!on!the!co ncept!of! reactive! bond! order! to! describe! chemical! reactions.! Application! of! the! four! interaction!potentials!will!be!discussed!in!the!subsequent!chapters.! ) 3.1 Interatomic)interactions)and)Lennard PJones)potential) ) Depending! on! the! atomic! species! and! interaction s! between! different! species,! interatomic!interactions!could!be!very!complicated.!This!variability!comes!from!the! quantum!mechanical!motion!and!interaction!of!electrons! (Cohen!1984).!The!fitting! of! interatomic! interactions! should! be! based! on! the! Schr Ödinger’s! equation! for! interacting!electrons,!which!is!usually!referred!to!as!the!first!principle!or! ab)initio! theory.!! ! 60! ! However,!calculations!based!on!first!principles!are!very!expensive!and!can! only!deal!with!a!small!number!(~!100)!of!atoms.!For!systems!t hat!involve!at!least! many! thousands! of! atoms,! we! can! only! apply! less! sophisticated! but! more! computationally!efficient!empirical!models.!Such!an!empirical!model!usually!uses!an! analytical!functional!form!for!the!potential!energy!of!a!set!of!atoms,! ! ! U({ r i })≡U(r 1 ,r 2 ,...,r N )! ! ! ! ! ! !!!!!!!! (Eq.!3.1)! where! r i !is!the!position!vector!of!atom! i!and!N!is!the!total!number!of!atoms.!The! force!on!an!atom!is!negative!derivative!of!the!potential!function!with!respect!to!its! position,!! ! f j =− ∂U({ r i }) ∂r j ! ! ! ! ! ! ! !!!!!!!! (Eq.!3.2)! Such!functions!are!called!interatomic!potentials.! ! ! One!of!most!basic!atomic!interactions!would!be!repulsive!forces!when!atoms! are!in!close!proximity.!Another!important!aspect!is!that!atoms!attract!each!other!at! longer! distance.! A! simple! but! important! mode l! that! covers! both! short Yrange! repulsion!and!longYrange!attraction!is!the!Lennard YJones!(LJ)!potential!proposed!by! Sir!John!Edward!LennardYJones!in!1924!(Jones!1924).!The!LJ!potential!is!of!the!form !! ! φ(r)= 4ε 0 r σ 0 ! " # $ % & −12 − r σ 0 ! " # $ % & −6 ( ) * * + , - - , !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! (Eq.!3.3)! as!plotted!in!Figure!3.1.!Here,!ε 0 !is!the!depth!of!the!energy!well!and! σ 0 !can!be! considered!as!the!size!of!atoms.!The!first!term!is!dominant!at!shorter!distances.!Its! ! 61! physical! origin! comes! from! the! Pauli! exclusion! principle.! The! second! term! is! dominant!at!larger!distances!and!describes!van!der!Waals!dispersion!forces!arising! from!induced!dipoleYdipole!interactions.!The!interaction!energy!between!two!atoms! takes!the!minimum!value!at!distance! r!=!2 1 6 σ 0 ,.!! ! Figure!3.1!LennarYJones!potential!(from!http://www.wikipedia.org/)! ! LJ!potential!is!a!type!of!pair!potentials!because!it!only!considers!interactions! between!pairs!of!atoms.!Other!pair Ywise!potentials!include!Coulomb!interaction! between!atomic!charges,!Morse!potential!(bondi ngYtype!potential)!and!Buckingham! potential!(also!called!6Yexp!potential).!For!a!number!of!materials,!a!pair!potential! cannot!reproduce!accurate!physical!properties.!For!example,!a!pair!potential!model! cannot!differentiate!between!shear!and!tensile!so!that !it!is!bound!to!produce!an! identical!value!for!C 12 !and!C 44 ,!which!are!two!different!elastic!constants!for!cubic! crystals.!This!is!certainly!not!true!for!most!cubic!semiconductors!and!metals.! ! ! 62! ! In!this!dissertation,!LJ!potential!is!used!in! CHAPTER!4!for!a!simple!model! system!to!validate!our!hypothesis!regarding!amorphization!of!binary!alloys.! ! 3.2 ManyPbody)potentials)) As!is!stated!in!the!previous!section,!we!need!to!establish!ways!to!go!beyond!pair! potentials.!One!approach!is!to!represent!the!many Ybody!potential!energy!as!a!sum!of! twoYbody,!threeYbody,!fourYbody!and!all!the!way!up!to! NYbody!terms,!i.e.,!! ! U({ r i })= φ( r i − r j ) i<j ∑ + U 3 ( r i , r j , r k ) i<j<k ∑ + U 4 ( r i , r j , r k , r l )+.... i<j<k<l ∑ ! !!!!!!!! (Eq.!3.4)! ! To!simulate!the!interatomic!interactions,!a!low!order!truncation!is!often!used! to! achieve! a! balance! between! accurate! description! of! the! interactions! and! computational!cost.!! 3.2.1 StillingerPWeber)potential) ) StillingerYWeber! (SW)! potential! for! semiconductor! sili con! was! one! of! the! first! potentials!for!diamond!lattices!(Si,!GaAs,!Ge,!C),!introduced!in!1985! (Stillinger!and! Weber!1985)!by!Frank!H.!Stillinger!and!Thomas!A.!Weber!in!AT&T!Bell!Lab.!An! objective!of!this!model!was!to!describe !the!interaction!in!solid!and!liquid!forms!of!Si.! Tetrahedral!semiconductors!Si!and!Ge!shrink!when!they!melt !(Waseda!and!Suzuki! 1975).! The! melting! causes! a! partial! collapse! of! this! structure! a nd! increase! the! coordination!number!from!4!to!more!than!6. ! ! The!potential!contains!twoYbody!and!threeYbody!terms,! ! U(1,...,N)=v 2 (r ij )+h(r ij ,r ik ,θ jik )+h(r ji ,r jk ,θ ijk )+h(r ki ,r kj ,θ ikj )!! !!!!!!!!!! (Eq.!3.5)! ! 63! where!the!twoYbody!(v 2 )!and!threeYbody!(h)!terms!are!given!by ! ! v 2 (r ij )= A(Br −p −r −q )exp[(r−a) −1 ],! ! ! ! ! !!!!!!!!!! (Eq.!3.6)! ! h(r ij ,r ik ,θ jik )=λexp[γ(r ij −a) −1 +γ(r ik −a) −1 ]×(cosθ jik + 1 3 ) 2 .!! !!!!!!!!!! (Eq.!3.7)! ! The!twoYbody!term!is!designed!to!achieve!a!satisfactory!description!of!the! shortYrange!order!in!the!liquid!phase,!and!of!the!atom Yexchanging!diffusive!motions! that!occur!continuously!in!the!liquid!phase.!The!cut Yoff!length!of!this!term!is! a.!The! same!cutoff!length!is!extended!to!three Ybody!interactions.!The!3Ybody!term!of!SW! potential!is!proportional!to! ! ! (cosθ jik + 1 3 ) 2 ! ! ! ! ! ! ! ! !!!!!!!!!! (Eq.!3.8)! where!θ jik !is!the!angle!between!bond! ij)and!bond!ik,!as!shown!in!Figure!3.2.!If!either! r ij !or!r ik !is!larger!than!a,!h!vanishes!identically.!The!“ideal”!angle! θ t !is!such!that!! ! cosθ t =− 1 3 ,!! ! ! ! ! ! ! ! !!!!!!!!!! (Eq.!3.9)! so!that!the!this!term!penalizes!the!deviation!of!the!bond!angle!away!from!the!ideal! angle!and!therefore!helps!stabilize!t he!diamondYcubic!structure.!Physically,!this! threeYbody! term! reflects! the! strong! bond! directionality! of! most!tetrahedrally! bonded!covalent!semiconductors.! ! 64! ! ! Figure!3.2!Diamond!cubic!structure.!The!angle!between!any!two!bonds!involving!the! same!atom!is!θ jik =109.47°!i.e.,!cosθ t =−1/3.! 3.2.2 Effective)force)field)for)ZnO )) In!this!dissertation,!MD!simulations!on!ZnO!material!have!been!performed!using!an ! effective! force! field,! whose! functional! form! originated! from! the! work! by! Priya! Vashishta!et)al.!in!1989!(Vashishta,!Kalia!et!al.!1990 ).!The!original!potential!was! made!to!capture!the!richness!in!the!structure!of!SiO 2.!The!form!of!the!original! potential!was!soon!adapted!into!different!materials! (Ebbsjo,!Kalia!et!al.!2000 ).!The! parameter! set! used! in! this! dissertation! was! developed! by! José! P.! Rino)et) al.! (Vashishta,!Kalia!et!al.!1997 ).!The!potential!consists!of!two Ybody!and!threeYbody! interactions:!! !!!!!!!!!!! U= U ij (2) (r ij )+ U jik (3) (r ij ,r ik ) i,j<k ∑ i<j ∑ .!! ! ! ! ! !!!!!!! (Eq.!3.10)! The!twoYbody!contribution!to!the!interaction!potential!describes!the!effects!of!steric! repulsion!due!to!atomic!sizes,!coulomb!interactions!resulting!from!charge!transfer,! chargeYdipole!interaction!to!include!the!effects!of!lar ge!electronic!polarizability!of! anions!and!van!der!Waals!interactions: ! 0PS ! 65! !! U ij (2) (r ij )= H ij r ij n ij + Z i Z j r ij e −r ij a − D ij r ij 4 e −r ij b − W ij r ij 6 .! ! ! ! !!!!!!!! (Eq.!3.11)! In!Eq.!(3.11),!Hij!and!nij,!are!the!strengths!and!exponents!for!steric!repulsion,! Zi!is!the! effective! atomic! charge! for! the! Coulomb! interaction,!a)and! b)are! screening! parameters,)Dij!is!the!strength!of!charge Ydipole!interaction,!and!Wij!is!a!parameter!in! the!induced!dipoleYdipole!interaction.!The!exponential!screening!term!in!the!charge Y dipole!interaction!provides!a!reasonable!cutoff!for!the! r −4 !interaction.!! ThreeYbody!term!incorporates!covalent!effects!through!bond Ybending!and! bondYstretching!terms!and!is!given!by ! ! U (3) jik (r ij ,r ik )=R (3) (r ij ,r ik )⋅P (3) (θ jik ),! ! ! ! ! !!!!! !(Eq.!3.12)! where!! ! R jik (3) (r ij ,r ik )= B jik exp γ r ij −r 0 + γ r ik −r 0 " # $ $ % & ' ' Θ r 0 −r ij ( ) Θ r 0 −r ik ( ) !!!!!!!!!!!!!!!!!!!!!!!!!!!!! (Eq.!3.13)) and!! ! € P (3) (θ jik ) = cos(θ jik )− cos(θ jik ) ( ) 2 1+C jik cos(θ jik )− cos(θ jik ) ( ) 2 !.! ! ! !!!!!!! (Eq.!3.14) ! ! Although! there! are! six! three Ybody! terms! in! the! interaction! potential! between! species!A!and!B!(e.g.,!A!=!zinc!and!B!=!oxygen),!the!two!most!important!terms!ar e!AM BMA!and!BMAMB!because!AMB!is!the!shortest!bond!with!the!strongest!attractive!energy.! The!threeYbody!AMBMA!and!BMAMB!interactions!depend!on!variations!of!AMB!bond! length!and!AMBMA!and!BMAMB)bond!angles.!In!Eq.!( 3.13)!and!(3.14),!Bjik!is!the!strength! ! 66! of!the!interaction,!the!function!R!represents!the!effects!of!bond!stretching,! € r 0 !is!the! cutoff!for!these!interactions,! € θ jik !is!the!angle!formed!by! € r ij and! € r ik ,!Cjik!and! € θ jik !are! constants,!and! € Θ(r 0 − r ij )!is!the!step!function.! ! The!experimental!values!for!lattice!constant,!cohesive!energy,!bulk!modulus! and!some!elastic!constants!are!used!to!determine!the!parameters!in!the!effective! interaction!potential!listed!in!Figure!3.1,!and!Figure!3.2!shows!the!calculated!elastic! constants!along!with!experimental!values.!Calculation!of !several!other!properties,! not!used!in!the!parameterization,!will!be!discussed!in!the!following!sections. ! Table!3.1!TwoY!and!threeYbody!parameters!in!the!interaction!potential!for!ZnO. !José! P.!Rino)et)al.!(Vashishta,!Kalia!et!al.!1997 )! ! ZZn!(e)! ZO!(e)! l!(Å)! € ζ!(Å)! rc!(Å)! e!(C)! ! 0.90269! Y0.90269! 5.0! 3.0! 6.65! 1.602x10 Y19 ! TwoYbody! ! € η ij ! Hij!(eV!Å h ) ! Dij(eV!Å 4 )! Wij(eV!Å 6 )! ! ZnYZn! 7! 15.9634! 0! 0! ! ZnYO! 9! 399.5976! 17.6227! 4.5301! ! OYO! 7! 988.0706! 35.2455! 0! ! ThreeYbody! ! Bjik!(eV)! € θ jik (deg)! Cjik! r0(Å)! ! ZnYOYZn! 0.65543! 109! 10! 3.0! ! OYZnYO! 0.65543! 109! 10! 3.0! ! ! ! 67! Table!3.2!Experimental!and!calculated!values!from!MD!simulations!of!the!elastic! constants!for!zinc!oxide.! José!P.!Rino)et)al.!(Vashishta,!Kalia!et!al.!1997 )! • Bulk!modulus!(GPa)! • Elastic!constants!(GPa)! • B!!!!!!!!!B ’ !!!!!!!! <B>! • C11! • C12! • C13! • C33! • C44! • C66! • Ref.! ! 143.51! 143.7! 209.7! 121.1! 105.1! 210.9! 42.7! 44.3! 300K! ! 147.64! 147.7! 206! 117! 118! 211! 44.3! 44.6! 300K! ! 113.30! 114.7! 157! 89! 83! 208! 38! 34! 300K! ! 128.31! 128.4! 190! 110! 90! 196! 39! 40! 300K! ! 142.53! 142.6! 207! 117.7! 106.1! 209.5! 44.8! 44.6! 300K! 146! 142.71! 146.1! 289.9! 96.0! 64.5! 285.5! ! ! MDYT=0! 142.6! 139.72! 143! 224.2! 118.8! 95.3! 219.4! ! ! MDYT=300K! ! In!Figure!3.2,!the!elastic!modulus!B’!was!computed!using!the!definition,! (Landau!and! Lifshi*t*s!1970;!Golovchan!1998)!B ' = (C 11 +C 12 ) C 33 − 2.C 13 2 2.C 33 +C 11 +C 12 − 4C 13 ,!and!the!average!bulk! modulus,! for! hexagonal! symmetries,! relates! to! the! elastic! constants! by! € <B >= 2 9 (C 11 +C 12 + 2C 13 +C 33 /2).! As! expected,! at! zero! temperature! the! elastic! moduli!are!larger!than!those!at!300!K,!which! are!in!good!agreement!with!the! experimental!values.! ! With!the!interaction!potential!discussed!above,!the!total!energy!per!particle! was!calculated!as!a!function!of!the!volume!per!particle!for!hexagonal,!zinc Yblende! and!rockYsalt!structures!of!ZnO,!as!shown! in!Figure!3.3.!The!hexagonal!(wurtzite)! ! 68! phase!is!the!most!stable!phase!followed!by!zinc Yblende!and!rockYsalt!structures.!The! energies!of!the!zincYblende!and!hexagonal!ZnO!structures!differ!by!0.031!eV!per! atom.!! ! Figure!3.3!Energy!per!particle!as!a!function!of!the!volume!per!particle!for!hexagonal,! zincYblende!and!rockYsalt!ZnO.! From!the!common!tangent!between!the!energies!of!the!wurtzite!and!rock Y salt!phases,!we!obtained!the!pressure!of!structural!transformation!to!be!4.8!GPa.! Experimentally,! there! is! a! large! hysteresis! for! the! wurtzite YtoYrockYsalt! transformation.!Under!upYstroke!pressure,!the!wurtzite!transforms!to!rock Ysalt!at! around!9!GPa,!while!at!down Ystroke,!it!returns!to!its!original!structure!below!2!GPa.! We!can!define!the!transformation!pressure!as!the!middle!point!between!these!two! values,!Ptrans!~!5.5!GPa.!The!calculat ed!transformation!pressure!is!in!good!agreement! with!this!experimental!value.! -3.8 -3.75 -3.7 -3.65 -3.6 -3.55 -3.5 7 8 9 10 11 12 13 14 Zinc-blende Wurtzite Rock-salt E/N (eV) V/N (Å 3 ) ! 69! The! ZnO! interatomic! potential! is! used! in!CHAPTER! 5! to! study! thermomechanical!properties!of!ZnO!nanowires,!where!validation!of!other!physical! properties!(e.g.!surface!relaxation)!relevant!for!the!study!is!presented. ! 3.2.3 Effective)force)field)for)GaAs) ) The!potential!form!for!the!effective!force!field!for!GaAs,!like!the!potential!for!ZnO, ! was!adapted!from!Priya!Vashishta’s!interatomic!potential!developed!for!SiO 2!in! 1989.!The!original!GaAs!potential!using!this!form!was!used!to!study!amorphous! gallium!arsenide!(aYGaAs)!(Ebbsjo,!Kalia!et!al.!2000 )!and!structure!transformation!of! GaAs!(Rino,!Chatterjee!et!al.!2002 ).!The!potential!reproduce!well!the!experimental! crystalline! lattice! constant,! cohesive! energy,! elastic! constants,! and! melting! temperature.!The!comparison!between!the!MD!calculation!and!experiment!is!listed! in!Table!1.1.!The!energy!barrier!associated!with!the!transformation!between!zinc Y blende!structure!and!rockYsalt!structure!is!estimated!to!be!~!0.06!eV.! ! Table!3.3!Elastic!constants,!C11,!C12,!and!C!44,!and!bulk!modulus,!B,!of!crystalline!and! amorphous!GaAs!from!old!data!set.! José!P.!Rino)et)al.!(Vashishta,!Kalia!et!al.!1997 )! ) ! 70! ! Although! the! original! parameter! set! can! produce! the! structure! transformation!under!pressure,!it!gives!an!incorrect!energy!difference!between! zincYblende!and!wurtzite!structures!–!wurtzite!structure!is!more!stable!than!zinc Y blende!structure!under!room!tempe rature,!as!shown!in!the!calculated!volume! vs.! energy! curves! for! zincYblende,! rockYsalt! and! wurtzite! structures! in !Figure!3.4.! According!to!first!principle!calculation s!(Yeh,!Lu!et!al.!1992),!however,!the!wurtzite! structure!has!12.02!±!0.35!meV!higher!energy!per!atom!than!zinc Yblende.!! ! 71! ! Figure! 3.4! Energy! vs.! volume! relation! for! zinc Yblende,! wurtzite! and! rockYsalt! structures!of!GaAs.! ! ! To!correct!this!subtle!energy!difference!between!the!two!structures,!a!new! set!of!parameters!was!produced!with!a!smaller!cut Yoff!length,!as!shown!in !Table!3.4.! The!new!interatomic!potential!gives!an!energy!difference!of!about!10!meV/atom! ! 72! between!the!wurtzite!and!zinc Yblende!structures,!which!is!in!agreement!with!the! first!principles!calculation.!The!energy!vs.!volume!curve!calculated!with!the!new! interatomic!potential!is!shown! in!Figure!3.4.!! Table!3.4!TwoY!and!threeYbody!parameters!of!the!interaction!potential!for!GaAs! used!in!this!dissertation.! ! ZGa!(e)! ZAs!(e)! l!(Å)! € ζ!(Å)! rc!(Å)! e!(C)! ! 0.929762! Y0.929762! 5.0! 3.0! 6.2! 1.602x10 Y19 ! TwoYbody! ! € η ij ! Hij!(eV!Å h ) ! Dij(eV!Å 4 )! Wij(eV!Å 6 )! ! GaYGa! 7! 145.8875! 0! 0! ! GaYAs! 9! 5152.8259! 0.8644! 4.8876! ! AsYAs! 7! 3436.1885! 1.7289! 0! ! ThreeYbody! ! Bjik!(eV)! € θ jik (deg)! Cjik! r0(Å)! ! GaYAsYGa! 0.7619! 109.47122! 12! 3.45! ! AsYGaYAs! 0.7619! 109.47122! 12! 3.45! ! ! ! 73! 10 15 20 25 30 &3.5 &3.0 &2.5 &2.0 (ZB (WZ (RS E/N((eV) V/N((A 3 ) P~18(GPa E ZB (minimum)=&3.3600 E WZ (minimum)=&3.3506 ! Figure!3.5!Energy!per!particle!as!a!function!of!the!volume!per!particle!for!hexagonal,! zinc!blende!and!rockYsalt!GaAs.!José!P.!Rino)et)al.!(Vashishta,!Kalia!et!al.!1997 )! The! GaAs! interatomic! potential! is! used! in!CHAPTER! 6! to! study! defect! formation!in!GaAs!nanowires,!where!validation!of!other!physical!properties!( e.g.! surface!relaxation)!relevant!for!the!s tudy!is!presented.! 3.3 Reactive)force)field) ) MD!simulations!of!sulfur!(S)!embrittlement!of!nickel!nanocrystal!are!carried!out! with!a!reactive!force!field!(ReaxFF)! (van!Duin,!Dasgupta!et!al.!2001 ;!Strachan,!van! Duin!et!al.!2003),!which!can!describe!chemical!reactions!and!dynamic!properties!for! largeYscale! reactive! systems.! ReaxFF! features! both! bonded! and! nonbonded! interactions.!For!bonded!interactions,!a!general!relationship!between!bond!length! and!bond!order!is!used!to!obtain!smooth!transitions!betwee n!different!types!of! bonds!and!dissociation!of!bonds.!In!the!force!field,!valence!and!torsion!angles!are! ! 74! formulated!in!terms!of!bond!order!to!ensure!continuous!energy!distribution!upon! dissociation! of! bonds! during! reactions.! For! nonbonded! interactions,! Rea xFF! includes!shielded!van!der!Waals!and!Coulomb!interactions!at!short!range,!which! become!constant!as!the!distance!between!a!pair!of!atoms!goes!to!zero.!The!total! energy!of!the!system!is!composed!of!various!partial!energy!contributions!as!the! following:! ! € E = E bond + E lp + E over + E under + E val + E pen + E coa + E tors + E conj + E hbond + E vdWaals + E Coulomb (Eq.!3.15)! where!Ebond!is!the!bond!energy,!Elp!is!the!lone!pair!energy,!Eover!and!Eunder!denote!the! overY!and! underY!coordination)energy,!Eval!is!the!valence! angle! energy,! penalty! function!is!included!in!Epen,)Ecoa!describes!threeYbody!conjugation!energy,!torsion! angle!energy!is!Etors,!the!fourYbody!conjugation!energy)is)Econj,!and!the!hydrogen! bond!interaction!energy!is!included!in )Ehbond,,!nonbonded!interactions!EvdWaals!and! ECoulomb!are!the!van!der!Waals!and! Coulomb!energies,!respectively.!! 3.3.1 Bond)order)and)bond)energy ) The!bond!order,!BOij,!a!fundamental!element!of!ReaxFF,!is!obtained!in!terms!of! interatomic!distance!rij!between!a!pair!of!atoms .!Calculations!of!BOij!compose!of! three!exponent!terms,!which!are!contributions!from! σ,!π!and!doubleYπ!bonds.!! ! ! ! " # $ $ % & ' ' ( ) * * + , + ! ! " # $ $ % & ' ' ( ) * * + , + ! ! " # $ $ % & ' ' ( ) * * + , = - + - + - = - bo6 bo5 bo4 bo3 bo2 bo1 exp exp exp p r r p p r r p p r r p O B O B O B O B o ij o ij o ij ij ij ij ij ππ π σ ππ π σ ,(Eq.!3.16)! ! 75! where! € r ij =|r ij |=|r i −r j |!is! the! interatomic! distance! between! atom s! i! and)j,! and! 6 5 4 , 3 , 2 1 , , , bo bo bo bo bo bo P P P P P P ,! are! fitted! parameter! for! the! dependence! of! the! bond! order!on!σ,!π!and!doubleYπ!bonds.! The! uncorrected! bond! order,! BO%ij! is! used! to! obtain! the! uncorrected! overcoordination! term,!Δ%i,! which! is! the! degree! of! deviation! of! the! sum! of! the! uncorrected!bond!orders!around!an!atomic!center!from!its!valency! Vali,!! ! i i n j ij i Val BO − = Δ ∑ ∈ ) ( ' .!!! ! ! ! ! ! !!!!!!! (Eq.!3.17)! The!corrected!bond!order!term,! BOij!is!then!calculated!as! ! € BO ij σ = BO ij 'σ ⋅ f 1 (Δ i ' ,Δ j ' )⋅ f 4 (Δ i ' ,BO ij ' )⋅ f 5 (Δ j ' ,BO ij ' ),! ! € BO ij π = BO ij 'π ⋅ f 1 (Δ i ' ,Δ j ' )⋅ f 1 (Δ i ' ,Δ j ' )⋅ f 4 (Δ i ' ,BO ij ' )⋅ f 5 (Δ j ' ,BO ij ' ),!! ! € BO ij ππ = BO ij 'ππ ⋅ f 1 (Δ i ' ,Δ j ' )⋅ f 1 (Δ i ' ,Δ j ' )⋅ f 4 (Δ i ' ,BO ij ' )⋅ f 5 (Δ j ' ,BO ij ' ),!! ! ! ! € BO ij = BO ij σ + BO ij π + BO ij ππ .! ! ! ! ! ! !!!!!!!! (Eq.!3.18)! Here,!the!functions!f1,!f2,)f3,)f4,!f5,!are!defined!as!follow s:! ! € f 1 (Δ i ,Δ j ) = 1 2 Val i + f 2 (Δ i ' ,Δ j ' ) Val i + f 2 (Δ i ' ,Δ j ' ) + f 3 (Δ i ' ,Δ j ' ) + Val j + f 2 (Δ i ' ,Δ j ' ) Val j + f 2 (Δ i ' ,Δ j ' ) + f 3 (Δ i ' ,Δ j ' ) # $ % % & ' ( ( ,! ! ) exp( ) exp( ) , ( 2 j boc1 i boc1 j i p p f Δ − + Δ − = Δ Δ ,! ! € f 3 (Δ i ,Δ j ) =− 1 p boc2 ln 1 2 exp −p boc2 Δ i ( ) + exp −p boc2 Δ j ( ) [ ] $ % & ' ( ) ,! ! ) ) ( exp( 1 1 ) , ( boc5 boc boc4 boc3 4 p BO BO p p BO f i ij ij ij i + Δ − − + = Δ ,! ! 76! ! ) ) ( exp( 1 1 ) , ( boc5 boc boc4 boc3 5 p BO BO p p BO f j ij ij ij j + Δ − − + = Δ .!!!!!!!!!!!!!!!!!! (Eq.!3.19)! boc i Δ !is!a!correction!for!atoms!with!lone!electron!pairs,! ! ! ∑ ∈ + − = Δ ) ( boc boc i n j ij i i BO Val ,! ! ! ! ! ! !!!!!!! (Eq.!3.20)! where!n(i)!is!the!set!of!neighbor!atoms!of!i)that!are!within!a!single!covalent Ybond! cutoff!distance.!The!valency!of!atom! i)is!calculated!from!the!corrected!bond!order! term,!! ! ∑ ∈ + − = Δ ) (i n j ij i i BO Val .! ! ! ! ! ! !!!!!!!! (Eq.!3.21)! The!contribution!of!bond!energy!Ebond,!which!represents!the!strength!of!a!covalent! bond!between!atomic!pairs,!includes!the!s,!p!and!doubleYp !!bond!terms,!is!computed! as! !! € E bond = −D e σ ⋅BO ij σ ⋅exp p be1 1− BO ij σ ( ) p be2 ( ) % & ' ( ) * −D e π ⋅BO ij π −D e ππ ⋅BO ij ππ % & ' ( ) * j ∑ i ∑ (Eq.!3.22)) where! e be be D p p , , 2 , 1 , !are!additional!bond!parameters.) 3.3.2 Lone)pair)energy ) The!valence!angles!around!atoms!are!affected!by!the!lone!electron!pairs,!which! in! turn! influence! the! presence! of! overcoordinated! and! undercoordinated! atoms ,! especially!oxygens!and!nitrogens.!The!valency!term,! € Δ i e ,!is!given!by!the!number!of! outer!shell!electrons!and!a!summation!of! the!full!bond!order,! ! 77! ! € Δ i e =−Val i e + BO ij j∈n(i) ∑ .! ! ! ! ! ! !!!!!!! (Eq.!3.23)! The!number!of!lone!electron!pairs!for!atom! i,! € n lp,i ,!is!as!follows:! ! € n lp,i =int Δ i e 2 # $ % & ' ( exp −p lp1 2 +Δ i e −2int Δ i e 2 # $ % & ' ( # $ % & ' ( 2 * + , , - . / / .! ! ! !!!!!!!! (Eq.!3.24)! Once!the!bond!order!and!the!number!of!lone!electron!pairs!are!known,!the! penalty! energy!function!for!atoms!with!lone!electron!pairs!exceeding!the!optimal!number!is ! ! € E lp = p lp2 Δ i lp 1+exp −75Δ i lp ( ) i ∑ ,! ! ! ! ! ! !!!!! (Eq.!3.25)! where! € Δ i lp !is!the!deviation!from!the!optimal!number!of!lone!electron!pair: ! ! € Δ i lp = n lp,opt − n lp,i .! ! ! ! ! ! !!!! !!!!!!!! (Eq.!3.26)! 3.3.3 Overcoordination)and)undercoordination)energy ) For!the!effect!of!overcoordination!(Δi)>)0),!a!penalty!term!is!imposed!in!energy! contribution.!For!undercoordination!(Δi)<)0)!the!energy!contribution!of!resonance!of! the!πYelectron!between!attached!undercoordinated!atomic!centers!is!taken!into! account.! The! coordination! difference!Δi! decreases! when! a! lone! electron! pair! appears.!The!corrected! € Δ i lpcorr !is!! ! € Δ i lpcorr =Δ i − Δ i lp 1+ p ovun3 exp p ovun4 Δ j −Δ j lp ( ) (BO ij π + BO ij ππ ) j∈n(i) ∑ ' ( ) ) * + , , .! !!!!!!!! (Eq.!3.27)) ! 78! Energies!for!the!overcoordinated!and!undercoordinated!atoms!are!calculated!in! terms!of!the!corrected! € Δ i lpcorr ,! ! € E over = p ovun1 D e σ BO ij j∈n(i) ∑ Δ i lpcorr +Val i Δ i lpcorr 1 1+exp p ovun2 Δ i lpcorr ( ) & ' ( ( ) * + + i ∑ ,! ! ! € E under = (−p ovun5 ) 1−exp p ovun6 Δ i lpcorr ( ) 1+exp(−p ovun2 Δ i lpcorr ) × i ∑ 1 1+ p ovun7 exp p ovun8 Δ j −Δ j lp ( ) (BO ij π +BO ij ππ ) j∈n(i) ∑ ( ) * * + , - - .! ! !!!!!!!! (Eq.!3.28)! 3.3.4 Valence)angle)energies ) The! energy! contribution!arising! from!the! deviation! of! valence! angle!Qijk!from! equilibrium!is!described!by! Eval.!The!equilibrium!angle! Q0!is!obtained!by! ! € SBO j = BO jm π +BO jm ππ ( ) m∈n( j) ∑ + 1− exp −BO jm 8 ( ) m∈n( j) ∏ ' ( ) ) * + , , −Δ j angle − p val8 n lp,j ( ) ,! !! ! € Δ j angle =−Val j angle + BO jm m∈n(i) ∑ ,! ! ! € SBO2 = 0 (SBO≤ 0) SBO p val9 (0 <SBO <1) 2− (2−SBO) p val9 (1<SBO < 2) 2 (SBO > 2) $ % & & ' & & ,) ) ! € Θ 0 (SBO) =π−Θ 0,0 1−exp −p val10 2−SBO2 ( ) [ ] { } .! ! ! !!!!!!!! (Eq.!3.29)! ! The!function!Eval!goes!to!zero!as!the!bond!order!within!an!atomic!triplet!goes! to!zero:! ! 79! ! € E val =∑ i ∑ j ∑ k f 7 (BO ij )f 7 (BO jk )f 8 (Δ j )× p val1 − p val1 exp −p val2 Θ 0 SBO j ( ) −Θ ijk ( ) 2 ' ( ) * + , - . / 0 1 2 ,! ) € f 7 (BO ij ) =1−exp−p val3 BO ij p val4 ( ) ,) ! € f 8 (Δ j ) = p val5 − p val5 −1 ( ) 2 +exp p val6 Δ j angle ( ) 1+exp p val6 Δ j angle ( ) +exp−p val7 Δ j angle ( ) .!!!!!!!!!!!!!!!!!!! (Eq.!3.30)! ! When! two! double! bonds!share!an! atom! in! a! triplet,! such! as! allene,!an! additional!penalty!term!is!introduced :) ! € E pen = p pen1 f 9 Δ j ( ) exp −p pen2 BO ij −2 ( ) 2 [ ] exp −p pen2 BO jk −2 ( ) 2 [ ] k ∑ j ∑ i ∑ ,! ! € f 9 Δ j ( ) = 2 +exp−p pen3 Δ j ( ) 1+exp−p pen3 Δ j ( ) +exp p pen4 Δ j ( ) .! ! ! ! !!!!!! (Eq.!3.31)! ThreeYbody!conjugation!energy,!Ecoa,!has!the!following!form:! ! € E coa = p coa1 1 1+exp p coa2 Δ j val ( ) k ∑ j ∑ i ∑ ×exp −p coa3 −BO ij + BO im m∈n(i) ∑ ' ( ) ) * + , , 2 - . / / 0 1 2 2 exp −p coa3 −BO jk + BO km m∈n(k) ∑ ' ( ) ) * + , , 2 - . / / 0 1 2 2 ×exp −p coa4 BO ij −1.5 ( ) 2 [ ] exp −p coa4 BO jk −1.5 ( ) 2 [ ] .! !!!!!!!! (Eq.!3.32)! 3.3.5 Torsion)angle)energies ) The! torsion! angle! energy!Etors! is! calculated! from! 4Ybody! dihedral! angles.! The! dihedral!angle!wijkl!is!the!angle!between!the!two!planes!defined!by!two!atom!triplets! ! 80! (i,!j,!k)!and!(j,!k,!l).!The!torsion!angle!energy!Etors!decreases!as!the!valence!angles! approaches!π,!or!as!the!bond!orders!goes!to!zero,!that!is ) ! € E tors = 1 2 f 10 (BO ij ,BO jk ,BO kl )sinΘ ijk sinΘ jkl l ∑ k ∑ j ∑ i ∑ × V 1 (1+cosω ijkl ) +V 2 exp p tor1 2−BO jk π − f 11 (Δ j ,Δ k ) ( ) 2 { } 1−cos2ω ijkl ( ) +V 3 1+cos3ω ijkl ( ) ) * + + + + , - . . . . ,! ! € f 10 (BO ij ,BO jk ,BO kl ) = 1−exp −p tor2 BO ij ( ) [ ] 1−exp −p tor2 BO jk ( ) [ ] 1−exp −p tor2 BO kl ( ) [ ] ,! ! € f 11 (Δ j ,Δ k ) = 2 +exp −p tor3 Δ j angle +Δ k angle ( ) [ ] 1+exp −p tor3 Δ j angle +Δ k angle ( ) [ ] +exp p tor4 Δ j angle +Δ k angle ( ) [ ] .!!!!!!(Eq.!3.33)! ! The!4Ybody!conjugation!energy!Econj!is!given!by! ! € E conj = p cot1 f 12 (BO ij ,BO jk ,BO kl ) 1+ cos 2 ω ijkl −1 ( ) sinΘ ijk sinΘ jkl [ ] l ∑ k ∑ j ∑ i ∑ ,! ! € f 12 (BO ij ,BO jk ,BO kl ) = exp−p cot 2 BO ij −1.5 ( ) 2 [ ] exp −p cot 2 BO jk −1.5 ( ) 2 [ ] exp −p cot 2 BO kl −1.5 ( ) 2 [ ] .! ! ! ! ! ! !! ! ! ! ! !!!!!!! (Eq.!3.34)! ! 81! 3.3.6 Hydrogen)bond)energy ) The!hydrogen!bond!energy!Ehbond!is!in!a!form!of!the!bond!order!and! ijk Θ of!atomic! triplets.!Ehbond!is!thus!calculated!from!a !triplet!iMjMk,!where!atom!j)is!hydrogen,!atom!i! is!chosen!from!j’s!neighbor,!and!atom!k!is!selected!within!the!finite!cutoff!radius! (~10!Å)!from!atom! i:! ! € E hbond = p hb1 1−exp p hb2 BO ij ( ) [ ] exp p hb3 r hb o r jk + r jk r hb o −2 # $ % % & ' ( ( ) * + + , - . . sin 8 Θ ijk 2 # $ % & ' ( k ∑ j ∑ i ∑ (Eq.!3.35)! 3.3.7 Van)der)Waals)and)Coulomb)energies ) In!ReaxFF,!van!der!Waals!and!Coulomb!forces!are!calculated!among!all!pairs! of! atoms!to!account!for!Pauli!exclusion!principle!for!short!interatomic!distances!and! dispersion!for!long!interatomic!distances.!For!van!der!Waals!interaction,!a!shielded! interaction!is!included!in!a!distance Ycorrected!MorseYpotential!as!follows:! ! € E vdWaals = D ij Tap(r ij ) exp α ij 1− f 13 (r ij ) r vdW $ % & ' ( ) * + , - . / −2exp 1 2 α ij 1− f 13 (r ij ) r vdW $ % & ' ( ) * + , - . / 0 1 2 3 2 4 5 2 6 2 j ∑ i ∑ ,! ! € f 13 (r ij ) = r ij p vdW1 + 1 γ w # $ % & ' ( p vdW1 ) * + + , - . . 1/ p vdW1 ,! ! ! ! ! !!!!! (Eq.!3.36)! where!Tap!has!the!following!definition:) ! € Tap(r ij ) = 20 R cut 7 r ij 7 - 70 R cut 6 r ij 6 + 84 R cut 5 r ij 5 - 35 R cut 4 r ij 4 + r ij 0 ,!! ! ! !!!!!! (Eq.!3.37)! where!Rcut!is!the!nonYbonded!cutoff!radius.!! ! The!Coulomb!potential!is!expressed!as :! ! 82! !! E Coulomb = 1 2 q i q j j ∑ i ∑ Tap(r ij ) r ij 3 + 1 γ ij " # $ $ % & ' ' 3 ( ) * * + , - - 1/3 (1−δ ij ),! ! !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! (Eq.!3.38)! !! where!γ ij !is!a!parameter!for!the!smeared!Co ulombic!function(Nomura,!Kalia!et!al.! 2008).!! ! In! ReaxFF,! atomic! charges! are! updated! dynamically! during! force Yfield! optimization! procedure! using! the! electronegativity! equilibration! method! (EEM)! (Nomura,!Kalia!et!al.!2008 )!.!! 3.3.8 Validation)results) Table!3.5!shows!that!the!elastic!constants!and!bulk!modulus!of!Ni!face Ycentered! cubic! (fcc)! crystal! calculated! with! ReaxFF! agree! well! with! experimental! results! (Simmons!and!Wang!1971).!! Table!3.6!compares!S!impurity!energies!in!Ni!fcc!crystal!calculated!with!ReaxFF!with! DFT!results.!Here,!we!calculate!the!energy!to!substitute!one!S!to!one!of!the!Ni!atoms! in!the!fcc!crystal.!We!also!ca lculate!interstitial!energies!by!adding!one!S!in!either! tetrahedral!or!octahedral!site!of!the!Ni!fcc!crystal.!ReaxFF!results!reproduce!the! order!of!the!three!energies,!i.e.,!the!substitutional!and!octahedral!energies!are!the! highest!and!lowest,!respectively.! ! ! ! ! ! 83! ! Table!3.5!Elastic!constants!of!nickel!fcc!crystal. ) ! Experiment! ReaxFF! C11!(GPa)! 247! 247! C12!(GPa)! 147! 152! C44!(GPa)! 125! 161! B!(GPa)! 181! 184! ! Table!3.6!Comparison! of! ReaxFF! and! DFT! calculations! for! calculations! for! the! binding!energy!of!a!sulfur!impurity!at!various!locations!in!nickel!fcc!crystal. ) ! • ReaxFF! • DFT! • (100)!surface—top!(eV)! 3.82! • 3.76! • (100)!surface—bridge!(eV)! 4.59! • 4.77! (111)!surface—top!(eV)! 4.05! 3.86! • (111)!surface—bridge!(eV)! 4.50! • 5.09! • Σ5(210)!grain!boundary!(eV)! 5.22! • 5.24! Subsurface—octahedral!(eV)! 4.65! 4.12! ! ! 84! CHAPTER)4 ) ) ) ) ) ) ))))))) TRANSFORMATION) OF) GRAIN) BOUNDARIES) IN) NANOCRYSTALLINE) NICKEL) NEAR) PERCOLATION) THRESHOLD) This!chapter!mainly!investigates!into!the!sharp!amorphization!transition!that!occurs! in!a!nanocrystalline!grain!boundary!(GB)!within!a!narrow!range!of!impurity!atom! concentration.!Nickel!(Ni)!and!sulfur!(S)!were!chosen!to!be!the!materials!to!study ,! because!the!fundamental!understanding!and!control!of!embrittlement!mechanisms! in!the!NiYS!system!are!important!not!only!scientifically!but!also!technologically,! e.g.,! for!the!development!of!next Ygeneration!nuclear!reactors!(Allen,!Sridharan!et!al.! 2008).!Background!and!motivation!of!the!simulation!is!described!in!Section!4.1 .!In! section!4.2,!we!discuss!a!series!of!molecular!dynamics!simulations!using!ReaxFF!to! capture!the!structural!and!thermodynamic!p roperties!of!Ni!nanocrystalline!system! with!S!impurity!atoms!at!different!concentration s.!The!effect!of!impurity!atoms!is! discussed!in!Section!4.3,!where!a!hypothesis!regarding!the!atomistic!mechanism! underlying!the!sudden!change!in!physical!properties! is!proposed.!The!hypothesis!is! further!validated!in!Section!4.4 !using!a!simple!model!system .! ! 85! 4.1 Background)and)motivation ) 4.1.1 Sulfur)segregation)in)nickel)nanocrystalline) system)) The!modification!of!chemical!bonds!due!to!a!small!amount!of!impurities!segregated! to!GBs!essentially!controls!the!mechanical!properties!of!materials.!In!the!case!of!S! segregationYinduced!embrittlement!of!Ni,!Heuer!et)al.!performed!tensile!tests!of! notched!specimens!with!varying!amount!of!S!segrega tion!to!GBs!(Heuer,!Okamoto!et! al.!2002).!Upon!fracture!of!each!specimen,!S!concentrations!at!GBs!were!measured ! by! Auger! electron! spectroscopy! of! fracture! surfaces,! and! the! percentage! of! intergranular!fracture!was!determined!using!a!relative!area!method. !As!the!GB!S! concentration! increases,! a! transition! from! transgranular! ductile! tearing ! to! intergranular!brittle!fracture!was!observed,!and!the!critical!S!concentration!for!50%! intergranular! fracture! was! determined! to! be ! 15.5! ±! 3.4! %.! Heuer!et) al.! also! measured! the! critical! S! concentration! for! S Yinduced! amorphization! of! Ni!faceY centered!cubic!(fcc)!crystal!implanted!with!S + !ions!using!Rutherford!backscattering! spectrometry(Heuer,! Okamoto! et! al.! 2002 ).! BrightYfield! transmission! electron! micrographs! and! associated!selected! area! diffraction! patterns! confirmed! the! amorphization!of!specimens!with!higher!ion!doses. !To!quantify!the!formation!of! amorphous! phases,! progression! of! dechanneling! along ! the! € [110]!direction! was! monitored.!The!critical!S!concentra tion!of!14.2!±!3.3!%!required!to!induce!50%! amorphization!thus!determined!coincides!with!the!critical!S!concentration!for!GB! embrittlement!within!experimental!error.! ! ! 86! In!the!recent!theoretical!study!conducted!by!Chen! et)al.,)a!simulation!containing! 48,393,282! atoms! was! conducted! using! ReaxFF! MD! for! two! systems:! pure! Ni! nanocrystal!(ncYNi)!and!Ni!nanocrystal!with!20%!S Ydoped!GB!layers!of!thickness!5!Å! (ncYNi+20%S)!(Chen,!Kalia!et!al.!2010 ).!Simulation!results!show!two!crossovers:! from! mixed! transgranular! and! intergranular! fracture! in! nc YNi! to! purely! intergranular! fracture! in! ncYNi+20%S,! and! from! opening! mode! (or! mode! I)! predominantly!in!ncYNi!to!mixture!of!sliding!mode!(or!mode!II)!in!nc YNi+20%S.!The! mode!II!(slidingYmode)!cracks!in!ncYNi+20%S!imply!the!existence!of!shear!stress! acting!on!them.!To!study!the!shear!stress,!generalized!stacking!fault!energy!(GSFE)! and!shear!stress!are!calculated!to!slide!Ni!∑5!(012)!GB! with!different!concentration! of!segregated!S!atoms!on!the!GB!interface ,!as!illustrated!in!! Figure! 4.1.! An! orderYofYmagnitude! reduction! of! grainYboundary! shear! strength! was! observed! due! to! amorphization,! combined! with! tensile Ystrength! reduction,!which!allows!the!crack!tip!to!always!find!an!eas y!propagation!path.!This! study!provided!a!missing!link!between!GB!amorphization!and!embrittlement,!but! left!one!remaining!question:!what!causes!the!sudden!amorphization!at!GB! at!14%!of! S!concentration?!! ! 87! ! Figure!4.1!Generalized!stacking!fault!energy!(GSFE)!of!Ni!∑5(012)!GB!without!S!(A),! with!a!monolayer!of!segregated!S!(B),!and!with!amorphous!sulfide!GB!phases!(C).! Also! shown! are! shear! stress! calculated! from! the! derivative! of! the! general ized! stacking!fault!energy!with!respect!to!displacement!along!the! [100]!direction!(Chen,! Shi!et!al.).! ! 88! 4.1.2 Percolation)and)physical)properties) ) Percolation,! originally! studied! as! a! mathematical! subject,! found! its! broad! applications! in! materials! research! as! a! powerful! tool! to! account! for! physical! properties! of! heterogeneous! materials! (Zallen! and! Wiley! 1983;! CeYWen! 1993;! Stauffer!and!Aharony!1994).!The!significance!of!the!percolation!transition!lies!in!the! fact!that!a!continuous!cluster!extends!throughout!the!system !as!the!volume!fraction! f!of!this!minor!phase!approaches!a!critical!value! fc,!i.e.,!the!percolation!threshold.! Percolation!can!result!in!dramatic!changes!in!physical!properties!of!composites!near! fc,!as!indicated!in!the!many!examples!in!Figure!4.2!Instead!of!following!a!linear!rule! of!mixture,!the!direct!global!connection!immediately!results!in!a!nonlinear!scaling!of! transport!properties.!! In!this!dissertation,!we!use!MD!simulation!to !reveal!that!the!large!steric!size! of! sulfur! impurity! causes! strong! sulfur Ysulfur! interaction! mediated! by! lattice! distortion! up! to! the! next! nearest Yneighbor! lattice! sites! and! that! amorphization! occurs! at! the! percolation! threshold! of! the! sulfur Ysulfur! network! with! the! next! nearestYneighbor!connectivity.!Furthermore,!the!generality!of!the!amorphization! mechanism!due!to!the!percolation!of!an!impurity!network!is!confirmed!for!a!model! binary!material.! ! 89! !! !!! !! ! Figure!4.2!(a)!Electrical!conductivity!σ c !of!polymer!based!composites!of!carbon! nanotubes!as!a!function!of!the!filler!volume!fraction! f!(Bryning,!Islam!et!al.!2005 );! (b)! electrical! conductivity! σ c !of! polymer! based! composites! of! grapheme/polystyrene!as!a!function!of!the!filler!volume!fraction! f!(Stankovich,!Dikin! et!al.!2006);!(c)!variation!of!the!relative!dielectric!constant!κ!of!Ni/BaTiO 3!+!PVDF! [poly(vinylidene!fluoride)]!threeYphase!composites!with!the!volume!fraction! f!of!Ni! particles!(Dang,!Shen!et!al.!2002 ).!! ! 90! 4.2 Structural)and) thermodynamic)properties)of)Ni PS)system) ) ReaxFF!MD!simulations!with!validated!potential!on!Ni YS!reactions!(see!Section!3.3!)! are!carried!out!to!understand!S Yinduced!amorphization!of!Ni.!A!set!of!cubic!cells!of! size! 35.2! Å! containing! 4000! atoms! is! used! in! this! calculation,! where! periodic! boundary!conditions!are!applied!in!all!three!Cartesian!directions.!Ni!atoms!are! randomly!substituted!by!S!atoms!in!the!range!of!8!~!18% !(Heuer,!Okamoto!et!al.! 2002).! For! each! S! concentration,! the! system! is! relaxed! by! a! steepest Ydescent! procedure.!Subsequently,!the!system!is!thermalized!for!5!ps,!then!gradually!heated! to!300K!and!got!fully!relaxed!there.! ! To!study!the!effect!of!S!substitution!on!structural!correlations,!we!calculate! the! pair! distribution! functions! (PDF)!g(r)!(see! Section!2.2.2)!using! the! atomic! coordinates!from!the!MD!simulations.!Figure!4.3(a)!shows!the!partial!PDF,! gNiYNi(r),! between!Ni!atoms!for!three!different!S!concentrations:! cS!=!8%,!14%,!and!18%.!We! observe!that!the!peak!heights!in!gNiYNi(r)!decrease!for!increased!S!concentration,! signifying!increasing!disorder.!We!also!note!that! gNiYNi(r)!at!distance!r!>!7!Å!has! clearly!separated!peaks!at!cS!=!8%,!whereas!they!merge!together!i nto!broader! features!above!cS!=!14%,!which!indicates!the!disappearance!of!the!longer Yrange! structural!order!in!Ni!at! cS!>!14%.!! ! 91! ! Figure!4.3!(a)!NiYNi!partial!pair!distribution!function!gNiYNi(r)!of!NiYS!system!at! different!S!concentration!cS;!(b)!full!width!at!half!maximum!of!the!first!peak!of! gNiY Ni(r)!as!a!function!of! cS;!(c)!height!of!the!second!peak!of! gNiYNi(r)!as!a!function!of! cS;! (d)!coordination!number!NNiYNi(r)!of!NiYS!system!at!different!S!concentration!cS;! To!quantify!the!broadening!of!the!peaks,!we!calculate!the!full!width!at!half! maximum!(FWHM)!of!the!first!peak!in! gNiYNi(r)!by!Gaussian!fitting,!and!plot!it!as!a! function!of!cS!from!8!to!18%!in!Figure!4.3(b).!We!see!that!the!FWHM!is!well! described!by!a!linear!function!of! cS!before!and!after!a!critical!concentration!of!~!14! %,!as!shown!by!the!linear!fits!of!the!FWHM! vs.!cS!by!dashYdotted!and!dashed!lines! below!and!above!14!%,!respectively,!in!Figure!4.3(b).!The!curve!thus!exhibits!a! discontinuity!at!cS!~!14%.! ! 92! Figure!4.3(c)!plots!the!height!of!the!second!peak!in! gNiYNi(r)!as!a!function!of! cS.! The!behavior!of!the!second Ypeak!height!changes!dramatically!before!and!after! cS!=! 14%.!Namely,!it!is!a!decre asing!function!of!cS!before!14%,!while!it!takes )a!constant! value!for!cS!>!14%.!All!these!observations!indicate!sudden!disordering!of!structure! at!a!critical!S!concentration!of!~!14%,!which!agrees!well!with!the!experimental! amorphization!threshold!of!14.2±3.3!%!(Heuer,!Okamoto!et!al.!2002 ).! Besides!the!intensity!of!the!peak!height!in! gNiYNi(r),!the!change!in!bond!length! is!also!reflected!by!a!change!in!the!position!of!the!first!peak!in! gNiYNi(r)!and!at!the! first!shell!of!neighbors!in!coordination!number:!shorte r!NiYNi!bonds!were!observed! with!the!increase!of!S!concentration!while!th e!coordination!number,!in! Figure!4.3(d)! displays! a! small! change! from! 9! to! 11! at! the! first! shell.! The! behavior! of! Ni! coordination! number! change! and!gNiYNi(r)! can! be! attributed! to! the! size! effect! (S.G.Hao!2010).!The!distinct!difference!of!radius!between!Ni!and!S!drives!the!first! and! second! peaks! change! systematically!(Yamaguchi,! Shiga! et! al.! 2005).! The! increased! larger! S! atoms! tend! to! squeeze! and! push! the! surrounding! Ni! atoms,! resulting!in!loss!of!the!co ordination!number!at!the!first!nearest!neighbor!distance.! ! The!structural!change!as!a!function!of!S!concentration!is!also!manifested!in! bond!angle!distribution!(BAD,!see!section!2.2.2)!as!shown!in!Figure!4.4(a),!where! the!NiYNiYNi!BAD!is!plotted!for! cS!=!8,!14,!and!18%.!As! cS!increases!from!14!to!18%,! the!position!of!the!first!peak!at!~!60°!systematically!shifts!to!a!larger!angle!by!three! degrees!as!a!result!of!smaller!coordination!numbers,!accompanied!by!dramatic! decrease!of!its!height!and!broadening.! ! ! 93! !!!!! ! Figure!4.4!(a)!NiYNiYNi!bond!angle!distribution!(BAD)!at!S!concentrations! cS!=!8,!14,! and!18%.!(b)!Comparison!of!the!BAD! cS!=!18%!with!that!of!amorphous!Ni. ! ! The!structural!disorder!manifested!as!the!broadening!of!BAD!at! cS!=!18%!is! comparable! to! that! of! amorphous! Ni! prepared! by! a! melt Yquench! procedure! as! shown!in!Figure!4.4!(b).!There!are!peaks!at!only!60˚!and!its!supplementary!angle! 120˚,!which!correspond!to!the!nearest!neighbor!bond!angle.!The!peaks!around!90˚! and!180˚!in!the!fcc!crystal! disappeared.!This!indicates!a!total!disordered!structure.! ! ! 94! To!investigate!the!behavior!of!thermodynamic!quantities!of!the!Ni!crystal! induced!by!S!impurities,!we!calculate!the!equilibrium!volume!of!the!system!for! different!S!concentrations.!In!Figure!4.5,!the!solid!circles!represent!the!calculation! results,!and!the!dashYdotted!and!dashed!curves,!respectively,!are!the!best!fits!below! and!above!cS!=!14!%.!Below!the!critica l!concentration,!the!volume!is!a!linear!function! of!the!S!concentration.!In!contrast,!the!volume!expansion!due!to!S!impurity!becomes! much!greater!and!nonlinear!above!14%,!with!a!visible!gap!at!~!14%,!indicating!a! firstYorder!phase!transition.! ! Figure!4.5.!Equilibrium!volume!as!a!function!of!S!concentration. ! 4.3 Interaction)range)of)sulfur)in)nickel)crystal)and)percolation)analysis )) In! section! 4.2,! impurityYatom! induced! amorphization! was! confirmed! and! characterized.!Our!next!goal!is!to!answer!the!question:!what!causes!the!structural! transition!at!Cs!~14%?!The!first!step!here!is!to!isolate!and!to!quantify!the!effect!of! the!network!of!impurity!atoms!in!the! Ni!crystalline!structure.!! ! 95! ! To!estimate!the!SYS!interaction!range!in!Ni!fcc!crystal!with!S!in!substitutional! sites,!we!first!prepared!a!Ni!perfect!crystal!containing!5324!atoms!(11×11×11!unit! cells)!and!calculate!the!bulk!energy!E bulk !using!ReaxFF.!Then,!we!substitute!one!Ni! atom!in!the!crystal!with!one!S!atom.!The!new!system!is!relaxed!by!ReaxFF YMD,!and! the!final!energy!of!the!system!E 1 !is!calculated.!In!this!new!system,!another!Ni!is! substituted!with!S!at!a!lattice!site!at!distance! d!from!the!first!S!atom,!and!we! calculate!the!final!energy!of!the!system!with!two!S!atoms! E 2 !after!full!relaxation.! The!calculation!procedure!is!s chematically!described!in!Figure!4.6,!from!which!the! following!relations!are!derived:! ! E 1 =E bulk +ΔE insert (S)+E 1S ,! ! ! ! ! ! !!!!!!!!! (Eq.!4.1)! ! € E 2 =E 1 +ΔE insert (S)+E 1S +ε S−S,! ! ! ! ! !!!!!!!!! (Eq.!4.2)! where!ΔE insert (S)!is!the!energy!of!inserting!one!S!in!Ni!crystal,! E 1S !is!the!energy!of!an! isolated!S!atom!(=!0!in!ReaxFF),! ΔE insert (S)!is!the!energy!of!inserting!another!S!atom! in!Ni!crystal,!and!ε S−S !is!the!interaction!energy!between!the !two!S!atoms.!Subtracting! Eq.!4.2!from!Eq.!4.1,!the!interaction!energy!between!two!S!atoms!is!obtained!from! the!calculated!quantities!as!! ! € ε S−S =E 2 −2E 1 +E bulk.! ! ! ! ! ! !!!!!!!!!! (Eq.!4.3)! ! 96! ! Figure!4.6!Schematic!of!the!calculation!procedure!for!the!lattice!distortion Ymediated! SYS!interaction!energy.! ! Figure!4.7!shows!the!calculated!interaction!energy!between!two!S!atoms! € ε S−S! in!the!Ni!fcc!lattice!as!a!function!the!S YS!distance!d.!ε S−S !takes!the!minimum!value!at! d!equal!to!the!distance!d2!between!the!next!nearestYneighbor!lattice!sites,!which!is! close!the!lattice!constant!(LC)!=!3.52!Å.!Therefore,!two!S!atoms!in!Ni!crystal!are!most! likely!to!occupy!the!next!nearest Yneighbor!sites.!We!also!see!that!two!S!atoms!do!no t! have!any!interaction!beyond!twice!the!LC,! i.e.,!7.04!Å.! ! 97! ! Figure!4.7!SYS!interaction!energy!in!Ni!fcc!crystal!as!a!function!S YS!distance.! ! ! When!the!concentration!of!impurities!is!low,!most!impurity!atoms!a re!likely! to! stay! outside! of! the! interaction! ranges! of! other! impurity! atoms.! As! the! concentration!gets!higher,!the!influence!of!impurity!atoms!starts!to!overlap!to!form! clusters,!but!the!total!effect!of!each!cluster!is!still!localized. !At!a!certain!critical! density,!the!network!of!S YS!interaction!is!expected!to!extend!throughout!the!entire! crystal.!Such!a!critical!concentration!can!be!calculated!based!on!the!percolation! theory!(Aharony!and!Aharony!1994!)!(Zallen!1983;!Aharony!and!Aharony!1994!)! (Zallen! 1983).! The! site! percolation! model! involves! the! random! occupation! (S! impurity!substitution!in!our!case)!of!lattice!sites!of!a!regular!lattice!(fcc!lattice!in!our! case).!Each!site!of!the!regular!lattice!is!then!randomly!occu pied!with!probability!p.) ! 98! In!addition,!the!model!should!define!the!connectivity!between!the!occupied!sites.! Here,!we!define!two!occupied!sites!within!distance! d!to!be!connected.!The!critical! occupation! probability!pc,! called! percolation! threshold,! is! then! d efined! as! the! minimum! concentration! of! the! occupied! sites! at! which! an! infinitely! connected! cluster!is!formed!(Zallen!1983).!! We!perform!a!site!percolation!simulation!to!test!our!assumption.!A!system! consisting! of! 100×100×100! fcc! unit! cells! is! simulated! with! periodic! boundary! conditions!in!three!Cartesian!directions.!In!our!simulation,!lattice!sites!are!randomly! occupied!with!probability!p.!Adjacent!sites!within!the! xYth!nearest!neighbors!(x!=!1Y 4)!are!defined!to!be!connected!to!each!other,!and!we!enumerate!all!clusters!of!such! connected!sites!using!a!find/union!algorithm! (Newman!and!Ziff!2001).!Figure!4.8! shows!the!largest!cluster!size!(in!terms!of!the!number!of!sites!in!the!largest!cluster)! as!a!function!of!occupation!probability!p!for!different!connectivity!x.!For!each!x,!we! observe!a!critical!occupation!probability!pc,!above!which!the!largest!cluster!size! becomes!comparable!to!the!total!system!size.!The!resulting!percolation!threshold!is! 0.20,!0.136,!and!0.06,!respectively,!with!the!nearest Yneighbor!(n.n),!next!nearestY neighbor!(n.n.n),!and!the!fourth!nearest!neighbor!connectivity,!in!agreement!with! previous!percolation!study!(Pike!and!Seager!1974).!Our!percolation!analysis!on!fcc! lattice!thus!shows!that!the!critical!S!concentration!for!amorphization!(0.14)!is!close! to!the!percolation!threshold!with!n.n.n!connectivity.! Note!that!the!n.n.n!connectivity! is! consistent! with! the! most! favorable! S YS! distance! due! to! Ni! lattice! distortion Y mediated!interaction!as!shown!in! Figure!4.7.!This!serves!as!a!possible!explanation!of! ! 99! the!amorphization!transition:!For!small!impurity!concentration,!the!influence!of!the! impurity!atoms!on!structure!is!localized!around!each!impurity.!As!the!impurity! concentration! increases,! the! interaction! range! of! the! impur ity! atoms! starts! to! overlap,!and!finally!becomes!global!at!the!percolation!threshold. ! ! Figure!4.8!The!largest!cluster!size!as!a!function!of!the!occupation!probability!for!fcc! lattice!considering!the!1 st ,!2 nd ,!and!4 th !nearest!neighbor!connectivity.! 4.4 Thresholds)in)binary)Lennard PJones)systems) To!test!the!general!validity!of!the!amorphization!mechanism!based!on!the!atomic! size! effect! and! percolation! of! impurity! interactions! explained! in! section! III,! we! perform!a!similar!analysis!for!a!binary!Lennard YJones!(LJ)!system!consisting!of!two! types!of!atoms!with!different!atomic!sizes,!which!has!been!used!extensively!to!study! amorphization.! Binary! LJ! systems! are! chosen! because! they! are! simple! to! ! 100! characterize!and!have!been!extensively!studied!(Li!and!Johnson!1992)!(Shimizu,! Ogata!et!al.!2006)!(Hsieh!and!Yip!1989)!(Tang!and!Yip!1995).!The!LJ!interatomic! potentials!are!given!by! ! ,! ! ! ! ! !!!!!!!!!! (Eq.!4.4)! where!rij!is!the!interatomic!distance!between!iYth!and!jYth!atoms,!and! € α!and!β! represent!the!species!of!the!two!atoms!(denoted!A!and!B).!We!define!the!parameters! of!the!three!interactions,! € φ AA,! € φ AB!and! € φ BB!as! ! εAA!=!εBB!=!εAB!=!ε,! ! ! ! ! ! ! !!!!!!!!!! (Eq.!4.5)! ! € σ AB =(σ AA +σ BB )/2 .! ! ! ! ! ! !!!!!!!!!! (Eq.!4.6)! Thus,!all!three!potentials!are!taken!to!be!the!same!depth,!and!atoms!A!and!B!differ! only!in!size.! We!use!the!conjugateYgradient!method!to!perform!a!similar!procedure!as!we! did!in!determining!the!S YS!interaction!range!to!find!the!relationship!between!the! atomic!size!and!interaction!range.!The!system!is!an!fcc!lattice!consisting!of!6912!(=! 12×12×12×4)!atoms.!Two!B!atoms!are!kept!apart!a t!substitutional!sites!in!different! distance!in!the!〈111〉!direction,!and!BYB!interaction!energies!are!calculated!as!a! function!of!the!distance!for!different!atomic Ysize!ratios,!from!σAA/σBB!=!0.7!to!0.84.! Figure!4.9(a)!shows!the!calculated!interaction!energy!as!a!function!of!the!distance! between!two!B!impurity!atoms!in!A!crystal!with!different!atomic!size!ratios! σAA/σBB.! We!see!that!the!interaction!range !varies!due!to!the!size!effect.!Here,!we!calculate!the! € φ αβ (r ij ) = 4ε σ αβ r ij ' ( ) ) * + , , 12 - . / / − σ αβ r ij ' ( ) ) * + , , 6 1 2 3 3 ! 101! interaction!range!to!be!the!impurity Ypair!distance!at!which!the!interaction!energy! becomes!0.!Figure!4.9(b)!plots!the!interaction!range!Rint!between!two!impurity! atoms!as!a!function!of!the!atomic!size!ratio! σAA/σBB.! !!!!!! ! Figure!4.9!(a)!Interaction!energy!vs.!distance!between!two!B!impurity!atoms!in!A! crystal!in!binary!LennardYJones!systems!with!different!atomic!size!ratios! σAA/σBB.! (b)!Interaction!range!for!different!atomic!size!ratio.!(c)!Amorphization!threshold!in! a!binary!LYJ!system!with!different!atomic!size!ratios! σAA/σBB.!! ! 102! ! Phase! diagram! in! the! space! of! atomic! size! ratio!σBB/σAA! and! solute! concentration!cS!for!binary!LJ!solid!solution!has!been!obtained!by!Li! et)al.!using!MD! simulations!(Li!and!Johnson!1992).!The!simulations!were!performed!at!a!constant! pressure!and!temperature!(set!to!0!and!0.3,!respectively,!in!reduced!LJ!units).!The! corresponding! glass! transition! temperature! in! LJ! liquid! is! Tg!~! 0.4.! The! phase! diagram!shows!the!transition!from!fcc!crystal! to!amorphous!structure!at!elevated! cS,! with! different! amorphization! threshold! € c s * !for! different! atomic! size! ratio.!Figure! 4.9(c)!plots! € c s * !as!a!function!of!σBB/σAA.!! From!the!mapping!between!the!atomic!size!ratio!and!the!impurity!interaction! range,!Rint(σAA/σBB)!in!Figure!4.8(b)!and!that!between!the!atomic!size!ratio!and!the! amorphization!threshold! € c s * (σAA/σBB)!in!Figure!4.9(c),!we!can!derive!a!new!mapping! between! the! amorphization! threshold! (the! critical! concentration! where! amorphization!occurs)!and!the!interaction!range,! € c s * (Rint)!as!shown!in!Figure!4.8.! The!figure!also!shows!percolation!thresholds!obtained!by!using!the!1 st ,!2 nd ,!and!the! 4 th !nearestYneighbor!site!distances!for!site!connectivity.!Namely,!the!percolation! threshold! for! fcc! lattice! considering! the! 1 st ,! 2 nd ,! and! 4 th ! nearestYneighbor! connectivity!is!0.2,!0.14,!and!0.06,!respectively,!as!explained!in!section! 4.3.!Figure! 4.10!shows!that!the!crystalYtoYamorphous!transition!in!binary!LJ!system!occurs!at! lower!concentrations!for!larger!interaction!ranges!between!impurity!atoms.!This!is! consistent!with!the!decreasing!percolation!threshold!as!a!function!of!the!interaction! range!in!Figure!4.10,!which!supports!the!conjecture!presented!in!section! 4.3,!i.e.,! ! 103! amorphization! occurs! at! the! percolation! threshold! of! the! impurity Yimpurity! interaction!network.! ! Figure! 4.10! Amorphization! threshold! for! binary! LennardYJones! system! and! percolation!threshold.! ! ! ! 104! CHAPTER)5 ) ) ) ) ) ) ) ) ))))))))))) CORE/SHELL) STRUCTURE) AND) NOVEL) THERMOP MECHANICAL)BEHAVIORS)IN)NANOWIRES) ) In!this!chapter,!novel!thermoYmechanical!behaviors!in!nanowires!(NW)!are!studied.! A!prototypical!example!zinc!oxide!(ZnO)!is!chosen!to!be!the!research!material,! because!of!its!broad!applications!in!electronics,!photonics,!acoustics,!and!sensing.! Various!unique!nanomechanical!properties!have!been!reported !(Wang!2004;!Elias,! LevyYClement!et!al.!2010).!In!theoretical!study,!novel!structural!transformations!in! ZnO!NW!are!reported!(Dai,!Cheong!et!al.!2010 )!but!have!never!been!observed!in! experimental!work!(Agrawal,!Peng!et!al.!2009 ).!The!ductile!failure!of!ZnO!NW!has! been!observed!both!computationally!and!experimentally ,!but!the!complete!size!and! temperature!dependent!failure!behavior!mapping!is!missing.!! In!this!dissertation,!a!nanoYthermoYmechanical!phase!diagram!is!constructed! using!a!series!of!MD!simulation! studies!on![0001]Yoriented!ZnO!NW!under!tension .! Novel!transitions!from!brittle!cleavage!to!structural!transformation Ymediated!brittle! cleavage! to! ductile! failure! for! smaller! diameters! and! higher! temperatures !are! revealed.! The! diagram! also! explains! why! theoretically! proposed! structural! transformations!have!not!been!observed!in!experiments ,!and!resolves!the!brittle!vs.! ductile!controversy!between!experiments!and!theory .!Atomistic!mechanisms!of!the! unique! nanoYthermoYmechanical! behavior! are! elucidated! as! a! consequence! of! ! 105! ‘incipient’!surfaceYstructural!relaxation,!which!exists!even!i n!the!absence!of!tension.! This!in!particular!predicts!spontaneous!formation!of!‘intrinsic’!core/shell!structures! under!tension,!as!opposed!to!extrinsic!cor e/shell!structures(Lauhon,!Gudiksen!et!al.! 2002)!constituted!of!different!atomic!species. ! In!section!5.1,!background!knowledge!about!Z nO!material!and!current!study! are!provided,!as!well!as!detailed!information!on!the!analysis! method,!slip!vector! field.!Surface!relaxation!of!NW!is!characterized!in!section!5.2,!and!the!structure!of! the!NW!is!characterized!in!section!5.3.!In!section!5.4,!tensile!tests,!which!exhibit! novel! thermoYmechanical! behaviors! of! NW,! are! presented,! and! the! atomistic! foundation!is!investigated!in!section s!5.5!and!5.6.!The!complete!thermoYmechanical! phase!diagram!is!concluded!in!the!last!section.! ! ! ) ! 106! 5.1 Background)and) motivation)) 5.1.1 ZnO) ZnO!is!a!semiconductor!material!with!a!direct!wide!band!gap!energy!( Eg!=!3.37!eV)! and!a!large!exciton!binding!energy!(60!meV)!at!room!temperature! (Yang,!Li!et!al.! 2008).!Some!optoelectronic!applications!of!ZnO!overlap!with!that!of!GaN,!another! wideYgap!semiconductor!(Eg!~3.4!eV!at!300!K) ,!which!is!widely!used!for!production! of!green,!blueYultraviolet,!and!white!lightYemitting!devices.!However,!ZnO!has!some! advantages!over!GaN!among!which!are!the!availability!of!fa irly!highYquality!ZnO! single!crystals.!ZnO!crystal!can!be!produced!by!simpler!crystalYgrowth!techniques,! resulting!in!a!potentially!lower!cost!for!ZnO Ybased!devices.!Besides!that,!ZnO!is!also! biocompatible,! biodegradable,! and! biosafe! for! medical! and! environmental! applications! (Zhou! and! Huang! 2004).! Among! the! oneYdimensional! (1D)! nanostructures,! ZnO!NW!(or! nanorods)! have! been! widely! studied! due! to! their! unique!material!properties!and!remarkable!performance!in!elec tronics,!optics,!and! photonics,! and! have! become! one! of! the! most! important! nanomaterials! for! nanotechnology!in!today’s!research!(Wang!2009).!! ! ZnO!crystallizes!in!two!main!forms,!hexagonal!wurtzite!(WZ)!and!cubic!zinc! blende!(ZB).!The!rocksalt!(RS)!structure!can!exist!at!relatively!high!pressures.!At! ambient!conditions,!ZnO!exhibits!a!hexagonal!WZ!structure.!The!ZB!ZnO!structure! can!be!stabilized!only!by!growth!on!cubic!substrates.! The!crystalline!nature!of!ZnO! could!be!indexed!to!known!structures!of!hexagonal!ZnO,!with!a!=!0.32498!nm,!b!=! ! 107! 0.32498!nm!and!c!=!5.2066!nm!(Mehrabian!and!YousefiYKoma!2011).!The!ratio!of! c/a!of!about!1.60!is!close!to!the!ideal!value!for!a!hexagonal!cell! c/a!=!1.633.!The! structure!of!ZnO!could!be!described!as!a!number!of!alternating!planes!composed!of! tetrahedrally!coordinated!O 2Y !and!Zn 2+ !stacked!alternately!along!the!cYaxis!(Figure! 5.1),!which!corresponds!to!ABABAB!stacking!pattern .!! !!!!!! ! Figure!5.1!ZnO!structure:!(a)!the!wurtzite!unit!cell!(b)!the!wurtzite!structure!model;! (from!http://www.wikipedia.org/).!! ! The!groundYstate!total!energies!of!ZnO!in!WZ,!ZB!and!RS!structures! have! been!calculated!as!a!function !of!unitYcell!volume!using!a!first Yprinciple!periodic! HartreeYFock!(HF)!calculations!based!on!linear!combination!of!atomic!orbitals.!The! total!energy!in!WZ!was!calculated!to!be! Y5.658!eV,!Y5.606!eV!for!ZB,!and!Y5.416!eV! for!RS.!The!total!energy!(cohesive!energy!per!bond)!vs.!volume!for!the!three!phases! is!presented!in!Figure!5.2!(Jaffe!and!Hess!1993).!The!corresponding!calculation! using!our!force!field!is!shown!in!( Section!3.2.2).!Using!GGA!calculation!technique,! the!equilibrium!cohesive!energy!of!ZnO!was!reported!as !Y7.692,!Y7.679,!and!Y7.455! ! 108! eV!for!WZ,!ZB!and!RS,!respectively.!This!is!by!far!the!best!agreement!with!the! experimental! value! of! Y7.52eV! (Weast! and! Astle! 1993).! Although! in! both! calculations,!the!energy!difference!between!WZ!phase! and!ZB!phase!is!small!~!tens! of!meV,!the!transition!between!two!phases!has!been!proved!to!be!highly!controllable! during!NW!growth!(Dai,!Pan!et!al.!2001 ;!Duan,!Huang!et!al.!2001 ;!Huang,!Mao!et!al.! 2001;!Wang!2004).!Accordingly,!pure!crystalline!NW!can!be!obtained.! ! !!!!!!!!!! ! Figure!5.2!Total!energy!vs.!volume!(both!per!ZnO !formula!unit)!for!the!three!phases:! ZB!(squares),!WZ!(diamonds),!and!RS!(circles).! The!zero!of!energy!is!the!sum!of!the! total!energy!of!an!isolated!Zn!and!isolated!O!atom. !(Weast!and!Astle!1993).! On! the! thermoYmechanical! studies! of! ZnO! NW,! various! unique! nanomechanical!properties!have!been!reported !(Wang!2004;!Elias,!LevyYClement!et! al.!2010).!Both!experimental!(see!Figure!5.3)!and!theoretical!studies!have!shown!the! stiffening!of!NW!for!smaller!diameters(Chen,!Shi!et!al.!2006 ;!Agrawal,!Peng!et!al.! ! 109! 2008;!Liu,!Li!et!al.!2009 ).!MD!simulation!has!shown!that![0001] Yoriented!ZnO!NWs! transform! from! WZ! to! body YcenteredYtetragonal! (BCT)! structure! under! uniaxial! tension(Wang,! Kulkarni! et! al.! 2008 ),! which! leads! to! unique! fracture! behavior! (Agrawal,!Peng!et!al.!2009 ).!Structural!transformation!to!graphitic!structure!has!also! been!predicted!theoretically!below!a!critical!diameter!(Zhang!and!Huang!2007).! However,! none! of! these! structural! transformations! has! been! observed! experimentally!(Agrawal,!Peng!et!al.!2009 ),!and!this!controversy!demonstrates!the! lack!of!our!understanding!of!the!mechanical!properties!of!ZnO!NWs. ! ! 110! ! Figure!5.3!(a)!Young’s!modulus!as!a!function!of!NW!diameter!(Wen,!Sader!et!al.! 2008);!(b)!fracture!strain!as!a!function!of!NW!diameter !(Desai!and!Haque!2007).! ! A!major!unsolved!problem!is!the!size!and!temperature!dependence!of!the! failure! behavior! of! ZnO! NWs.! For! example ,! brittle! cleavage! has! been! observed! ! 111! experimentally!(Agrawal,!Peng!et!al.!2009 ),!while!an!MD!simulation!shows!super Y ductility!(Dai,!Cheong!et!al.!2010 ).!A!complete!sizeYtemperature!‘phase!diagram’!for! the!mechanical!response!of!a!ZnO!NW!under!tensio n!would!be!a!valuable!tool!for! understanding! nanomechanical! experiments! and! for! better! reliability! and! manufacturability!of!devices!utilizing!NWs.!In!this!Chapter,!our!MD!simulation!of! [0001]Yoriented!ZnO!NW!under!tension!provides!such!a! nanoYthermoYmechanical! phase!diagram,!which!reveals!novel!transitions!from!brittle!cleavage!to!structural! transformationYmediated!brittle!cleavage!to!ductile!failure!for!smaller!diameters! and! higher! temperatures.! The! phase! diagram! also! explains! why! theoret ically! proposed! structural! transformations! have! not! been! observed! in! experiments,(Agrawal,! Peng! et! al.! 2009 )! and! resolves! the! brittle!vs.! ductile! controversy! between! experiments(Agrawal,! Peng! et! al.! 2009 )! and! theory.(Dai,! Cheong!et!al.!2010)!Furthermore,!atomistic!mechanisms!of!the!unique!na noYthermoY mechanical! behavior! are! elucidated! as! a! consequence! of! ‘incipient’! surface Y structural! relaxation,! which! exists! even! in! the! absence! of! tension,! which! in! particular!predicts!spontaneous!formation!of!‘intrinsic’!core/shell!structures!under! tension,!as!opposed!to!extrinsic!core/shell!structures (Lauhon,!Gudiksen!et!al.!2002 )! constituted!of!different!atomic!species. ! 5.1.2 Slip)vector) for)the)characterization)of) plastic)deformation) Plastic! deformation! is! an! important! concept! in! understanding! and! quantifying! mechanical!behavior!of!materials.!In!crystalline!materials,!the!plastic!flow!is!related! ! 112! directly!to!the!presence!of!dislocation!and!to!their!response!to!applied!stresses.! The! most!useful!definition!of!a!dislocation!is!given!in!terms!of!the!Burgers! circuit.!A! Burgers!circuit!is!any!atomYtoYatom!path!taken!in!a!crystal!containing!dislocations ,! which!forms!a!closed!loop.!Such!a!path!is!illustrated!in! Figure!5.4!(c).!The!vector! required!to!complete!the!circuit!is!called!the!Burgers!vector !(marked!as!b)in!Figure! 5.4).! There! are! three! types! of! dislocations! depending! on!the! angle! between! dislocation!line!and!Burgers!vector:!an!edge!dislocation!is!normal!to!the!dislocation! line;!a!screw!dislocation!is!parallel!to!the!dislocation!line;!and!a!mixed!dislocation! gives!an!angle!in!between!0˚!and!90˚ !(Hull!and!Bacon!2001).!The!dislocation!line!and! Burgers!vector!of!the!three!typ es!of!dislocations!are!shown!in !Figure!5.4!(d),!Figure! 5.4(e),!and!Figure!5.4!(f),!respectively.!! ! There!are!two!basic!types!of!dislocation!movement :!(i)!glide!or!conservative! motion,!in!which!the!dislocation!moves!in!the!surface! that!contains!both!its!line!and! Burgers!vector;!and!(ii)!climb!or!nonYconservative!motion,!in!which!the!dislocation! moves! out! of!the! glide! surface! normal! to! the ! Burgers! vector.! Glide! of! many! dislocations! results! in! slip,! which! is! the! most! common! manifestation! of! plastic! deformation!in!crystalline!solids.!Dislocation!lines!move!on!a!slip!plane,!which!is! uniquely!defined!as!the!plane!that!contains!both!line!and!the!Burgers!vector!of!the! dislocation.!The!slip!plane!is!marked!as!A!in!two!dimensional!graph!or!color! coded! as!gray!in!threeYdimensional!representation!in!Figure!5.4.!! ! 113! ! Figure!5.4!(a)!A!perfect!simpleYcubic!crystal!(reference!system);!(b)!displacement!of! halfYcrystals!along!cut!plane!A!by!lattice!vector! b!results!in!two!surface!steps!but! does!not!alter!the!atomic!structure !inside!the!crystal;!(c)!t he!same!procedure!limited! to!a!part!of!cut!plane!A!introduces!as!an!edge!dislocation;!(d)!an!edge!dislocation! created!by!inserting!a!halfYplane!of!atoms;!(e)!a!s crew!dislocation!in!which!the!slip! vector!is!parallel!to!the!dislocation!line;!the!slipped!area!of!the!cut!plane!is!shown!in! dark!gray!and!the!un Yslipped!area!is!shown!in!light!gray;!the!dislocation!line!is! marked!by!the!solid!line;!(f)!a!curved!dislocatio n!line!with!an!edge!orientation!at!one! end!(on!the!left)!and!a!screw!orientation!at!the!other!end!(on!the!right) !(Bulatov!and! Cai!2006).!! ! 114! ! As!explained!in!(Section!2.2.4),!slip!vector!analysis!can!be!used!to!analyze! structural!change!during!deformation!and!failure!of!NWs.! The!slip!vector!of!atom! € α! is!defined!as!(Same!as!Eq.!2.62)! ! s α =− 1 n s ( r αβ − R αβ ) β(≠α) n ∑ ,! ! ! ! ! ! !!!!!!! !!(Eq.!5.1)! where!n!is!the!number!of!nearest Yneighbor!atoms!β!to!atom! € α,!ns!is!the!number!of! slipped!neighbors,!and! ! r αβ = r α − r β ,!! ! ! ! !!!!!!!!!!!!!!!!! (same!as!Eq.!2.63)(Eq.5.2) ! R αβ = r α0 − r β0 ,! ! ! ! ! !!!(same!as!Eq.2.64)(Eq.!5.3)!! with! r α !and! r α0 !being! the! current! and! reference! positions! of! the! atom!α,! respectively!(Zimmerman,!Kelchner!et!al.!2001 ).!Here,!α!and!β!are!indices!of!a!pair! of! atoms! that! form! a! bond! in! the! reference! configuration.!The! reference! configuration!is!the!original!system!without!defect!or!atomic!displacement.!In!the! slip!vector!field!calculation,!we!compute! s α ! vectors!for!all!atoms!pairs!that!are! nearest!neighbors.!The!result!can!be!generalized!into!3 !general!situations:!A!slip! vector!that!is!perpendicular!to!slip!plane;!a!vector!that!is!parallel!to!the!plane;!or!a! vector!sits!in!between.!The!first!two!cases!are!illustrated!in!Figure!5.5.!If!a!slip!plane! cuts!through!the!bond!between!atoms! α!and!β,!then! s α !vector!corresponds!to!the! Burgers!vector,!which!lies!on!the!slip!plane !(Figure!5.5!(a)).!The!only!difference! between! slip! vector! and! Burgers! vector! is! that! slip! vecto r! accumulates! the! displacement!of!the!atoms!while!Burgers!vector!shows!periodicity.! Therefore,!if!a! ! 115! slip!vector!lies!on!the!slip!plane!and!shows!a!larger!magnitude!than!Burgers!vector,! it!indicates!a!slip!process.!The!slip!vectors!are!parallel!to!Burgers!vector.! When!the! slip!vectors!are!smaller!than!Burgers!vector,!their!directions!will!the!local!atomic! symmetry!change!and!the!magnitude!is!proportional!to!the!local!elastic!energy ! (Kang!and!Cai!2010).!If!a!crack!cuts!through!the!bond!between!atoms !α!and!β,!then! s α !vector!corresponds!to!the!crack!opening!displacement,!which!is!perpendicular!to! the!crack!plane!for!a!Mode!I!crack !(Figure!5.5!(b)).!! ! Figure! 5.5! A! bond! that! connects! a! pair! of! atoms!α !and!β !in! the! reference! configuration!(drawn!in!a!dashed!line).! The!displacement! s α !can!indicate!(a)!slip!if! the!slip!plane!cuts!the!bond!between!atoms! α!and!β,!or!(b)!bond!breaking!by!crack.! ! 5.2 Surface)relaxation)) To!capture!the!nanostructure!of!ZnO!NW,!w e!first!use!the!MD!method!to!study! structural! relaxation! of! the! nonpolar!(0110) !surface! of! WZ! crystal,! which! constitutes!all!six!sidewalls!of!the![0001] Yoriented!ZnO!NWs!studied!in!this! Chapter.! ! 116! A!slab!consisting!of!40!atomic!layers!was!prepared!and!relaxed!using!the!steepest Y decent!method.! ! The! calculated! surface! relaxation! involves! inward! movement! of! surface! cation!(Zn),!with!anion!(O)!staying!on!top!of!the!surface,!resulting!in!a!tilt! of!the! surface!ZnYO!dimers;!see!the!inset!in!Figure!5.6!(a).!In!general,!the!magnitude!of! such! relaxation! is! determined! by! a! competition! between! rehybridization! and! chargeYtransfer! effects!(Meyer! and! Marx! 2003).!On! one! hand,! reduction! of! the! coordination! number! of! the! surface! atoms! from! four! to! three! leads! to! rehybridization!of!covalent!bonds!from!sp 3 !to!sp 2 !character,!resulting!in!a!tilt!of!the! surface!anionYcation!dimer!with!only!small!change!of!the!bond!length .!On!the!other! hand,!for!ionic!bonding,!electrostatics!causes!inward!movement!of!both!anion!and! cation!to!gain!better!screening,!resulting!in!small!tilt!of!the!dimer!but!with!reduced! bond!length.!Therefore,!relaxation!of!the!surface!dimers!reflects!the!cov alency!or! ionicity! of! the! chemical! bonding.! The! ionicity! of! ZnO! resides! at! the! borderline! between! covalent! and! ionic! semiconductors,! for! which! surface! relaxation! is! nontrivial.(Ozgur,!Alivov!et!al.!2005 )!To!characterize!surface!relaxation,!we!thus! calculate!the!tilt!angle!and!bond!length!of!Zn YO!dimers!and!validate!them!against! quantumYmechanical!(QM)!calculations.! ! Figure!5.6!(a)!shows!the!calculated!tilt!angle!as!a!function!of!surface!depth.! The!firstYlayer!tilt!angle!is!about!5º,!and!the!second!layer!tilt!angle!is! −4º.!While!the! 3 rd Y!and!4 th Ylayer!tilt!angles!are!significantly!reduced!to!less!than!1º,!the!deviation! from!the!bulk!structure!can!be!seen!as!deep!as!five!to!six!layers!below!the!surface.! ! 117! This!is!in!accord!with!QM!calculation (Meyer!and!Marx!2003)!based!on!density! functional!theory!using!the!generalized!gradient!approximation!by!Perdue,!Burke,! and!Ernzerhof!(PBE),!also!shown!in! Figure!5.6!(a)!(Perdew,!Burke!et!al.!1996 ).! ))))) ))))))) ) Figure!5.6!Structural!relaxation!at!the!nonpolar!(1010)!surface!as!a!function!of!the! distance!(in!atomic!layers)!from!the!surface.!(a)!Tilt!angle!(in!degrees)!as!defined!in! the!inset.!(b)!The!change!in!Zn YO!bond!length!(in!%)!from!the!bulk!value.!MD!results! (blue!lines)!are!compared!with!QM!results (Meyer!and!Marx!2003)!based!on!the!PBE! approximation!(black!dot!lines).! ! 118! ! The!calculated!change!in!Zn YO!bond!length!from!the!bulk!value!is!shown!in! Figure!5.6!(b).!While!bonds!at!the!topmost!layer!contract,!bonds!at!the!second!layer! are!elongated!at!a!reduced!magnitude.!We!can!see!that!the!bond!lengths!converge!to! the!bulk!value!at!the!6 th !layer,!which!is!in!agreement!with!the!PBE!result. (Perdew,! Burke!et!al.!1996)!It!should!be!noted!that!QM!results!on!the!surface Ybond!tilt!and! length!vary!in!magnitude!depending!on!the!approximations!used.!For!example,!QM! calculation!based!on!the!Hartree YFock!approximation!shows!even!closer!agreement! with!our!MD!results!(Jaffe,!Harrison!et!al.!1994 ).! 5.3 Nanowire)core/shell)structure)and)system)preparation) ) In!order!to!study!the!property!of!ZnO! NW!structure,!we!perform!MD!simulation!of! an![0001]Yoriented!ZnO!NW!with!six !{0110}!sidewalls!(Figure!5.7!(a)).!The!NW! diameter!is!D!=!2.5,!5,!and!10!nm,!with!an!aspect!ratio! L/D!=!8!(L!is!the!NW!length).! A!smaller!aspect!ratio!of!L/D!=!4!did!not!provide!converged!thermo Ymechanical! properties,!which!may!be!attribut ed!to!the!absence!of!long Yrange!phonons!in!the! axial!direction.!The!atomic!positions!are!first!relaxed!by!the!steepest Ydecent!method! to!obtain!a!local!minimum Yenergy!configuration,!where!period!boundary!condition! is!applied!along!the!NW!axis.!Subsequent ly,!NWs!are!slowly!heated!up!to!a!target! temperature!and!thermalized!for!10!ps.!In!total,!11!samples!are!prepared!for!each! diameter!at!temperatures!ranging!from!100!to!1400!K.!Detailed!information!on!the! MD!simulation!is!provided!in! Section!3.2.2.!! ! 119! ! The!surface!relaxation!of!the!{0110}!sidewalls!mentioned!above!leads!to! unique!structural!features!of!the!NWs.!To!show!these!features,! Figure!5.7!(b)!and!(c)! visualize!the!change!of!surface!bonds.! Figure!5.7!(b)!is!the!top!view!of!slip Yvector! field!of!the!smallest!NW!(2.5!nm!in!diameter).!As!described!in !section!5.1.2,!the!slip! vector! quantifies! the! displacement! of! atoms! relative! to! their! neighbors! in! the! reference!system!(i.e.!a!system!without!defect),!and!contains!information!about!the! slip!plane!and!Burgers!vector !(Zimmerman,!Kelchner!et!al.!2001 ).!Figure!5.7!(c)! shows!the!bondYnetwork!structure!of!the!same!NW,!where!different!levels!of!bond! contraction!are!colorYcoded.!The!outermost!layer!of!the!NW!has!the!shortest!bond! length,!which!is!colored!black.!The!third Ylayer!atoms!have!the!2 nd !shortest!bond! length!(colored!blue).!The!3 rd !shortest!bond!length!is!colored!green,!and!normal Y length!or!elongated!bonds!are!in!red.!From!the!slipYvector!field!and!the!bond Y network!structure,!we!can!see!that!the!first!layer!of!atoms!moves!towards!the!inne r! core!of!the!NW!as!a!result!of!the!bond!contraction.!At!the!same!time,!the!atomic! bonds!in!the!next!layer,!which!moves!outward!with!less!magnitude,!are!stretched.! Similar!behavior!is!exhibited!for!the!subsequent!atomic!bilayers,! i.e.,!the!3 rd !and!4 th ! layers,!and!the!5 th !to!6 th !layers,!with!less!magnitude.!This!‘incipient’!core/shell! structure,!with!a!sixYlayerYthick!shell!of!different!symmetry!from!the!core,!will!play!a! major!role!in!the!mechanical!response!of!the!NWs!as!shown!below. ! ! 120! ! Figure!5.7!Schematic!of!simulated!ZnO!nanowire!with!diameter!D.!(b)!and!(c)!show! the!incipient!coreYshell!structure!of!ZnO!nanowire!with!D!=!2.5!nm!at!temperature!5! K.!(b)!Top!view!of!color Ycoded!slipYvector!field.!(c)!Top!view!of!bond!network,! where!the!bonds!are!colored!in!black!(shortest!bond),!blue!(2 nd !shortest),!and!green! (3 rd !shortest).!Normal!or!elongated!bonds!are!colored!in!red. ! ! 121! 5.4 Tensile)test) A!series! of!tensile! tests! are! performed! after! thermalization!of! the! nanowires.! Uniaxial!tensile!tests!are!applied!along!the![0001]!direction,!in!which!the!NWs!are! elongated!by!0.1!%!at!every!15!ps!until!they!break. )(As!shown!in!Ref.!(Kang!and!Cai! 2010),!Young’s!modulus!and!other!mechanical!properties!are!rather!insensitive!to! the!strain!rate,!except!for!the!fracture!strength. ))StressYstrain!curves!of!ZnO!NWs!of! different!diameters!at!temperature!100!K!in! Figure!5.8!exhibit!the!size!effect!on! fracture!strength!(i.e.!the!stress!at!which!the!NW!fractures) !(Hoffmann,!Ostlund!et!al.! 2007).!The!facture!strength!increases!from!8.5!to!15!GPa!as!the!diameter!of!the!NW! decreases!from!10!to!2.5!nm.!The!strength!of!the!N W!thus!decreases!by!43!%!as!the! diameter!increases!by!fourYfold.!The!slope!of!the!stress Ystrain!curve!also!changes! systematically!with!the!diameter.!Young’s!modulus!E〈0001〉!is!obtained!by!linear!fitting! of!the!stressYstrain!curve!in!the!strain!range!of! 0!~!1!%.Stress!is!calculated!using!the! Virial!formula![see!e.g.!M.!Parrinello!and!A.!Rahman,!J.!Appl.!Phys.!52!(12),!7182 Y 7190!(1981)],!reflecting!the!volume!change!due!t o!elongation!and!Poisson!effect.! Accordingly,! the! calculated! stress! is! the! true! value. ! The! calculated!E〈0001〉! is! a! decreasing!function!of!the!diameter:! 129,!120,!and!113!GPa! for!D!=!2.5,!5,!and!10!nm,! respectively.!The!calculated!stiffening!of!NWs!with! decreasing!diameter!is!in!accord! with!previous!theoretical!and!experimental!observations,!and!can!be!explained!by! the!larger!ratio!of!the!volume!of!the!bond Ycontracted!(and!thus!stiffer)!outer!shell!to! that!of!the!normally!bonded!inner!core!(see! Figure!5.7!(c))(Chen,!Shi!et!al.!2006 ;! Agrawal,!Peng!et!al.!2008 ;!Liu,!Li!et!al.!2009 ).! ! 122! ! Figure!5.8!Tensile!tressYstrain!curves!of!ZnO!NW!of!diameter!D!=!2.5,!5,!and!10!nm,! at!temperature!100!K.! ! We!next!study!the!effect!of!temperature!on!the!stress Ystrain!curve.! The! stressYstrain! curves! of! the! smallest! NW! of! diameter! 2.5! nm! at! different! temperatures!between!100!and!1400!K!are!shown!in! ! Figure!5.9!(a).!Fracture!strength!decreases!as!increased!temperature.!Also,! Young’s!modulus!decreases!by!12!%!(from!129!to!114!GPa) !from!temperature!100! to!500!K.!The!NW!fails!by!cracking!at!strains!of!~!9!and!7!%!at!100!and!500 !K,! respectively.!At!600!K,!the!stress Ystrain!curve!exhibits!a!distinct!feature,!i.e.,!it!is! divided!into!2!stages.!The!first!stage!is!below!strain!~!9%,!and!the!second!stage!is! between!strain!9!and!12%.!Similar!two Ystage!stressYstrain!curves!were!observed!in! previous!MD!simulations.(Wang,!Kulkarni!et!al.!2008 ;!Agrawal,!Peng!et!al.!2009 )!In! spite!of!this!novel!behavior,!however,!the!NW!still!fails!by!cleavage!at!600!K,!and!the! ! 123! stress!drops!to!0!sharply!at!the!failure!strain.! Above!1200!K,!on!the!other!hand,!the! stress!gradually!decreases!at!large!strains!but!does!not!become!0,!indicating !the! onset!of!ductility.!Similar!behavior!is!observed!for!NWs!of!diameter!5!and!10!nm.!As! shown!in!! Figure!5.9!(b)!and!(c),!the!onset!of!the!two Ystage!behavior!is!1000!and!1200!K!for!D! =!5!and!10!nm,!respectively.!The!onset!of!ductility!for! D!=!5!nm!is!1400!K,!whereas! the! 10! nm! NW! still! fails! in! a! brittle! manner! even! at! the! highest! simulated! temperature!of!1400!K.!It!should!be!noted!that!at!room!temperature!no!simulated! NW!exhibits!the!twoYstage!stressYstrain!curve,!in!accord!with!existing!exp erimental! studies,!thus!resolving!the!previous!controversies!between!experiments!and!theory! mentioned!before(Agrawal,!Peng!et!al.!2009 ;!Dai,!Cheong!et!al.!2010 ).!Especially,! ZnO!NW!exhibits!both!brittle!and!ductile!beha viors,!as!well!as!the!two Ystage!stressY strain!relation,!depending!on!the!size!and!temperature.!The!mechanical!response!of! ZnO! NW! is! a! sensitive! function! of! the! size! and! temperature,! and! comparison! between!experiment!and!theory!must!be!made!carefully!at!equ ivalent!conditions.!! ! ! 124! !!!!!!!! ! Figure!5.9!Temperature!dependence!of!the!tensile!stress Ystrain!curve!of!ZnO!NW! with!diameter!D!=!2.5!nm!(a),!5!nm!(b),!and!10!nm!(c )! ! 125! 5.5 Core/shell)structure)and)n ovel)structural) transformation)) In!order!to!understand!the!atomistic!mechanisms!underlying!the! twoYstage!stressY strain!curves!in!Figure!5.9,!we!analyze!the!atomic!configurations!of!th e!NW!before! and!after!the!transition!to!the!two Ystage!stressYstrain!behavior.!Figure!5.10!shows! slipYvector!fields!and!atomic!structures!during!the!ascending!part!of!the!second! stage!in!the!stressYstrain!curve!for!diameter!5!nm!at!temperature!1000!K.! Figure! 5.10!(a)!is!the!top!view!of!the!slip Yvector!field,!and!Figure!5.10!(b)!is!the!side!view!of! a!1.5!nm!slab!marked!by!black!dashed!lines!in! Figure!5.10!(a).!Figure!5.10!(a)!shows! that!atoms!in!the!outermost!layer!(with!the!shortest!bonds)!move!radially!o n!the! (0001)!plane,!while!the!second Ylayer!atoms!move!axially!along!the![0001]!direction! with!a!larger!magnitude.!Similar!behavior!is!observed!for!the!3 rd !and!4 th !layers!as! well!as!for!the!5 th !and!6 th !layers.!These!six!surface!layers!correspond!to!those!in !the! incipient!shell!in!the!initially!relaxed!NW!( Figure!5.7!(b)!and!(c)).!In!the!inner!core!of! the! NW,! in! contrast,! the! radial! movements! of! the! odd Ynumbered! layers! are! disordered,!while!the!vertical!movements!of!the!even Ynumbered!layers!are!still! pronounced.!The!corresponding!atomic!bond!structures!are!shown!in! Figure!5.10!(c)! and!(d),!which!respectively!show!the!top!and!side!views.!In! Figure!5.10!(c),!the!shell! loses!the!hexagonal!symmetry!of!the!original!WZ!crystal.!In! Figure!5.10!(d),!the!shell! atoms!form!alternating!8Y!and!4Yatom!rings,!instead!of!the!6Yatom!rings!in!WZ,! which!is!a!consequence!of!the!alternating!layer YbyYlayer!bondYlength!contraction! and!elongation!under!tension!explained!above.!This!structural!transformation!in!the! NW!shell!under!tension!is!identical!to!the!WZ YtoYBCT!transformation!reported!in! ! 126! previous!simulations.(Wang,!Kulkarni!et!al.!2008 ;!Agrawal,!Peng!et!al.!2009 )!As!for! the!inner!core!structure,!it!still!retains!the!original!hexagonal!(HX)!symmetry!in!the! (0001)!plane!and!the![0110]!direction.!With!its!characteristic!atomic!movements! (every!even!layer!moves!axially),!the!core!forms!a!graphite Ylike!structure!reported! in!a!tensile!test!along!the![0110]!direction. (Zhang!and!Huang!2007)!Our!simulation! thus!predicts!spontaneous!formation!of!a!core/shell!structure!under!tension.!This! structural!transformation!is!accompanied!by!a!large!stress!relaxation,!resulting!in! the!twoYstage!stressYstrain!response.!Similar!NW!failure!mediated!by!structural! transformation! has! been! reported! for! SiSe 2(Li,! Kalia! et! al.! 1996 ).! It! should! be! pointed!out!that!the!‘intrinsic’!core/shell!structure!here!(as!opposed!to!extrinsic! core/structures!constituted!of!heterogeneous!atomic!species (Lauhon,!Gudiksen!et!al.! 2002))!originates!directly!from!the!incipient!core/shell!structure!in!the!uns tretched! NW!shown!in!Figure!5.7.! ! 127! ! Figure!5.10!Intrinsic!coreYshell!structure!of!the!NW!under!tension!for!diameter!5!nm! at!temperature!1000!K.!(a)!Top!view!of!slip Yvector!field.!(b)!Side!view!of!slip Yvector! field!of!the!slab!shown!in!dashed!black!lines!in!(a).!(c)!Atomic!structure!of!the!same! NW!as!shown!in!(a).!Magenta!dashed!line!delineates!BCT Y!and!HXYtransformed! zones.!(d)!Atomic!structure!of!the!same!NW!slab!as!in!(b).!Magenta!dashed!lines! delineate!the!BCT!and!HX!zones. ! ! ! It! is! worth! mentioning! that! the! surface Ystructural! relaxation! induced! mechanical!response!is!most!pronounced!in!thin!NWs.!For!example,!the!volume! ! 128! ratio!of!the!shell!to!the!inner!core!is!8/1!for! D!=!2.5!nm,!and!is!1/2!for! D!=!10!nm.! The!ratio!becomes!smaller!for!larger!diameters,!where!the!intrinsic!core/shell!effect! is!less!significant.!In!addition!to!the!temperature!effect!explained!above,!this!size! dependence!may!partially!explain!existing!controversies!between!experiment!and! theory.(Agrawal,! Peng! et! al.! 2009 )! Furthermore,! the! origin! of! the! core/shell! mechanism!suggests!that!it!should!be!sensitive!to!surface!functionalization. ! 5.6 Size)and)temperature)dependent)brittle PtoPductile)transition)) In!order!to!understand!the!second!transition!at!a!higher!temperature,! we!again!use! the!slipYvector!analysis,!which!can!distinguish!whether!the!NW!fails!by!cleavage! fracture!or!ductile!failure!( i.e.,!whether!the!initial!event!responsible!for!the!failure!is! the! nucleation! of! a! crack! or! a! dislocation) (Kang! and! Cai! 2010).!If! the! vector! displacement!relative!to!the!reference!system!w ith!no!defect!(same!direction!as! Burgers’!vector)!is!parallel!to!the!slip!plane,!it!corresponds!to!a!dislocation.!On!the! other!hand,!vector!displacement!predominantly!perpendicular!to!a!plane!signifies! modeYI!crack!opening.) ! Figure!5.11!(a)!shows!a!side!view!of!slip Yvector!field!in!the!5!nm!NW!at! temperature!100!K,!for!which!the!stress Ystrain!curve!exhibits!brittle!behavior.!The! vectors!in!the!central!core!are!mostly!perpendicular!to!the!(0001)!crack!plane,! indicating! modeYI! fracture.! The!snapshots! of! atomic! positions! before! and! after! failure!in!Figure!5.11!(c)!and!(d)!indeed!confirm!the!brittle!cleavage.!It!should!be! noted!that!the!crack!initiates!in!the!inner!core!of!the!NW!and!then!propagates!to!the! ! 129! stiffer!bondYcontracted!shell.!A!similar!core/shell!model!has!been!proposed!for! explaining!the!sizeYdependent!Young’s!modulus!(Chen,!Shi!et!al.!2006 ;!Agrawal,! Peng!et!al.!2008;!Liu,!Li!et!al.!2009 ),!and!our!simulation!results!for!the!first!time! show!that!the!core/shell!structure!essentially!dictates!the!failure!mode!of!the!NW.! The!slipYvector!field!in!Figure!5.11!(b)!shows!that!the!NW!fails!in!a!ductile!manner! at!1000!K,!where!the!vectors!deviating!from!the![0001]!direction!indicate!mixed Y mode!fracture.!The!corresponding!snapshots!of!atomic!positions!in! Figure!5.11!(e)! and!(f)!show!necking,!which!is!a!signature!of!ductile!failure. ! ! ! ! ! ! 130! ! Figure!5.11!Snapshot!of!slipYvector!field!of!the!5!nm!NW!right!before!failure!at! temperature!100!K!(a)!and!1000!K!(b).!Corresponding!atomic!positions!before!(c)! and!after!(d)!failure!at!100K!(d),!and!those!before!(e)!and!after!(f)!failure!at!1000!K.! SlipYvector!field!in!a!15!nm Ythick!slab!at!the!center!of!the!NW!is!shown. ! 5.7 ThermoPmechanical)phase)diagram ) The! above! analysis! confirms! the! existence! of! brittle YtoYductile! transition! (BDT),(Östlund,!RzepiejewskaYMalyska!et!al.!2009;!Kang!and!Cai!2010)!in!addition! to! the! surfaceYmediated! structural! transformation! under! tension! as! described! before.!For!example,!the!structural!transformation!occurs!at!600!K,!followed!by!BDT! at!1000!K,!in!the!smallest!NW!( D!=!2.5!nm).!Figure!5.12!summarizes!the!sizeY ! 131! dependence!of!the!two!transformation!temperatures.!This!size Ytemperature!phase! diagram!can!be!used!to!understand!nanomechanical!experiments!on!ZnO!NWs. ! ) Figure!5.12!SizeYtemperature!phase!diagram!for!the!mechanical!response!of![0001] Y oriented!ZnO!NW,!where!the!temperature!is!scaled!by!the!melting!temperature. !The! calculated!melting!temperture,!2050K,!agrees!well!with!the!experimental!value,! 2248K! The! circles! and! crosses! denote! response! without! and! with! structural! transformation,! respectively.! The! squares! denote! ductile! failure,! otherwise,! the! fracture!is!brittle.! ! ! 132! CHAPTER)6 ) ) ) ) ) ) ) ) ) STACKINGPFAULT) GENERATION) IN ) ZINCBLENDE) NANOWIRES) Nanowires!(NWs)!composed!of!semiconductor!such!as!gallium!arsenide!(GaAs)!have! broad!applications!including!solar!cells (Czaban,!Thompson!et!al.!2009 ).!GaAs!NWs! have!traditionally!been!grown!by!the!vapor YliquidYsolid!(VLS)!technique!(Persson,! Larsson!et!al.!2004).!Recently,!an!alternative!growth!method!called!selective!area! metal!organic!vaporYphase!epitaxy!(SAYMOPVE)!has!attracted!much!attention!due!to! its!ability!to!form!atomically!sharp!interfaces!to!fabricate!axial!hetrostructures!such! as!tandem!NW!solar!cells !(Tomioka,!Ikejiri!et!al.!2011 ).!The!catalystYfree!SAYMOPVE! method!also!avoids!metal!impurities!that!degrade!VLS Ygrown!NWs!(Tomioka,!Ikejiri! et!al.!2011).!However,!a!major!problem!of!non YVLSYgrown!NWs!is!a!large!number!of! stacking! faults! (SFs)! that! may! alter! electronic! properties! and! degrade! device! performance!(Rudolph,!Hertenberger!et!al.!2011 ;!Tomioka,!Ikejiri!et!al.!2011 ).!In!the! case!of!VLS!growth,!previous!works!have!pointed!out!the!importance!of!the!edge!of! the!NW!top!surface!in!dictating!the!grown!crystalline!phase !(Glas,!Harmand!et!al.! 2007).!Though!semiYempirical!explanations!based!on!surface!energies!have!been! attempted,! atomistic! mechanisms! underlying! the! edge Ymediated! SF! nucleation! remain!elusive,!especially!for!SA YMOPVE!(Glas,!Harmand!et!al.!2007 ).! ! 133! ! In! this!chapter,! molecularYdynamics!(MD)!simulation! of! a! [111]Yoriented! gallium!arsenide!NW!reveals!an!atomistic!mechanism!of!SF!generation.! In!section! 6.2!and!6.3,!setup!and!characterization!of!the! MD!simulation!system!are!discussed.!A! novel!core/shell!structure!is!revealed,!which!is!crucial!for!the!understating!of! SF.!In! section!6.4,!adatom!energetics!on!the!(111)B!top!surface!of!a![111] Yoriented!GaAs! NW!associated!with!the!core/shell!structure!is!calculated,!indicating!that!SFs!are! preferentially!nucleated!in!the!shell.!A!nucleation!growth!model! is!proposed!in! section!6.5!and!6.6,!which!suggests!optimal!growth!temperature!and!pressures!for! minimizing!SFs!as!well!as!a!novel!size Ycontrol!growth!method!for!eliminating!them. ! ! ) ! 134! 6.1 Background)and)motivation) ) 6.1.1 Crystal)structure)of) GaAs)) GaAs!is!a!IIIYV!semiconductor,!which!has!been!widely!used!in!the!manufacturing!of! devices.!This!material!has!three!advantages:!(1)!a!wide!direct!bandgap!(comparing! to! Si)!of! 1.42! eV! at! room! temperature ,! which!makes! it! an!ideal!material!for! photovoltaic!device;!(2)!less!sensitive!to!heat ,!which!allows!it!to!operate!at!higher! temperatures;!(3)!resistant!to!radiation!damage !(Anspaugh!1996).!Due!to!these! features,!GaAs!is!a! very!good!photovoltaic!material.!! ! The!natural!crystal!arrangement!for!GaAs,!like!most!IIIYV!semiconductor!bulk! materials,!is!usually!zincblende!(ZB)!crystal!structure.!ZB!structure!is!in!the!cubic! crystal!system.!The!structure!could!be!visua lized!as!two!faceYcenteredYcubic!(FCC)! lattice!with!a!displacement!of!¼!lattice!constant!(LC) !and!two!different!base!atoms ,! as!shown!in!Figure!6.1.!In!some!cases,!however,!the!WZ!structure!has!been !reported! (Yeh,!Lu!et!al.!1992;!Wang!and!Ye!2002;!McMahon!and!Nelmes!2005).!At!room! temperature,!GaAs!lattice!constant!a)is!5.65325!Å!(Blakemore!1982).!The!closely! related!hexagonal!counterpart!has!the!parameter!set:!a)=!3.912!Å,!c!=!6.441!with!a! c/a! ratio! 1.647! (Thim! 1980).! The! experimentally! measured!energy! difference! between!the!two!phases!is:!ΔE−ex wz-zB =12.02(meV/atom)(Thim!1980).!This!energy! difference!is!somewhat!larger!than!those!in!other!IIIYV!systems,!which!makes!it! difficult!to!see!WZ!phase!GaAs!bulk!structure !(Yeh,!Lu!et!al.!1992).!The!energy! difference!calculated!by!out!potential!is! about!10!meV/atom,!as!shown!in!Figure!3.5.! ! 135! ! Figure!6.1!Crystal!structure!of!zincblende!lattice!at!(100)!direction! ! ! The!atomic!arrangements!of!ZB!and!WZ!phase s!are!illustrated!in!Figure!6.2! (McMahon!and!Nelmes!2005).!Both!phases!are!presented!in!similar!directions.! ZB! phase!is!put!in!<111>!direction!and!WZ!phase!is!put!in!<0001>!direction.! The!ZB! stacking!could!be!described!as:!…AaBbCcAaBbC c…!or!…ABCABC….!Here,!each!pair!of! capital!and!lowerYcase!letters!represents!a! stacking! of!two! lattice!layers!(each! containing!group!III!or!V!atoms)!in!the!lattice.!Alternatively,!each!pair!can!also!be! replaced!by!a!single!capital!letter.!Each!stacking!layer!is!displaced!by!1/3!of!the! lattice!constant!comparing!to!the!previous!one.!In!WZ!structure,!there!are!only !two! stacking! types:! …AaBbAaBb…! or! …ABAB…,.! In! Figure! 6.2,! the! two! different! structures!are!also!shown!in! hexagonal!settings!for!comparison.! ! ! !! ! 136! ! Figure!6.2!Crystal!structure!of!zincblende!(left)!and!wurtzite !(right)!respectively! (McMahon!and!Nelmes!2005;!Dick,!Caroff!et!al.!2010 ).!! ! 137! 6.1.2 Stacking)fault) in)GaAs)NWs ) Although!WZ!phases!rarely!appear!in! GaAs!bulk!material,!they!often!appear!in!GaAs! NWs,! along! with! various! intermediate! phases! compositions! such! as! faulted/incomplete! WZ! stacking,! mixed! WZ/ZB! stacking,! randomly! distributed! twins,!etc.!GaAs!NWs!are!usually!g rown!in!a!<111>!direction,!resulting!in!arsenic!(As)! atom!terminated!(111)!plane!to!be!the!growth!plane.!This!is!shown!to!be!the!most! favored!growth!direction!for!both!VLS!and!selective Yarea!growth!methods.!The! associated!As!terminated!plane!is!by!conventi on!noted!as!(111)B!plane!and!the! growth!orientation!to!be!<111>B!direction.!If!each!capital/small!letter!pair!(IIIYV! elements)!in!the!AaBbCc!sequence!continue!to! repeat,!the!NW!will!form!perfect!ZB! crystal!in!the!absence!of!SF,!as!shown!in!Figure!6.3!(a).!On!top!of!a!(111)B!plane,!the! next!group!III!atoms!are!always!located!directly!on!top!of!the!group!V!atoms !along! the!<111>!direction.!However,!the!nex t!group!V!layer!can!be!placed!in!either!of!2! alternate!positions!limited!by!{111}!planes.!This!gives!rise!to!a!possibility!for!fault! stacking,! which! is! marked! as! red! segments! in!Figure! 6.3! (b)! and! (c)! and!is! underlined!by!the!top!blue!line.! This!could!be!the!origin!of! either!WZ!phase!or!a!twin,! depending!on!where!the!next!group!V!(small!letter)!atom s!are!placed.!If!the!growth! sequence!goes!back!to!the!perfect!ZB!stacking:!…ABCABC…,!a!twin!is!created.!The! twin!plane!is!located!right!below!the!fault!stacking!plane.! On!the!other!hand,!WZ! phase!will!appear!if!the!new!sequence!continues!to!alternate!between!‘right’!and! ‘fault’!stacking,!which!is!associated!with!…ABAB… !phase.!Intrinsic!stacking!fault!is! considered!as!an!incomplete! WZ!phase.!! ! 138! ! Figure! 6.3! 2D! projection! in! the! 110 !viewing! direction! of! possible! stacking! sequences!during!nanowire!growth.!(a)!shows!zincblende!stacking,!whereas!in!(b)! and!(c),!the!first!fault!plane!on!top!of!a!ZB!region!is!indicated!in!red.!If!the!stacking! sequence!continues!with!layer!A!and!C! as!in!(b),!a!twin!segment!is! formed!and!the! stacking!sequence!is!mirrored!around!the!C!layer!under!the!fault!plane.!On!the!other! hand,!another!fault!plane!formed!on!top!of!the!red Ymarked!B!layer!will!result!in!WZ! phase!(c)!(Johansson,!Karlsson!et!al.!2008 ).!! ! Since! GaAs! has! two! different! crystal! orientations! along! 111! direction — (111)A!and!(111)B,!two!different!types!of!twin!can!be!created.!As!indicated!in! Figure! 6.4,! forming! either! GaYGa! or! AsYAs! bonds,! which! usually! require! high! formation!energy,!or!GaYAs!bonds!at!the!twin!mirror!plane .!These!two!structures! correspond! to! para! and! ortho! twins ,! respectively.! Para! twin! will! result! in! a! mirroring!of!the!stacking!sequence,!as!well!as!the!bonds!across!the!twin!boundary.! The!formation!of!para!twin!requires!the!growth!direction!to!change!from!<111>B!to! <111>A!and!has!never!been!reported!even!in!bulk!material !(Lee,!Lee!et!al.!1990 ).! Ortho!twin!(see!Figure!6.4),!on!the!other!hand,!can!be!seen!as! 60˚!rotation!of!the!NW! ! 139! segment!around!the!twin!mirror!plane.!The!stacking!sequence!is!mirrored,!but!the! bond!orientation!stays!the!same.!Since!GaAs!has!3 Yfold!symmetry!along!the!<111>! direction,!the!symmetry!of!the!entire!segment!above!the!twin!mirror!plane!will!be! rotated,!which!may!result!in! the!change!of!strain!in!the!NW.! ! ! Figure!6.4!Two!possible!atomic!structures!of!twins!in!GaAs :!(a)!ortho!twin!projected! along![110]!direction;!and!(b)!para!twin!projected!along![110]!direction! (Lee,!Lee!et! al.!1990).!(c)!Ortho!twin!results!in!a!switch!from!ABCABC!to!a!CBACBA!stacking .!(d)! An!ortho!twin!crystallite!can!be!created!by!60˚!rotation!of! segments(Bolinsson,! Ouattara!et!al.!2009).!! ! 140! 6.2 Surface)relaxation) We!use!the!MD!method!to!study!structural!relaxation!of!the!(110)!surface!of!GaAs! ZB!crystal,!which!constitutes!all!six!sidewall!surfaces!of!the![111] Yoriented!NW! studied!in!this!paper.!To!characterize!surface!relaxation,!we!calculate!the!Ga YAs! bond!length!near!the!surface!and!validate!them!against! quantumYmechanical!(QM)! calculation!based!on!density!functional!theory!(DFT)!using!the!generalized!gradient! approximation!by!Perdue,!Burke,!and!Ernzerhof!(PBE) (Perdew,!Burke!et!al.!1996 ).! The!system!is!a!slab!consisting!of!40!atomic!monolayers!with!area!35! ×!38!Å 2 ,!which! has!free!(110)!top!and!bottom!surfaces.!Periodic!boundary!conditions!are!applied!to! the! lateral! directions,! and! we! relax! the! system! to! obtain! the! minimum Yenergy! atomic!configuration.! ! Figure!6.5!shows!the!bond!length!as!a!function!of!surface!depth!in!terms!of! the!number!of!atomic!layers.!While!the!bonds!at!the!topmost!two!layers!contrac t,! the!bonds!at!the!third!layer!are!elongated!at!a!reduced!magnitude.!The!bond!length! converges!to!the!bulk!value!at!the! 7th!layer!within!0.5!%.!The!MD!calculation!is!in! reasonable! agreement! with! the! QM! calculation! as! shown! in ! Table! 6.1,! which! compares!the!bond!lengths!in!the!top!three!atomic!layers!of!the!(110)!surface! calculated!by!the!MD!and!DFT!methods. ! ! 141! ! Figure!6.5!GaYAs!bond!length!as!a!function!of!the!distance!(in!atomic!layers)!from! the!(110)!surface!of!GaAs!ZB!crystal.! ! Table!6.1!GaYAs!bond!length!in!the!top!three!atomic!layers!of!the!(110)!surface!of! GaAs!ZB!crystal.! Atomic!layer! 1 st !layer! 2 nd !layer!! 3 rd !layer! DFT!bond!length!(Å)! 2.433! 2.424! 2.459! MD!bond!length!(Å)! 2.389! 2.397! 2.449! 6.3 Nanowire)core/shell)structure)and)system)preparation) ) We!perform!MD!simulation!of!a!GaAs!NW!with!diameter! Dw!~!30!Å.!The!NW!axis!i s! the![111]!orientation!of!(ZB)!crystal,!whereas!the!hexagonal!NW!has!six! {110} ! sidewalls;!see!Figure!6.6!(a).!The!bond!contraction!at!the! {110} !sidewalls!leads!to! a!core/shell!structure!of!the!NW.!To!show!this,! the!NW!is!relaxed!using!steepest Y descent!method!with!the!periodic!bo undary!condition!(PBC)!applied!only!to!the! ! 142! [111]!direction.!Figure!6.6!(b)!and!(c)!show!the!core/shell!structure!with!three Yfold! symmetry.!Figure!6.6!(b)!is!the!top!view!of!the!atomic!structure!of!the!NW,!where! atomic!bonds!are!colorYcoded!according!to!the!bond!length.! In!particular,!three!of! the!six!corners!have!the!shortest!bonds!(colored!in!black),!while!the!bond!lengths!at! the!other!3!corners!(green)!are!slightly!longer!but!still!shorter!than!the!bulk!value.! All!bonds!along!the!edge!(blue)!have!the!length!between!the!two!contracted!corner! bond!lengths.!In!contrast,!all!bon ds!in!the!inner!core!of!the!NW!(red)!have!the!same! length!as!the!bulk,!while!some!bonds!in!the!transient!region!between!the!shell!and! the!core!are!longer!than!the!bulk!bond. !Figure!6.6!(c)!is!the!top!view!of!slip Yvector! field!of!the!same!NW.!As!described!in! Section!2.2.4!and!Section!5.1.2,!the!slip!vector! quantifies!the!displacement!of!atoms!relative!to!their!neighbors! in!the!reference! system!(i.e.!a!system!without!defect),!and!contains!information!about!the!slip!plane! and!Burgers!vector!(Zimmerman,!Kelchner!et!al.!2001 ).!From!the!slipYvector!field! and!the!bondYnetwork!structure,!we!can!see!that!the!outermost!layer!of!atoms! moves!towards!the!inner!core!of!the!NW!as!a!result!of!the!bond!contraction.!At!the! same!time,!the!atomic!bonds!in!the!next!layer,!which!moves!outward!with!less! magnitude,!are!stretched.!Similar!to!the!two!outermost!layers,!the!third!atomic!layer! moves!towards!the!inner!core!due! to!bond!contraction,!while!the!next!layer!moves! outwards.!From!Figure!6.6!(b)!and!(c),!the!shell!depth!is!estimated!to!be!7!atomic! layers!from!the!edge.!For!the!30!Å!NW,!the!core/shell!volume!ratio!is!as!large!as!1/8. !! ! 143! ! Figure!6.6!(a)!Schematic!of!simulated!GaAs!nanowire.!(b)!Top!view!of!bond!network! in!NW,!where!the!bond!length!is!color Ycoded.!(c)!Top!view!of!color Ycoded!slipYvector! field.! ! ! 144! ! The![111]Yoriented!ZB!nanowire!is!a!stack!of!three!types!of!GaAs!bilayers! denoted!by!ABCABC,!and!each!bilayer!exhibits!a!distinct!three Yfold!symmetry.! This!is!in!contrast!to!a!similar!core/shell!structure!found!in!a![0001] Yoriented! wurtzite!NW,!where!all!layers!have!the!identical!six Yfold!symmetry.!Figure!6.7!(a)! and!(b)!are!the!top!view!of!the!atomic!bond!structure!(where!the!bond!length!is! colorYcoded)!and!the!slip!vector!field,!respectively,!in!cA,!aB,!and!bC !bilayers!in!the! [111]Yoriented!GaAs!NW.!All!three!stacking!layers!show!the!core/shell!structure! described! in! the! paper,! only! with! small! differences! in! the! magnitude! of! bond! structure!change.!In!addition,!all!three!bilayers!have!three Yfold!mirror!symmetry.! However,!across!the!bilayers,!each!corner!exhibits!alternating!bond!contraction!and! elongation.!For!example,!bond!contraction!at!one!corner!in!aB!layer!is!followed!by! bond!stretching!at!the!same!corner!in!the!next!bC!bilayer. ! ! ! 145! ! Figure!6.7!Top!view!of!the!slip!vector!field!(a )!and!colorYcoded!atomic!bonds!(b)!in! the!Aa,!Bb,!and!Cc!bilayers!of!a![111] Yoriented!GaAs!NW.!! 6.4 Adatom)energy)map ) The!core/shell!structure!in!Figure!6.7!has!a!significant!effect!on!the!adsorption! energy!of!Ga!and!As!atoms!on!the!top!surface!of!the!NW.!To!quantify!this!effect,!we! remove!the!axial!PBC!to!relax!the!top!(111)B!surface!of!the![111] Yoriented!GaAs!NW.! We!then!calculate!the!energy!of!a!Ga!adatom!at!all!possible!adsorptions!sites!on!the! top!surface!(Figure!6.8!(a)).!Note!that!the!GaAs!(111)B!surface!is!As!terminated,!and! Ga!atoms!need!to!be!deposited!first!on!top!of!the!topmost!As!atoms!for!subsequent! layers!to!grow.!Figure!6.8!(b)!and!(c)!show!the!Ga!adatom!energy!along!the!blue!and! red!lines!marked!in!Figure!6.8!(a),!which!respectively!include!the!corner!and!edge! ! 146! sites!at!the!ends.!Figure!6.8!(d)!shows!the!spatial!distribution!of!the!adatom!energy! on!the!entire!NW!top.!The!adatom!ener gy!map!reflects!the!core/shell!structure!in! Figure!6.7.!Namely,!the!adatom!energies!are!lower!in!the!shell!compared!with!those! in!the!core.!In!particular,!three!corne rs!of!the!top!surface!have!the!largest!low! adsorption! energy! regions.! This! result! suggests! that! adatoms! are! adsorbed! preferentially!at!the!corners!and!edges!of!the!NW!top!surface. ! ! 147! ! Figure!6.8!(a)!Schematic!of!the!calculation!of!ad Yatom!energy!map.!(b)!and!(c)!Ga! adatom!energy!along!the!blue!and!red!lines!marked!in!(a),!respectively.!(d)!The! adatom!energy!map!of!the!entire!NW!top!surface. )) ! 148! 6.5 Adlayer)energy) ) Based!on!the!above!observation,!we!cons truct!a!simple!nucleation!growth!model ! (Glas,!Harmand!et!al.!2007 ),!in!which!an!island!of!a!GaAs!bilayer!is!nucleated!at!a! corner!of!the!NW!top!surface!( Figure!6.9!(a)).!For!simplicity,!we!choose!a!hexagonal! island!consisting!of!an!e qual!number!of!Ga!and!As!atoms:! NGa!=!NAs!=!N/2,!where!N!is! the!total!number!of!adatoms!in!the!island.!Here,!the!Ga!atoms!are!placed!on!top!of!As! atoms!on!the!NW!top!surface,!whereas!the!placement!of!the!As!atoms!are!at!either! ZB!or!fault!sites!( Figure!6.7).!Let!the!stacking!of!the!top!three!As!monolayers!(MLs)! be!ABC,!where!C!is!the!topmost!ML.!Then,!As!adatoms!in!a!ZB!island!are!placed!on! top!of!As!atoms!in!the!A YML,!i.e.,!(ABC)A,!whereas!As!atoms!in!a!fault!island!are!at! BYML!positions,!(ABC)B.!The!fault!stacking!in!the!latter!is!common!to!all!stacking! defects,!including!twins!as!well!as!intrinsic!and!extrinsic!SFs !(Glas!2008).!Figure!6.9! (b)!shows!the!energy!per!ad atom,!E(N)/N,!for!ZB!and!fault!islands!as!a!function!of! N.! The!core/shell!structure!shown!in!Figs.!1!and!2!is!reflected!here!in!the!lower!fault! island!energy!than!the!ZB!island!energy!for!small! N.!Figure!6.9!(b)!also!shows!the! asymptotic!energies!for!N!→!∞!(i.e.!infinite!adYlayer)!by!arrows,!where!the!fault! energy!is!higher!than!the!ZB!energy!by!7.8!meV!per!atom.!These!asymptotic!energies! have!been!computed!for!a!slab!with!(111)!free!surfaces!with!PBCs!in!the!lateral! directions,!where!either!a!ZB!( Figure!6.7!(a))!or!fault!(Figure!6.7!(b))!bilayer!is! deposited!on!top!of!the!(111)B!top!surface.!The!most!important!consequence!of!the! adatom!core/shell!structure!here!is!a!crossover!of!the!island!size!( N!~!70),!above! which!ZB!islands!become!energetically!more!stable!than!fault!islands.!Namely,!small! ! 149! adatom!islands!nucleated!at!NW!corners!are!likely!to!occupy!fault Ystacking!sites,! whereas!larger!islands!energetically!favor!regular!ZB!stacking. ! ! Figure!6.9!(a)!Schematic!of!a!GaAs!bilayer!island!(magenta)!nucleated!at!a!corner!of! the!top!surface!of!a!NW!(cyan).!(b)!ZB!(blue)!and!fault!(red)!inland!energies!per! atom!as!a!function!of!the!total!number!of!adatoms.!The!asymptotic!limi t!of!the!ZB! and!fault!energies!are!indicated!by!blue!and!red!arrows,!respectively .! 6.6 Nucleation)growth)model) ) We!consider!a!bilayer!island!on!the!(111)B!top!surface!of!the!GaAs!NW.!The!island!is! prepared!by!first!placing!a!monolayer!(ML)!of! NGa!Ga!atoms!on!top!of!the!topmost!As! atom!positions!on!the!NW.!We!denote!their!layer!to!be!C!within! ABCABC!stacking.! Then,!another!ML!consisting!of!Nas!As!atoms!is!added!either!at!A!or!B!positions,! respectively,!for!ZB!or!fault!islands!(see! Figure!6.7!(a)).!The!NW!+!island!system!is! relaxed!using!the!steepestYdescent!method!with!the!MD!interatomic!potential,!and! the!formation!energy!E(NGa,!NAs)!of!the!island!is! calculated!as!a!difference!between! ! 150! the!total!energy!of!the!relaxed!NW!+!island!system!and!that!of!the!NW!+!isolated!Ga! and!As!atoms.!The!change!of!Gibbs!free!energy!due!to!the!island!formation!is!given! by(Tatematsu,!Sano!et!al.!2008 ).! ! G=E(N Ga ,N As )−N Ga µ Ga gas −N As µ As gas .! ! ! ! !!!!!!!! (Eq.!6.1)! Here,!the!chemical!potential!of!the!i deal!gas!of!the!αYth!species!(α!=!Ga!or!As)!is ! ! µ α gas =−k B Tln g α k B T p α 2πm α k B T h 2 3/2 ,! ! ! ! !!!!!!!! (Eq.!6.2)! where!kB!is!the!Boltzmann!constant,!T!is!the!temperature,!and!h!is!the!Planck! constant.! In! Eq.! 6.1,!pα,!gα!and)mα!are! the! partial! pressure,! degeneracy! of! the! electronic!ground!state,!and!mass!for!the! αYth!species,!respectively.! ! The!crossover!of!the!island!energetics!in! Figure!6.9!has!major!consequences! on!the!NW!growth.!To!show!this,!we!calculate!the!change!of!Gibbs!free!energy!due!to! the!nucleation!of!an!island (Johansson,!Karlsson!et!al.!2006 ;!Ikejiri,!Sato!et!al.!2008 )! in!the!presence!of!vapor!of!reactants!by!subtracting!the!free!energy!of!th e!gas!from! the! island! energy! as! Eq.! 6.2.! Here,!µ Ga gas !and!µ As gas !are! the! gasYphase! chemical! potentials!of!Ga!and!As,!which!depend!on!the!growth!temperature!and!the!partial! vapor!pressures,!pGa!and!pAs,!of!Ga!and!As.!Figure!6.10!(a)!shows!G!as!a!function!of! the!island!diameter!D!at!two!temperatures,!T!=!1030!and!1050!K,!where!PGa!=! 2.7×10 −6 !bar!and!PAs!=!5.0×10 −4 !bar!(Ikejiri,!Sato!et!al.!2008 ).!These!temperatures!are! within! the! optimal! growth! temperature! window! that! is! sufficiently! high! for! detrimental!surface!AsYtrimers!to!be!desorbed,!but!is!low!enough!not!to!cause!the! ! 151! desorption!of!adatoms!(Tomioka,!Ikejiri!et!al.!2011 ).!The!critical!nucleus!size!D * !and! the!activation!barrier!G * !=!G(D * )!for!growth!are!obtained!by!the!peak!position!and! height!of!each!curve,!respectively!(indicated!by!the!arrows!in! Figure!6.10!(a)).!At!the! lower!temperature,!the!fault!island!has!the!lower! G * !(see!the!red!solid!line)!than!the! ZB!island!(see!the!blue!solid!line)!and!is!likely!to!grow.!At!the!higher!temperature,!in! contrast,!the!ZB!island!(see!the!blue!dashed!line)!is!likely!to!grow. ! ! We!calculate!the!critical!nucleus!sizes!of!ZB!and!fault!islands!at!different! temperatures.!The!result!in!Figure!6.10!(b)!exhibits!a!crossover!phenomenon:!At! lower!temperatures,!fault!islands!have!the!smaller!critical!diameters!than!ZB!islands,! D * fault!<!D * ZB,!while!at!higher!temperatures,! D * ZB!<!D * fault.!The!probability!of!fault Ylayer! generation! is! calculated! as! pfault! =! exp(−G * fault/kBT)/[exp(−G * fault/kBT)! +! exp(−G * ZB/kBT)],! where!kB!is! the! Boltzmann! constant(Johansson,! Karlsson! et!al.! 2009).!In!general,!the!larger!the!critical!nucleus,!the!larger!the!activation!barrier,! and!hence!the!slower!is!the!growth!rate.! Figure!6.10!(b)!indicates!that!fewer!SFs!are! generated!under!such!conditions.!This!analysis!shows!that!the!growth!temperature! and!partial!pressures!can!be!controlled!to!minimize!SFs.! In!addition!to!the!growth! temperature!and!pressure!control!of!SF!density,! Figure!6.10!(b)!suggests!a!novel! sizeYcontrol!growth!method!for!completely!eliminating!SFs.!Namely,!if!the!diameter! DW!of!the!NW!is!between!the!critical!nucleus!sizes!of!ZB!and!fault!islands,! i.e.,!D * ZB!<! DW!<!D * fault,!only!ZB!islands!but!no!fault!islands!can!grow!on!the!NW!top!surface.!The! SFYfree! growth! range! is! indicated! by! the! yellow! area! in! Figure!6.10!(b).! Our! simulation!results!suggest!a!possibility!of!SA YMOPVE!growth!of!SFYfree!NWs!with! ! 152! judicious!choice!of!diameter!size,!growth!temperature,!and!vapor!pressures. !We! should!note!that!the!Gibbs!free!energy!here!is!based!on!a!simple!ideal!gas!model,! and!more!realistic!estimate!should!include!reaction!chemistry,!which!is !beyond!the! scope!of!this!paper!(T.!R.!Omstead,!et!al.!1990).! ! Figure!6.10!(a)!Gibbs!freeYenergy!changes!of!ZB!(blue)!and!fault!(red)!islands!of!a! GaYAs!bilayer!adsorbed!on!the!GaAs!NW!as!a!function!of!the!island!diameter!at! temperature!T!=!1030!K!(solid!lines)!and!1050!K!(dashed!lines).!The!peak!po sition! (i.e.!the!critical!nucleus!size)!of!each!curve!is!indicated!by!an!arrow.!!(b)!Critical! nucleus!sizes!of!ZB!(blue)!and!fault!(red)!islands!as!a!function!of!the!temperature.! The!yellow!area!indicates!(DW,!T)!pairs,!for!which!NWs!are!grown!free!of!SF.! The! vapor!pressures!are!pGa!=!2.7 ×10Y6!bar!and!pAs!=!5 ×10Y4!bar.! ! 153! CHAPTER)7 )) ) ) ) ) ) ) )))))))) CONCLUSION) The!main!contribution!of!this!dissertation!is!fundamental!study!of!boundaries!and! their!effects!on!nanostructures!at!atomistic!level!using!molecular!dynamics!(MD)! simulation.!! ! The!first!problem!we!studied!was!the!structure!of!impurity!atoms! segregated! grain!boundaries!(GBs)!in!metal!nanocrystalline!system.!Using!reactive!force Yfield! MD!simulations,!we!have!found!an!atomistic!mechanism!of! sulfur!(S)!segregationY induced!amorphization!of!nickel!(Ni).!Namely,!the!large!steric!size!of! S!impurity! causes!strong!SYS!interaction!mediated!by!the!distortion!of! Ni!lattice!up!to!the!next! nearestYneighbor! lattice! sites,! and! amorphization! occurs! at! the! percolation ! threshold!of!the!sulfurYsulfur!network!with!the!next!nearest Yneighbor!connectivity! at! GB! interface.! The! generality! of! the! amorphization! mechanism! due! to! the! percolation!of!an!impurity!network!has!been!confirmed!for!a!binary!Lennard YJones! system.! ! In!the!second!and!third!problems,!we!have!focused!on!mysterious!behaviors! of! semiconductor! nanowires! (NWs).! MD! simulations! revealed! that! bond! environment!change!on!the!surface!of!NW!side!walls!can!go!up!to!3!to!4!nanometers! deep!due!to!relaxation.!This!lead s!to!a!novel!core/shell!structure!in!both!zincblende! (ZB)!and!wurtzite!(WZ)!NWs,!which!will! in!turn!cause!novel!behaviors!and!affect! properties!of!nanodevices.!! ! 154! In!our!studies!on!novel!thermal Ymechanical!properties!of!WZ!structured! zincYoxide!(ZnO)!NW,!rich!size Ytemperature!dependence!of!the!mechanical!response! of! ZnO! NW! under! tension! was! discovered ,! which! we! attribute! to!a! structure! transformation!originated!from!the!incipient!core/shell!structural!motifs!due!to! surface!relaxation!of!unstretched!NW.!The!size Ytemperature!phase!diagram!shows! novel!transitions!from!brittle!cleavage!to!structural!transformation Ymediated!brittle! cleavage!to!ductile!failure!for!smaller!diameters!and!higher!temperatures.!This! nanoYthermoYmechanical! phase! diagram!resolved!several! controversies! between! previous!nanomechanical!experiments!and!theory,!and!is!expected!to!help!design! and!understand!nanomechanical!experiments.!Furthermore,!the!p redicted!intrinsic! core/shell!structure!may!find!novel!device!applications!due!to!different!mechanical! (and!possibly!electronic)!properties!of!the!body!centered!tetragonal!(BCT)!shell!and! original!hexagonal!(HX)!core,!along!with!their!atomically!sharp!inte rface.!! In! our! studies! on! stacking! fault!(SF)!generation! during! NW! growth,! MD! simulation! has! revealed! that! the! intrinsic! core/shell! structure! due! to! surface! relaxation!will!leads!to!another!core/shell!structure!for!the!adatom!energetics!on! the! (111)B! growth! surface! of! a! gallium! arsenide! (GaAs)! NWs,! where! SF! are! preferentially!nucleated!in!the!shell.!Based!on!this!atomistic!understanding,!we!have! proposed!a!sizeYcontrol!growth!of!SFYfree!GaAs!NWs.!! The!researches!encompassing!several!different!nanostructures!described!in! this!dissertation!all!point!to!a!unified!theme:!an!essential!role!of!boundaries!in! nanostructures.!Here,!boundary!to!both!interior!boundary!inside!an!object!and! ! 155! exterior! boundary! of! an! object.! The! interior! boundary! cor responds! to! GB! in! nanocrystalline!structures,!while!the!exterior!boundary!can!be!a!surface!of!NW.!The! thin!boundaries!are!usually!treated!in!the!twoYdimensional!limit.!Most!theoretical! studies!on!nanostructures!are!also!mainly!focused!on!how!defects/atoms!behave! under!a!confined!volume!and!ignore!the!fact!that!the!third!dimension!of!a!boundary,! which!is!often!a!few!nanometers!in!length,!is!nonYnegligible!at!nanoscale.!In!our! research,!we!have!extended!the!concept!of!boundary!to!a!threeYdimensional!space! with!volume.!For!each!interface/surface!we!studied,!we!examined!and!characterize d! the!interface!volume!and!then!found!that!the!boundary !(e.g.!GB!phase!and!NW!shell)! in! nanostructures! can! play! a! key! role!in! determining! the! properties! and! performance!of!materials.!I!believe!that!the!boundaryYvolume!in!nanostructures!is! an!essential!element,!which!should!augment!existing! materials!models.!This!is!a!new! and!compelling!area.!I!look!forward!to!the!day!that! such!an!extended!model!would! help!the!advancement!of!modern!nanotechnology.! ! ! ! ! ! ! ! ! ! 156! ! ! References) Agrawal,!R.,!B.!Peng,!et!al.!(2009).!"Experimental Ycomputational!investigation!of!ZnO! nanowires!strength!and!fracture."! Nano!Letters!9(12):!4177Y4183.! Agrawal,! R.,! B.! Peng,! et! al.! (2008).! "Elasticity! size! effects! in! ZnO! nanowiresYa! combined!experimentalYcomputational!approach."!Nano!Letters!8(11):!3668Y3674.! Aharony,!S.!and!A.!Aharony!(1994!).! Introduction!to!Percolation!Theroy,!2nd!ed. ! London,!Taylor!and!Francis.! Allen,!M.!P.,!D.!J.!Tildesley,!et!al.!(1989 ).!"Computer!simulation!of!liquids."!Physics! Today!42:!71Y80.! Allen,!T.!R.,!K.!Sridharan,!et!al.!(2008).!"Materials!challenges!for!Generation!IV! nuclear!energy!systems."!Nuclear!Technology!162(3):!342Y357.! Anspaugh,!B.!(1996).!"GaAs!solar!cell!radiation!ha ndbook."! Bao,! J.,! D.! C.! Bell,! et! al.! (2008).! "Optical! properties! of! rotationally! twinned! InP! nanowire!heterostructures."!Nano!letters!8(3):!836Y841.! Binnig,!G.,!C.!F.!Quate,!et!al.!(1986).!"Atomic!force!microscope."! Physical!review! letters!56(9):!930Y933.! Binnig,! G.,! H.! Rohrer,! et! al.! (1982).! "Surface! studies! by! scanning! tunneling! microscopy."!Physical!review!letters!49(1):!57Y61.! ! 157! Bjork,! M.,! B.! Ohlsson,! et! al.! (2002).! "One Ydimensional! heterostructures! in! semiconductor!nanowhiskers."!Applied!Physics!Letters!80(6):!1058Y1060.! Blakemore,! J.! (1982).! "Semiconducting! and! other! major! properties! of! gallium! arsenide."!Journal!of!Applied!Physics!53(10):!R123YR181.! Bolinsson,!J.,!L.!Ouattara,!et!al.!(2009).!"Direct!observation!of!atomic!scale!surface! relaxation!in!ortho!twin!structures!in!GaAs!by!XSTM."! Journal!of!Physics:!Condensed! Matter!21(5):!055404.! Born,!M.!(1912).!Über!Schwingungen!in!Raumgittern,!Hirzel.! Brinckmann,!S.,!J.!Y.!Kim,!et!al.!(2008).!"Fundamental!differences!in!mechanical! behavior!between!two!types!of!crystals!at!the!nanoscale."! Physical!review!letters! 100(15):!155502.! Bryning,!M.!B.,!M.!F.!Islam,!et!al.!(2005).!"Very!Low!Conductivity!Threshold!in! Bulk! Isotropic!SingleYWalled!Carbon!NanotubeYEpoxy!Composites."!Advanced!Materials! 17(9):!1186Y1191.! Bulatov,!V.!and!W.!Cai!(2006).! Computer!simulations!of!dislocations,!OUP!Oxford.! Cai,!W.,!V.!V.!Bulatob,!et!al.!(2003).!"Periodic!image!effects!in!dislocat ion!modelling."! Philosophical!Magazine!83(5):!539Y567.! Caroff,!P.,!K.!Dick,!et!al.!(2008).!"Controlled!polytypic!and!twin Yplane!superlattices!in! IIIYV!nanowires."!Nature!nanotechnology!4(1):!50Y55.! CeYWen,!N.!(1993).!"Physics!of!inhomogeneous!inorganic!mat erials."!Prog.!Mater.! Sci.!37(1):!1Y116.! Chang,!J.,!W.!Cai,!et!al.!(2002).!"Molecular!dynamics!simulations!of!motion!of!edge! and!screw!dislocations!in!a!metal."! Computational!materials!science!23(1):!111Y115.! ! 158! Chang,!P.!C.,!C.!J.!Chien,!et!al.!(2007).!"Fini te!size!effect!in!ZnO!nanowires."! Applied! Physics!Letters!90(11):!113101.! Chen,! C.! Q.,! Y.! Shi,! et! al.! (2006).! "Size! dependence! of! Young's! modulus! in! ZnO! nanowires."!Physical!Review!Letters!96(7):!075505.! Chen,!H.!P.,!R.!K.!Kalia,!et!al.!(2010).!"Embrittlem ent!of!metal!by!solute!segregation Y induced!amorphization."!Physical!Review!Letters!104(15):!155502.! Cohen,!M.!L.!(1984).!"Electronic!structure!of!solids."! Physics!Reports!110(5):!293Y 309.! Colombo,! C.,! M.! Heiß≤,! et! al.! (2009).! "Gallium! arsenide! radial! structures! for! photovoltaic!applications."!Applied!Physics!Letters!94(17):!173108.! Cui,! Y.,! Z.! Zhong,! et! al.! (2003).! "High! performance! silicon! nanowire! field! effect! transistors."!Nano!Letters!3(2):!149Y152.! Czaban,! J.! A.,! D.! A.! Thompson,! et! al.! (2009).! "GaAs! core Yshell! nanowires! for! photovoltaic!applications."!Nano!Letters!9(1):!148Y154.! Dai,! L.,! W.! C.! D.! Cheong,! et! al.! (2010).! "Molecular! dynamics! simulation! of! ZnO! nanowires:!size!effects,!defects,!and!super!ductil ity."!Langmuir!26(2):!1165Y1171.! Dai,!Z.,!Z.!Pan,!et!al.!(2001).!"Ultra Ylong!single!crystalline!nanoribbons!of!tin!oxide."! Solid!state!communications!118(7):!351Y354.! Dang,!Z.!M.,!Y.!Shen,!et!al.!(2002).!"Dielectric!behavior!of!three Yphase!percolative! Ni‚ÄìBaTiO!3/polyvinylidene!fluoride!composites."!Applied!physics!letters!81(25):! 4814Y4816.! Desai,!A.!and!M.!Haque!(2007).!"Mechanical!properties!of!ZnO!nanowires."! Sensors! and!Actuators!A:!Physical!134(1):!169Y176.! ! 159! Deshpande,!V.,!A.!Needleman,!et!al.!(2005). !"Plasticity!size!effects!in!tension!and! compression! of! single! crystals."!Journal! of! the! Mechanics! and! Physics! of! Solids ! 53(12):!2661Y2691.! Devincre,!B.,!L.!Kubin,!et!al.!(2001).!"Mesoscopic!simulations!of!plastic!deformation."! Materials!Science!and!Engineering:!A!309:!211Y219.! Dick,!K.!A.,!P.!Caroff,!et!al.!(2010).!"Control!of!III YV!nanowire!crystal!structure!by! growth!parameter!tuning."!Semiconductor!Science!and!Technology!25(2):!024009.! Dick,!K.!A.,!K.!Deppert,!et!al.!(2005).!"A!New!Understanding!of!Au YAssisted!Growth!of! IIIYV! Semiconductor! Nanowires."!Advanced! Functional! Materials!15(10):! 1603Y 1610.! Dong,!Y.,!B.!Tian,!et!al.!(2009).!"Coaxial!group!III! Y!nitride!nanowire!photovoltaics."! Nano!letters!9(5):!2183Y2187.! Duan,!X.,!Y.!Huang,!et!al.!(2001).!"Indium!phosphide!nanowires!as!building!blocks!for! nanoscale!electronic!and!optoelectronic!devices."! Nature!409(6816):!66Y69.! Ebbsjo,!I.,!R.!K.!Kalia,!et!al.!(2000).!"Topology!of!amorphous!gallium!arsenide!on! intermediate!length!scales:!A!molecular!dynamics!study."! Journal!of!applied!physics! 87(11):!7708Y7711.! Elias,! J.,! C.! LevyYClement,! et! al.! (2010).! "Hollow! urchin Ylike! ZnO! thin! films! by! electrochemical!deposition."!Advanced!Materials!22(14):!1607Y1612.! Espinosa,!H.,!M.!Panico,!et!al.!(2006).!"Discrete!dislocation!dynamics!simulations!to! interpret! plasticity! size! and! surface! effects! in! freestanding! FCC! thin! films."! International!journal!of!plasticity!22(11):!2091Y2117.! Farkas,!D.,!S.!Mohanty,!et!al.!(2007).!"Linear!grain!growth!kinetics!and!rotation!in! nanocrystalline!Ni."!Physical!review!letters!98(16):!165502.! Farkas,! D.,! S.! Van! Petegem,! et! al.! (2005).! "Dislocation! activity! and! nano Yvoid! formation!near!crack!tips!in!nanocrystalline!Ni."! Acta!Materialia!53(11):!3115Y3123.! ! 160! Frenkel,! D.! and! B.! Smit! (2001).!Understanding! molecular! simulation:! from! algorithms!to!applications,!Academic!press.! Gilman,!J.!J.!(1996).!"Mechanochemistry."! Science!274(5284):!65Y65.! Glas,! F.! (2008).! "A! simple! calculation! of! energy! changes! upon! stacking! fault! formation! or! local! crystalline! phase! transition! in! semiconductors."!Journal! of! Applied!Physics!104(9):!093520.! Glas,!F.,!J.!C.!Harmand,!et!al.!(2007).!"Why!does!wurtzite!form!in!nanowires!of!III YV! zinc!blende!semiconductors?"!Physical!Review!Letters!99(14):!146101.! Gleiter,!H.!(2000).!"Nanostructured!materials:!basic!concepts!and!microstr ucture."! Acta!materialia!48(1):!1Y29.! Golovchan,! V.! T.! (1998).! "Elastic! moduli! of! a! tungsten! monocarbide! crystal."! International!Applied!Mechanics!34(8):!755Y757.! Green,!M.!A.!(2003).!"Crystalline!and!thin Yfilm!silicon!solar!cells:!state!of!the!art!and! future!potential."!Solar!energy!74(3):!181Y192.! Greer,!J.,!D.!Jang,!et!al.!(2009).!"Emergence!of!New!Mechanical!Functionality!in! Materials!via!Size!Reduction."! Advanced!Functional!Materials!19(18):!2880Y2886.! Greer,!J.!R.!and!J.!T.!M.!De!Hosson!(2011).!"Plast icity!in!smallYsized!metallic!systems:! intrinsic!versus!extrinsic!size!effect."! Progress!in!Materials!Science!56(6):!654Y724.! Greer,!J.!R.!and!W.!D.!Nix!(2006).!"Nanoscale!gold!pillars!strengthened!through! dislocation!starvation."!Physical!review!B!73(24):!245410.! Haick,!H.,!P.!T.!Hurley,!et!al.!(2006).!"Electrical!characteristics!and!chemical!stability! of! nonYoxidized,! methylYterminated! silicon! nanowires."!Journal! of! the! American! Chemical!Society!128(28):!8990Y8991.! Haupt,!R.!L.!and!S.!E.!Haupt!(2004).! Practical!genetic!algorithms,!WileyYInterscience.! ! 161! Heuer,! J.! K.,! P.! R.! Okamoto,! et! al.! (2002).! "Disorder Yinduced! melting! in! to! intergranular!sulfur!nickel:!implication!embrittlement."! Journal!of!Nuclear!Materials! 301(2Y3):!129Y141.! Hoffmann,!S.,!F.!Ostlund,!et !al.!(2007).!"Fracture!strength!and!Young's!modulus!of! ZnO!nanowires."!Nanotechnology!18(20):!205503.! Hoffmann,!S.,!I.!Utke,!et!al.!(2006).!"Measurement!of!the!bending!strength!of!vapor Y liquidYsolid!grown!silicon!nanowires."! Nano!Letters!6(4):!622Y625.! Hohenberg,!P.!and!W.!Kohn!(1964).!"Inhomogeneous!electron!gas."! Physical!Review! 136:!B864YB871.! Holland,!J.!H.!(1992).!"Adaptation!In!Natural!And!Artificial!Systems:!An!Introductory! Analysis!With!Applications!To!Biology,!Control,!And!Artific." ! Hoover,!W.!G.!(1985).!"Canonical!dynamics:!Equilibrium!phase Yspace!distributions."! Physical!Review!A!31(3):!1695.! Hsieh,!H.!and!S.!Yip!(1989).!"Atomistic!simulation!of!defect Yinduced!amorphization! of!binary!lattices."!Physical!Review!B!39(11):!7476Y7491.! Huang,!H.!and!H.!Van!Swygenhoven!(2009).!"Atomistic!simulations!of!mechanics!of! nanostructures."!MRS!bulletin!34(03):!160Y166.! Huang,! M.! H.,! S.! Mao,! et! al.! (2001).! "Room Ytemperature! ultraviolet! nanowire! nanolasers."!science!292(5523):!1897Y1899.! Hull,! D.! and! D.! J.! Bacon! ( 2001).! Introduction! to! dislocations,! ButterworthY Heinemann.! Iijima,!S.!(1991).!"Helical!microtubules!of!graphitic!carbon."! nature!354(6348):!56Y 58.! ! 162! Ikejiri,!K.,!T.!Sato,!et!al.!(2008).!"Growth!characteristics!of!GaAs!nanowires!obtained! by! selective! area! metalYorganic! vapourYphase! epitaxy."!Nanotechnology!19(26):! 265504.! Jaffe,!J.!and!A.!Hess!(1993).!"Hartree YFock!study!of!phase!changes!in!ZnO!at!high! pressure."!Physical!Review!B!48(11):!7903.! Jaffe,!J.!E.,!N.!M.!Harrison,!et!al.!(1994).!"Ab Yinitio!study!of!ZnO!(1010)!surface! relaxation."!Physical!Review!B!49(16):!11153Y11158.! Johansson,!J.,!L.!S.!Karlsson,!et!al.!(2008).!"Effects!of!Supersaturation!on!the!Crystal! Structure!of!Gold!Seeded!III‚àí !V!Nanowires."!Crystal!Growth!and!Design!9(2):!766Y 773.! Johansson,!J.,!L.!S.!Karlsson,!et!al.!(2009).!"Effects!of!supersaturation!on!the!crystal! structure!of!gold!seeded!III YV!nanowires."!Crystal!Growth!&!Design!9(2):!766Y773.! Johansson,!J.,!L.!S.!Karlsson ,!et!al.!(2006).!"Structural!properties!of!111B Yoriented!IIIY V!nanowires."!Nature!Materials!5(7):!574Y580.! Jones,!J.!(1924).!"On!the!determination!of!molecular!fields.!I.!From!the!variation!of! the!viscosity!of!a!gas!with!temperature."! Proceedings!of!the!Royal!Society!of!London.! Series!A,!Containing!Papers!of!a!Mathematical!and!Physical!Character !106(738):! 441Y462.! Kang,! K.! W.! and! W.! Cai! (2010).! "Size! and! temperature! effects! on! the! fracture! mechanisms!of!silicon!nanowires:!molecular!dynamics!simulations."! International! Journal!of!Plasticity!26(9):!1387Y1401.! Keten,!S.,!Z.!P.!Xu,!et!al.!(2010).!"Nanoconfinement!controls!stiffness,!strength!and! mechanical!toughness!of!betaYsheet!crystals!in!silk."! Nature!Materials!9(4):!359Y367.! Kirkpatrick,!S.!(1984).!"Optimization!by!simulated!annealing:!Quantitative!studies."! Journal!of!statistical!physics!34(5):!975Y986.! ! 163! Kirkpatrick,! S.! and! M.! Vecchi! (1983).! "Optimization! by! simmulated! annealing."! science!220(4598):!671Y680.! Koch,! C.! (2007).! "Structural! nanocrystalline! mater ials:! an! overview."!Journal! of! Materials!Science!42(5):!1403Y1414.! Landau,!L.!D.!and!E.!M.!Lifshi*t*s!(1970).! Theory!of!elasticity.!Oxford,!New!York,,! Pergamon!Press.! Landau,!L.!D.,!E.!Lifshitz,!et!al.!(1976).! Mechanics.!Course!of!Theoretical!Physics,! Pergamon!Press.! Lauhon,!L.!J.,!M.!S.!Gudiksen,!et!al.!(2002).!"Epitaxial!core Yshell!and!coreYmultishell! nanowire!heterostructures."!Nature!420(6911):!57Y61.! Lee,!B.!T.,!J.!Y.!Lee,!et!al.!(1990).!"Atomic!structure!of!twins!in!GaAs."! Applied!physics! letters!57:!346.! Li,!D.,!Y.!Wu,!et!al.!(2003).!"Thermal!conductivity!of!individual!silicon!nanowires."! Applied!Physics!Letters!83(14):!2934Y2936.! Li,!J.!(2007).!"The!mechanics!and!physics!of!defect!nucleation."! MRS!bulletin!32(02):! 151Y159.! Li,! M.! and! W.! L.! Johnson! (1992).! "Instability! of! Metastable! Solid! Solutions! and! Crystal!to!Glass!Transition."! Physcial!Review!Letters!70(8):!1120.! Li,!W.,!R.!K.!Kalia,!et!al.!(1996).!"Amorphization!and!fracture!in!silicon!diselenide! nanowires:!a!molecular!dynamics!study."!Physical!Review!Letters!77(11):!2241Y 2244.! Liu,!X.!J.,!J.!W.!Li,!et!al.!(2009).!"Size Yinduced!elastic!stiffening!of!ZnO!nanostructures:! skinYdepth!energy!pinning."!Applied!Physics!Letters!94(13):!131902.! ! 164! Martyna,!G.!J.,!M.!L.!Klein,!et!al. !(1992).!"NoséYHoover!chains:!the!canonical!ensemble! via!continuous!dynamics."!The!Journal!of!Chemical!Physics !97:!2635.! Martyna,!G.!J.,!D.!J.!Tobias,!et!al.!(1994).!"Constant!pressure!molecular!dynamics! algorithms."!The!Journal!of!Chemical!Physics !101:!4177.! Martyna,!G.!J.,!M.!E.!Tuckerman,!et!al.!(1996).!"Explicit!reversible!integrators!for! extended!systems!dynamics."!Molecular!Physics!87(5):!1117Y1157.! McMahon,!M.!and!R.!Nelmes!(2005).!"Observation!of!a!wurtzite!form!of!gallium! arsenide."!Physical!review!letters!95(21):!215505.! Mehrabian,! A.! R.! and! A.! YousefiYKoma! (2011).! "A! novel! technique! for! optimal! placement!of!piezoelectric!actuators!on!smart!structures."! Journal!of!the!Franklin! Institute!348(1):!12Y23.! Messing,!M.!E.,!K.!Hillerich,!et!al.!(2009).!"T he!use!of!gold!for!fabrication!of!nanowire! structures."!Gold!bulletin!42(3):!172Y181.! Messmer,!R.!and!C.!Briant!(1982).!"The!role!of!chemical!bonding!in!grain!boundary! embrittlement."!Acta!Metallurgica!30(2):!457Y467.! Meyer,! B.! and! D.! Marx! (2003).! "Density Yfunctional! study! of! the! structure! and! stability!of!ZnO!surfaces."! Physical!Review!B!67(3):!035403.! Meyers,!M.!A.,!A.!Mishra,!et!al.!(2006).!"Mechanical!properties!of!nanocrystalline! materials."!Progress!in!Materials!Science!51(4):!427Y556.! Mingo,!N.!(2003).!"Calculation!of!Si!nanowire!thermal!conductivity!using!complete! phonon!dispersion!relations."!Physical!Review!B!68(11):!113308.! Morales,!A.!M.!and!C.!M.!Lieber!(1998).!"A!laser!ablation!method!for!the!synthesis!of! crystalline!semiconductor!nanowires."!Science!279(5348):!208Y211.! ! 165! Motayed,!A.,!M.!Vaudin,!et!al.!(2007).!"Diameter!dependent!transport!properties!of! gallium!nitride!nanowire!field!effect!transistors."!Applied!Physics!Letters!90(4):! 043104Y043104Y043103.! Nakano,!A.,!R.!K.!Kalia,!et!al.!(1994).!"Multiresolution!molecular!dynamics!algorithm! for! realistic! materials! modeling! on! parallel! computers."!Computer! Physics! Communications!83(2):!197Y214.! Nakano,!A.,!R.!K.!Kalia,!et!al.!(1999).!"Scalable!molecular Ydynamics,!visualization,!and! data!management!algorithms!for!materials!simulations."!Computing!in!science!&! engineering!1(5):!39Y47.! Newman,!M.!E.!J.!and!R.!M.!Ziff!(2001).!"Fast !Monte!Carlo!algorithm!for!site!or!bond! percolation."!Physical!Review!E!64(1):!016706.! Nicola,!L.,!E.!Van!der!Giessen,!et!al.!(2003).!"Discrete!dislocation!analysis!of!size! effects!in!thin!films."! Journal!of!Applied!Physics!93(10):!5920Y5928.! Nielson,!K.!D.,!A.!C.!T.!van!Duin,!et!al.!(2005).!"Development!of!the!ReaxFF!reactive! force!field!for!describing!transition!metal!catalyzed!reactions,!with!application!to! the!initial!stages!of!the!catalytic!formation!of!carbon!nanotubes."! Journal!of!Physical! Chemistry!A!109(3):!493Y499.! Nomura,!K.,!R.!K.!Kalia,!et!al.!(2008).!"A!scalable!parallel!algorithm!for!large Yscale! reactive! forceYfield! molecular! dynamics! simulations."! Computer! Physics! Communications!178(2):!73Y87.! Nomura,!K.,!R.!Seymour,!et!al.!(2009).! A!metascalable!computing!framework!for! large!spatiotemporalYscale!atomistic!simulations.!Parallel!&!Distributed!Processing,! 2009.!IPDPS!2009.!IEEE!International!Symposium!on,!IEEE. ! Nomura,!K.!I.,!R.!K.!Kalia,!et!al.!(2008).!"A!scalable!parallel!algorithm!for!large Yscale! reactive! forceYfield! molecular! dynamics! simulations."! Computer! Physics! Communications!178(2):!73Y87.! Nosé,!S.!(1984).!"A!molecular!dynamics!method!for!simulations!in!the!canonical! ensemble."!Molecular!Physics!52(2):!255Y268.! ! 166! Nosé,! S.! (1984).! "A! unifie d! formulation! of! the! constant! temperature! molecular! dynamics!methods."!The!Journal!of!Chemical!Physics !81:!511.! Östlund,! F.,! P.! R.! Howie,! et! al.! (2011).! "Ductile Ybrittle! transition! in! micropillar! compression!of!GaAs!at!room!temperature."! Philosophical!Magazine!91(7Y9):!1190Y 1199.! Östlund,!F.,!K.!RzepiejewskaYMalyska,!et!al.!(2009).!"Brittle YtoYductile!transition!in! uniaxial!compression!of!silicon!pillars!at!room!temperature."! Advanced!Functional! Materials!19(15):!2439Y2444.! Ozgur,!U.,!Y.!I.!Alivov,!et!al.!(2005).!"A!comprehensive!review!of!ZnO!materials!and! devices."!Journal!of!Applied!Physics!98(4):!041301.! Paek,! J.! H.,! T.! Nishiwaki,! et! al.! (2009).! "Catalyst! free! MBE YVLS! growth! of! GaAs! nanowires!on!(111)Si!substrate."! physica!status!solidi!(c)!6(6):!1436Y1440.! Palumbo,!G.,!S.!Thorpe,!et!al.!(1990).!"On!the!contribution!of!triple!junctions!to!the! structure!and!properties!of!nanocrystalline!materials."!Scr.!Metall.!Mater.!24(7):! 1347Y1350.! Pan,!C.!and!J.!Zhu!(2009).!"The!synthe ses,!properties!and!applications!of!Si,!ZnO,! metal,!and!heterojunction!nanowires."! J.!Mater.!Chem.!19(7):!869Y884.! Payne,!M.!C.,!M.!P.!Teter,!et!al.!(1992).!"Iterative!minimization!techniques!for!abinitio! totalYenergy!calculations!Y!molecularYdynamics!and!conjugate!gradients."!Reviews!of! Modern!Physics!64(4):!1045Y1097.! Perdew,!J.!P.,!K.!Burke,!et!al.!(1996).!"Generalized!gradient!approximation!made! simple."!Physical!Review!Letters!77(18):!3865Y3868.! Persson,!A.!I.,!M.!W.!Larsson,!et!al.!(2004).!"Solid Yphase!diffusion!mechanism!for! GaAs!nanowire!growth."!Nature!Materials!3(10):!677Y681.! Pike,!G.!and!C.!Seager!(1974).!"Percolation!and!conductivity:!A!computer!study.!I."! Physical!Review!B!10(4):!1421Y1434.! ! 167! Poncharal,!P.,!Z.!Wang,!et!al.!(1999).!"Electrostatic! deflections!and!electromechanical! resonances!of!carbon!nanotubes."! Science!283(5407):!1513Y1516.! Qi,! W.! (2005).! "Size! effect! on! melting! temperature! of! nanosolids."! Physica! B:! Condensed!Matter!368(1):!46Y50.! RafiiYTabar,!H.!(2000).!"Modelling!the!nano Yscale!phenomena!in!condensed!matter! physics!via!computerYbased!numerical!simulations."!Physics!Reports!325(6):!239Y 310.! Rice,!J.!and!J.!Wang!(1989).!"Embrittlement!of!interfaces!by!solute!segregation."! Materials!Science!and!Engineering:!A !107:!23Y40.! Rice,!J.!R.!and!J.!S.!Wang!(1989).!"Embrittlement!of!Interfaces!by!Solute!Segregation."! Materials!Science!and!Engineering!a YStructural!Materials!Properties!Microstructure! and!Processing!107:!23Y40.! Rino,!J.!P.,!A.!Chatterjee,!et!al.!(2002).!"Pressure Yinduced!structural!transformation! in!GaAs:‚ÄÉA!molecularYdynamics!study."!Physical!Review!B!65(19):!195206.! Roco,! M.! C.,! C.! A.! Mirkin,! et! al.! (2011).! Nanotechnology! research! directions! for! societal!needs!in!2020:!Retrospective!and!outlook ,!Springer.! Roco,!M.!C.,!S.!Williams,!et!al.!(1999).!"Nanotechnology!Research!Directions:!IWGN! Workshop!Report!Vision!for!Nanotechnology!Research!and!Development!in!the!Next! Decade."!WTEC,!Loyola!College!in!Maryland .! Rogers,! H.! C.! (1968).! "Hydrogen! Embrittlement! of! Me tals."!Science!159(3819):! 1057Y&.! Rowley,!R.!L.,!Y.!Yang,!et!al.!(2001).!"Determination!of!an!ethane!intermolecular! potential!model!for!use!in!molecular!simulations!from!ab!initio!calculations."! The! Journal!of!Chemical!Physics!114:!6058.! Rudolph,!D.,!S.!Hertenberger,!et!al.!(2011).!"Direct!observation!of!a!noncatalytic! growth!regime!for!GaAs!nanowires."! Nano!Letters!11(9):!3848Y3854.! ! 168! Ruoff,! R.,! D.! S.! Tse,! et! al.! (1993).! "Solubility! of! fullerene! (C60)! in! a! variety! of! solvents."!The!Journal!of!Physical!Chemis try!97(13):!3379Y3383.! S.G.Hao,! C.! Z.! W.,! M.J.Kramer! and! K.M.Ho! (2010).! "Microscopic! origin! of! slow! dynamics!at!the!good!glass!foming!composition!range!in!Zr1 YxCux!metallic!liquids."! Journal!of!Applied!Physics!107(5):!053511.! Sarid,!D.!(1994).!Scanning!force!microscopy:!with!applications!to!electric,!magnetic,! and!atomic!forces,!Oxford!University!Press,!USA. ! Schweinfest,!R.,!A.!T.!Paxton,!et!al.!(2004).!"Bismuth!embrittlement!of!copper!is!an! atomic!size!effect."!Nature!432(7020):!1008Y1011.! Seiler,!M.!C.!and!F.!A.!Seiler!(1989).!"Numerical!Recipes!in!C:!The!Art!of!Scientific! Computing."!Risk!Analysis!9(3):!415Y416.! Shimizu,!F.,!S.!Ogata,!et!al.!(2006).!"Yield!point!of!metallic!glass."! Acta!Materialia! 54(16):!4293Y4298.! Simmons,!G.!and!H.!Wang!(1971).!Single!crystal!elastic!constants!and!calculated! aggregate!properties:!a!handbook.!Cambridge,!Mass.,!M.I.T.!Press. ! Sinclair,!J.,!P.!Gehlen,!et!al.!(1978).!"Flexible!boundary!conditions!and!nonlinear! geometric!effects!in!atomic!dislocation!modeling."! Journal!of!Applied!physics!49(7):! 3890Y3897.! Sivanandam,!S.!and!S.!Deepa!(2007).!Introduction!to!genetic!algorithms,!Springer! Publishing!Company,!Incorporated.! Song,!J.,!X.!Wang,!et!al.!(2005).!"Elastic!property!of!vertically!aligned!nanowires."! Nano!letters!5(10):!1954Y1958.! Stankovich,!S.,!D.!A.!Dikin,!et!al.!(2006).!"Graphene Ybased!composite!materials."! Nature!442(7100):!282Y286.! ! 169! Stauffer,!D.!and!A.!Aharony!(1994).! Introduction!to!percolation!theory,!CRC.! Stiles,!M.!D.!and!D.!R.!Hamann!(1988).!"Ballistic!electron!transmission!through! interfaces."!Physical!Review!B!38(3):!2021Y2037.! Stiles,! M.! D.! and! D.! R.! Hamann! (1990).! "Electron! transmission! through! silicon! stacking!faults."!Physical!Review!B!41(8):!5280Y5282.! Stillinger,!F.!H.!and!T.!A.!Weber!(1985).!"Computer!simulation!of!local!order!in! condensed!phases!of!silicon."! Physical!Review!B!31(8):!5262.! Strachan,!A.,!A.!C.!T.!van!Duin,!et!al.!(2003).!"Shock!waves!in!high Yenergy!materials:! The!initial!chemical!events!in!nitramine!RDX."! Physical!Review!Letters!91(9):!Y.! Sun,!K.,!A.!Kargar,!et!al.!(2011).!"Compound!semiconductor!nanowire!solar!cells."! Selected!Topics!in!Quantum!Electronics,!IEEE!Journal!of !17(4):!1033Y1049.! Tang,!M.!and!S.!Yip!(1995).!"Atomic!size!effects!in!pressureYinduced!amorphization! of!a!binary!covalent!lattice."! Physical!review!letters!75(14):!2738Y2741.! Tatematsu,!H.,!K.!Sano,!et!al.!(2008).!"Ab!initio Ybased!approach!to!initial!growth! processes! on! GaAs(111)BY(2x2)! surfaces:! SelfYsurfactant! effect! of! Ga! adatoms! revisited."!Physical!Review!B!77(23):!233306.! Thelander,! C.,! P.! Caroff,! et! al.! (2011).! "Effects! of! crystal! phase! mixing! on! the! electrical!properties!of!InAs!nanowires."! Nano!letters!11(6):!2424Y2429.! Thim,!H.!(1980).!"Gallium!Arsenide!a nd!Related!Compounds."! Tian,!B.,!T.!J.!Kempa,!et!al.!(2009).!"Single!nanowire!photovoltaics."! Chemical!Society! Reviews!38(1):!16Y24.! Tomioka,!K.,!K.!Ikejiri,!et!al.!(2011).!"Selective Yarea!growth!of!IIIYV!nanowires!and! their!applications."!Journal!of!Materials!Research!26(17):!2127Y2141.! ! 170! Tomioka,!K.,!Y.!Kobayashi,!et!al.!(2009).!"SelectiveYarea!growth!of!vertically!alig ned! GaAs! and! GaAs/AlGaAs! coreYshell! nanowires! on! Si! (111)! substrate."! Nanotechnology!20(14):!145302.! van!Duin,!A.!C.!T.,!S.!Dasgupta,!e t!al.!(2001).!"ReaxFF:!A!reactive!force!field!for! hydrocarbons."!Journal!of!Physical!Chemistry!A !105(41):!9396Y9409.! Van!Swygenhoven,!H.!and!J.!R.!Weertman!(2006).!"Deformation!in!nanocrystalline! metals."!Materials!Today!9(5):!24Y31.! Van!Weert,!M.,!O.!Wunnicke,!et!al.!(2006).!"Large!redshift!in!photoluminescence!of! pYdoped!InP!nanowires!induced!by!Fermi Ylevel!pinning."!Applied!Physics!Letters! 88(4):!043109.! Vashishta,!P.,!R.!K.!Kalia,!et!al.!(1997).! Amorphous!Insulators!and!Semiconductors.! Dordrooht,!the!Netherlands,!Klewer!Academic!Publisher. ! Vashishta,!P.,!R.!K.!Kalia,!et!al.!(1990).!"Interaction!potential!for!SiO 2!Y!a!molecularY dynamics!study!of!structural!correlations."! Physical!Review!B!41(17):!12197Y12209.! Vashishta,!P.!and!A.!Rah man!(1979).!Nature!of!ionic!motions!in!AgI!and!CuI,!Argonne! National!Lab.,!IL!(USA).! Vieu,!C.,!F.!Carcenac,!et!al.!(2000).!"Electron!beam!lithography:!resolution!limits!and! applications."!Applied!Surface!Science!164(1):!111Y117.! Wang,!J.,!A.!J.!Kulkarni,!e t!al.!(2008).!"Novel!mechanical!behavior!of!ZnO!nanorods."! Computer!Methods!in!Applied!Mechanics!and!Engineering !197(41Y42):!3182Y3189.! Wang,!S.!and!H.!Ye!(2002).!"A!planeYwave!pseudopotential!study!on!III YV!zincY blende! and! wurtzite! semiconductors! under! pr essure."! Journal! of! Physics:! Condensed!Matter!14(41):!9579.! Wang,!Z.!L.!(2004).!"Nanostructures!of!zinc!oxide."! Materials!today!7(6):!26Y33.! ! 171! Wang,! Z.! L.! (2004).! "Zinc! oxide! nanostructures:! growth,! properties! and! applications."!Journal!of!PhysicsYCondensed!Matter!16(25):!R829YR858.! Wang,!Z.!L.!(2009).!"ZnO!nanowire!and!nanobelt!platform!for!nanotechnology."! Materials!Science!and!Engineering:!R:!Reports !64(3):!33Y71.! Waseda,!Y.!and!K.!Suzuki!(1975).!"Structure!of!molten!silicon!and!germanium!by!X Y ray!diffraction."!Zeitschrift!f√ºr!Physik!B!Condensed!Matter !20(4):!339Y343.! Weast,!R.!C.!and!M.!J.!Astle!(1993).!CRC!handbook!of!chemistry!and!physics!74th!ed,! Boca!Raton:!CRC!Press.! Weissmüller,!J.!and!J.!Markmann!(2005).!"Deforming!nanocrystalline!metals:!new! insights,!new!puzzles."!Advanced!Engineering!Materials!7(4):!202Y207.! Wen,! B.,! J.! E.! Sader,! et! al.! (2008).! "Mechanical! Properties! of! ZnO! Nanowires."! Physical!review!letters!101(17):!175502.! Wong,!E.!W.,!P.!E.!Sheehan,!et!al.!(1997).!"Nanobeam!mechanics:!elasticity,!strength,! and!toughness!of!nanorods!and!nanotubes."! Science!277(5334):!1971Y1975.! Wu,!Y.,!H.!Yan,!et!al.!(2002).!"Inorganic!semiconductor!nanowires:!rational!growth,! assembly,!and!novel!properties."!ChemistryYA!European!Journal!8(6):!1260Y1268.! Xia,!Y.!and!G.!M.!Whitesides!(1998).!"Soft!lithography."!Annual!review!of!materials! science!28(1):!153Y184.! Xu,!H.,!Y.!Guo,!et!al.!(2009).!"Effects!of!annealing!and!substrate!orientation!on! epitaxial!growth!of!GaAs!on!Si."! Journal!of!Applied!Physics!106(8):!083514Y083514Y 083514.! Yamaguchi,!M.,!M.!Shiga,!et!al.!(2005).!"Grain!boundary!decohesion!by!impurity! segregation!in!a!nickelYsulfur!system."!Science!307(5708):!393Y397.! ! 172! Yamakov,!V.,!D.!Wolf,!et!al.!(2002).!"Dislocation!processes!in!the!deformation!of! nanocrystalline! aluminium! by! molecularYdynamics! simulation."!Nature! Materials! 1(1):!45Y49.! Yang,!C.,!X.!Li,!et!al.!(2008).!"ZnO!based!oxide!system!with!continuous!bandgap! modulation!from!3.7!to!4.9!eV."!Applied!Physics!Letters!93(11):!112114Y112114Y 112113.! Yeh,!C.!Y.,!Z.!Lu,!et!al.!(1992).!"ZincYblendeYwurtzite!polytypism!in!semiconductors."! Physical!review!B!46(16):!10086.! Yoshida,!H.,!K.!Ikejiri,!et!al.!(2009).!"Analysis!of!twin!defects!in!GaAs !nanowires!and! tetrahedra! and! their! correlation! to! GaAs! (111)! B! surface! reconstructions! in! selectiveYarea! metal! organic! vapourYphase! epitaxy."!Journal! of! Crystal! Growth! 312(1):!52Y57.! Yu,!C.,!L.!Shi,!et!al.!(2005).!"Thermal!conductance!and!thermopower!of!an!individual! singleYwall!carbon!nanotube."!Nano!Letters!5(9):!1842Y1846.! Yu,!M.!F.,!O.!Lourie,!et!al.!(2000).!"Strength!and!breaking!mechanism!of!multiwalled! carbon!nanotubes!under!tensile!load."!Science!287(5453):!637Y640.! Zallen,!R.!(1983).!The!Physics!of!Amorphous!Solids .!New!York,!Wiley.! Zallen,! R.! and! J.! Wiley! (1983).!The! physics! of! amorphous! solids ,! Wiley! Online! Library.! Zhang,! L.! X.! and! H.! C.! Huang! (2007).! "Structural! transformat ion! of! ZnO! nanostructures."!Applied!Physics!Letters!90(2):!023115.! Zhou,!L.!and!H.!Huang!(2004).!"Are!surfaces!elastically!softer!or!stiffer?"! Applied! Physics!Letters!84(11):!1940Y1942.! Zimmerman,! J.! A.,! C.! L.! Kelchner,! et! al.! (2001).! "Surface! step! effects! on! nanoindentation."!Physical!Review!Letters!87(16):!165507.! !
Abstract (if available)
Abstract
This dissertation is focused on multimillion-atom molecular dynamics (MD) simulations of nanoscale materials. In the past decade, nanoscale materials have made significant commercial impacts, which will potentially lead to the next industrial revolution. The interest lies in the novel and promising features nanoscale materials exhibit due to their confined sizes. However, not all novel behaviors are understood or controllable. Many uncontrollable parameters, e.g. defects and dangling bonds, are known to hinder the performance of nanodevices. Solutions to these problems rely on our understanding of fundamental elements in nanoscience: isolated individual nanostructures and their assemblies. ❧ In this dissertation, we will address atomistic foundations of several problems of technological importance in nanoscience. Specifically, three basic problems are discussed: (1) embrittlement of nanocrystalline metal
Linked assets
University of Southern California Dissertations and Theses
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Asset Metadata
Creator
Yuan, Zaoshi
(author)
Core Title
Molecular dynamics simulations of nanostructures
School
Viterbi School of Engineering
Degree
Doctor of Philosophy
Degree Program
Materials Science
Publication Date
02/07/2013
Defense Date
12/10/2012
Publisher
University of Southern California
(original),
University of Southern California. Libraries
(digital)
Tag
brittle-to-ductile transition,embrittlement,GaAs,molecular dynamics,nanostructures,nanowire,OAI-PMH Harvest,percolation,phase transition,selective area growth,twin,ZnO
Language
English
Contributor
Electronically uploaded by the author
(provenance)
Advisor
Vashishta, Priya (
committee chair
), Goo, Edward K. (
committee member
), Nakano, Aiichiro (
committee member
), Rajiv, Kalia (
committee member
)
Creator Email
zaoshiyu@usc.edu,zaozaoyuan@gmail.com
Permanent Link (DOI)
https://doi.org/10.25549/usctheses-c3-219293
Unique identifier
UC11293774
Identifier
usctheses-c3-219293 (legacy record id)
Legacy Identifier
etd-YuanZaoshi-1431.pdf
Dmrecord
219293
Document Type
Dissertation
Rights
Yuan, Zaoshi
Type
texts
Source
University of Southern California
(contributing entity),
University of Southern California Dissertations and Theses
(collection)
Access Conditions
The author retains rights to his/her dissertation, thesis or other graduate work according to U.S. copyright law. Electronic access is being provided by the USC Libraries in agreement with the a...
Repository Name
University of Southern California Digital Library
Repository Location
USC Digital Library, University of Southern California, University Park Campus MC 2810, 3434 South Grand Avenue, 2nd Floor, Los Angeles, California 90089-2810, USA
Tags
brittle-to-ductile transition
embrittlement
GaAs
molecular dynamics
nanostructures
nanowire
percolation
phase transition
selective area growth
twin
ZnO