Structural and electronic properties of transition metal nanoalloys and magnetic compounds

S uc u al and elec onic p ope ies o ansi ion me al nanoalloys and magne ic compounds Genehmig e Abhandlung zu E langung des akademischen G ades eines Dok o s de Na u wissenscha en (D . e . na .) im P omo ionsp og amm Physik Weiche Ma e ie, Nich linea e Dynamik und Fes k¨o pe physik de Bay eu he G aduie enschule ¨u Ma hema ik und Na u wissenscha en on Linn Leppe gebo en in Chemni z 1. Gu ach e : P o . D . S ephan K¨ummel 2. Gu ach e : P o . D . Ma hias Schmid 3. Gu ach e : P o . D . Paul-Ge ha d Reinha d Tag de Ein eichung: 28. Mai 2013 Tag des Kolloqiums: 5. Sep embe 2013 I migh be w ong I could ha e swo n I saw a ligh coming on (Radiohead) Abs ac In ansi ion me al clus e s, po en ially p o i able echnological applica ions and ascina ing undamen al ques ions a e closely connec ed. Bime allic nanoalloys, e.g., ha e become inc easingly popula as hei pe o mance in ca alysis is o en supe io o hei pu e coun e pa s. Exempla y o his a e gold-pla inum (Au-P ) nanoalloys ha ha e been used as highly po en ca alys s in elec oca alysis and in a a ie y o oxida ion eac ions. Howe e , he me e exis ence o Au-P nanoalloys is as onishing, as Au and P canno be mixed in bulk o e a wide ange o composi ions. Fu he mo e, how a combina ion o Au and P in nanoalloys esul s in hei special p ope ies has no ye been de e mined conclusi ely. I has been shown in empi ical simula ions and i s -p inciples densi y unc ional heo y (DFT) calcula ions ha Au-P nanoalloys p e e ably a ange in a co e-shell mixing pa e n wi h Au o ming a shell a ound a P co e. This is in con adic ion o many expe imen al s udies ha epo he o ma ion o solid solu ions o Au and P . In he p esen wo k, his seeming disc epancy is add essed by simula ing x- ay di ac ion pa e ns ha a e expe imen ally used o cha ac e ize nanoalloys. I is shown ha he in e p e a ion o he di ac ion pa e ns elies on ques ionable assump ions and he e o e does no su ice as a de ini e cha ac e iza ion ool o Au-P nanoalloys. To shed ligh on he special ca aly ic p ope ies o Au-P nanoalloys unde a he di e en expe imen al condi ions, a ho ough in es iga ion o hei elec onic and s uc u al p ope ies has been ca ied ou . I is ound ha ea u es a o able o ca alysis in Au-P nanoalloys eme ge as a consequence o combining wo undamen al p ope ies: P con ibu es a high densi y o s a es close o he Fe mi le el, which p omo es chemical ac i i y. Au inc eases he s uc u al lexibili y o he Au-P sys em, which migh be bene icial o he o ma ion o ac i e and elemen -speci ic binding si es as well as egene a ion o he ca alys a e he eac ion. Al hough DFT o e s an a ac i e comp omise be ween compu a ional e o and accu acy o a heo e ical desc ip ion o Au-P nanoalloys, o he ansi ion me al compounds se e ely challenge exis ing DFT app oxima ions. Manganese (Mn) doped silicon (Si) clus e s ep esen an ideal model sys em o s udy he in e ac ion o a single magne ic impu i y wi h a semiconduc ing hos bo h expe imen ally and heo e ically. The ansi ion om exohed al (lowly coo dina ed) o endohed al (highly coo dina ed) doping ha occu s o Si clus e s wi h mo e han en a oms, is accompanied by comple e quenching o he magne ic momen o Mn. We show ha MnSi + 11 , he smalles endohed al clus e ound in expe imen , su e s s ongly om a well-known gene al p oblem o mos DFT app oxima ions: he sel -in e ac ion e o . Finally, a uni e sal co ela ion be ween magne ic momen and he coo dina ion o he Mn dopan is es ablished ha can be gene alized o ex ended sys ems and sugges s a ou e o s abilize he magne ic momen o bulk Mn-Si compounds. Ku zda s ellung In ¨ Ube gangsme allclus e n liegen gewinnb ingende echnische Anwendungen und undamen- ale F agen o mals nah beieinande . Dime allische Nanolegie ungen e euen sich beispiel- sweise g oße Belieb hei , weil sie in di e sen ka aly ischen Reak ionen den en sp ech-enden einen Me allen ¨ube legen sind. Gold-Pla in (Au-P ) Nanolegie ungen wu den als ¨auße s wi ksame Ka alysa o en in de Elek oka alyse und ¨u eine Reihe on Oxida ions eak ionen iden i izie . Genau genommen is jedoch die bloße Exis enz diese Nanolegie ungen e - s aunlich, da Au und P in ausgedehn en Sys emen kaum mischba sind. Bislang is zudem nich kla , au welche Weise eine Kombina ion on Au und P in eine Nanolegie ung zu speziellen Eigenscha en ¨uh . In empi ischen Simula ionen und nich -empi ischen Dich e unk ional heo ie (DFT) Rech- nungen konn e gezeig we den, dass Au-P Nanolegie ungen ein Au Schale P Ke n Mischungsmu- s e be o zugen. Dies s eh im Wide sp uch zu expe imen ellen S udien, in denen homogen gemisch e Clus e beobach e wu den. In de o liegenden A bei wi d diese scheinba e Disk epanz du ch Simula ion on R¨on genbeugungsmus e n un e such . Es wi d gezeig , dass die ¨ubliche Auswe ung diese Beugungsmus e au meh deu igen Annahmen be uh und somi nich zu alleinigen Cha ak e isie ung on Au-P Nanolegie ungen aus eich . Um Au schluss ¨ube die hohe ka aly ische Ak i i ¨a on Au-P Nanolegie ungen zu e langen, wu den ih e elek onischen und s uk u ellen Eigenscha en un e such . Es wi d gezeig , dass sich die g¨uns igen Eigenscha en on Au-P Nanolegie ungen als Konsequenz de Kombina ion zweie undamen ale Eigenscha en e geben k¨onnen: P ¨ag zu eine hohen Zus andsdich e am Fe mini eau bei, welche ¨o de lich ¨u die chemische Ak i i ¨a sein kann. Mi s eigendem Au-An eil s eig wiede um die s uk u elle Flexibili ¨a de Au-P Sys eme. Dies kann ¨u die Bildung ak i e und elemen spezi ische Ka alysezen en sowie zu Regene a ion des Ka alysa o s nach de Reak ion on Nu zen sein. DFT bie e ¨u die Behandlung on Au-P Nanolegie ungen einen gu en Komp omiss zwischen echne ischem Au wand und Genauigkei . Im zwei en Teil diese A bei geh es jedoch um eine ande e ¨ Ube gangsme all e bindung, welche exis ie ende N¨ahe ungen de DFT au eine ha e P obe s ell . Siliziumclus e (Si) do ie mi einem einzelnen Mangana om (Mn) sind ein ideales Modellsys em um die F age zu un e suchen, wie eine magne ische Ve un einigung mi einem halblei enden Wi sma e ial wechselwi k . De ¨ Ube gang on exohed ale (nied ige Koo dina ion) zu endohed ale (hohe Koo dina ion) Do ie ung inde ¨u Clus e mi meh als zehn Si-A omen s a und wi d on eine komple en Ausl¨oschung des magne ischen Momen s beglei e . In diese A bei wi d gezeig , dass MnSi + 11 , de kleins e expe imen ell iden i izie e endohed ale Clus e , besonde s s a k un e einem wohlbekann en P oblem iele DFT N¨ahe ungen leide : dem Selbs wechselwi kungs ehle . Abschließend wi d ein uni e selle Zusammenhang zwischen dem magne ischen Momen und de Koo dina ion des Mn-Do ie a oms gezeig , de auch au Fes k¨o pe sys eme ¨ube agen we den kann. Diese Zusammenhang e ¨o ne die M¨oglichkei das magne ische Momen on ausgedehn en Mn-Si Ve bindungen zu s abilisie en. Con en s 1 In oduc ion 1 2 Theo e ical and echnical amewo k 5 2.1 Founda ions o Densi y Func ional Theo y . . . . . . . . . . . . . . . . . . . 5 2.2 Exchange-co ela ion ene gy unc ionals . . . . . . . . . . . . . . . . . . . . 7 2.3 In e p e a ion o (gene alized) Kohn-Sham eigen alues . . . . . . . . . . . . 12 2.4 Pseudopo en ials ................................. 14 2.4.1 No m-conse ing pseudopo en ials . . . . . . . . . . . . . . . . . . . 15 2.4.2 Ene gy-adjus ed pseudopo en ials . . . . . . . . . . . . . . . . . . . . 16 2.4.3 P ojec o augmen ed wa es . . . . . . . . . . . . . . . . . . . . . . . 17 2.5 Me hods o geome y op imiza ion . . . . . . . . . . . . . . . . . . . . . . . 18 2.5.1 Simula ed annealing . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 2.5.2 The ”Big Bang” sea ch . . . . . . . . . . . . . . . . . . . . . . . . . 19 2.5.3 The Nudged Elas ic Band me hod . . . . . . . . . . . . . . . . . . . 20 3 The bime allic e ec in Au-P and Au-Pd nanoalloys 23 3.1 Au-P and Au-Pd nanoalloys in ca alysis . . . . . . . . . . . . . . . . . . . . 23 3.2 Geome ic s uc u e and mixing pa e ns o Au-P nanoalloys . . . . . . . . 25 3.3 Au-P alloys and Vega d’s law on he nanoscale . . . . . . . . . . . . . . . . 27 3.4 The in e play be ween luxionali y and elec onic s uc u e . . . . . . . . . 34 3.4.1 Elec onic s uc u e o Au-P nanoalloys . . . . . . . . . . . . . . . . 35 3.4.2 Elec onic s uc u e o Au-Pd nanoalloys . . . . . . . . . . . . . . . 39 3.4.3 The ole o Au in Au-P nanoalloys . . . . . . . . . . . . . . . . . . . 41 3.4.4 Combining elec onic s uc u e and luxionali y . . . . . . . . . . . . 45 4 Magne ic- o-nonmagne ic ansi ion in Mn-doped Si-clus e s 47 4.1 Dilu e magne ic semiconduc o s . . . . . . . . . . . . . . . . . . . . . . . . . 47 4.2 S ong co ela ion in Densi y Func ional Theo y . . . . . . . . . . . . . . . 49 4.3 A DFT-s udy o he magne ic- o-nonmagne ic ansi ion in Mn-doped Si- clus e s ...................................... 52 5 Summa y and ou look 61 5.1 Summa y ..................................... 61 6 Chap e 2 — Theo e ical and echnical amewo k he in e ac ion o he elec ons wi h all ex e nal po en ials and ˆ W = Pi<j w ( ˆ i,ˆ j ) is he elec on-elec on in e ac ion. The Hohenbe g-Kohn heo em can be subsumed in wo s a emen s: Fi s , o a gi en elec on-elec on in e ac ion ˆ W he e is a one- o-one mapping be ween he ex e nal po en ial ex (up o a physically i ele an cons an ), he (non-degene a e) g ound s a e |ψ0i esul ing om Sch ¨odinge ’s equa ion ˆ H|ψ0i = E0|ψ0i and he g ound s a e densi y n ( ). |ψ0i is a unique unc ional o he g ound s a e densi y, hence e e y g ound s a e obse able, and in pa icula he g ound s a e ene gy, is a densi y unc ional, oo. Second, using he Raleigh-Ri z a ia ional p inciple one can ob ain he exac g ound s a e densi y and ene gy co esponding o ex by minimizing he ene gy unc ional E[n] = F[n] + Z ex ( )n( )d3 . (2.1) F [ n ] = hψ [ n ] |ˆ T + ˆ W|ψ [ n ] i is a uni e sal unc ional, i.e., i is independen o ex ( ). This p o ides a simple and exac e o mula ion o Sch ¨odinge ’s equa ion. Howe e , as Hohenbe g and Kohn no ed al eady in 1964 [15]: “The majo pa o he complexi ies o he many- elec on p oblem a e associa ed wi h he de e mina ion o he uni e sal unc ional F[n].” The mos success ul app oach o de e mine F [ n ] was p oposed in 1965 by Kohn and Sham [16], who e o mula ed he ene gy unc ional o a sys em o N in e ac ing elec ons as E[n] = Ts[n] + EH[n] + Eex [n] + Exc[n],(2.2) whe e Ts [ n ] is he kine ic ene gy o a sys em o N non-in e ac ing elec ons. EH [ n ] is he classical elec os a ic Ha ee ene gy EH=e2 2Z Z n( )n( 0) | − 0|d3 d3 0(2.3) and Eex [n] desc ibes he in e ac ion be ween he elec ons and he ex e nal po en ial Eex [n] = Z ex ( )n( )d3 . (2.4) Exc [ n ], he exchange-co ela ion (xc) ene gy unc ional, is de ined as consis ing o e e y con ibu ion no ea ed by EHand Ts, i.e., Exc =T−Ts+W−EH.(2.5) He e, T is he ull kine ic ene gy o he in e ac ing elec on sys em and W is he elec on- elec on in e ac ion ene gy. Minimiza ion o Eq. (2.2) wi h espec o nσ , he o al spin densi y, and compa ison wi h he co esponding e m o a sys em o N non-in e ac ing elec ons leads o he same g ound s a e densi y i he N non-in e ac ing elec ons a e subjec o an e ec i e, local, mul iplica i e po en ial KS σ( ) = ex ( ) + H( ) + xc,σ( ),(2.6) in which H is he Ha ee po en ial H = e2Rn ( 0 ) /| − 0|d3 0 and xc,σ = δExc/δnσ he xc po en ial. Eq. (2.6) de ines he so-called KS po en ial. Rega ding he ques ion o 2.2. Exchange-co ela ion ene gy unc ionals 7 whe he such a po en ial exis s o all densi ies ( - ep esen abili y p oblem) he eade is e e ed o Re . [17] and e e ences he ein. In mos p ac ically ele an cases he densi y is - ep esen able. This means ha one can calcula e he g ound s a e densi y n ( ) o he many-elec on sys em in an ex e nal po en ial ex ( ) by sol ing he Sch ¨odinge -like one-pa icle equa ions −¯h2 2m∇2+ KS σ( )!ϕiσ( ) = εiσϕiσ( ),(2.7) and summing o e all occupied KS o bi als, ϕiσ( ), n( ) = X σ=↑,↓ nσ( ) = X σ=↑,↓ Nσ X i=1 iσ|ϕiσ( )|2.(2.8) The sum o e all occupa ion numbe s iσ yields he o al numbe o elec ons X σ=↑,↓ Nσ X i=1 iσ =N. (2.9) Equa ions (2.6) – (2.8) cons i u e he KS equa ions, ha ha e o be sol ed sel -consis en ly in mos p ac ical applica ions o DFT. The KS equa ions a e in p inciple an exac e o mula ion o he ull in e ac ing many- body p oblem o quan um mechanics. Howe e , all exchange and co ela ion e ec s ha go beyond he Ha ee ene gy EHand he non-in e ac ing kine ic ene gy Tsa e by de ini ion (Eq. (2.5) ) included in he xc ene gy Exc , which is in gene al no known exac ly. The mos impo an app oxima ions o Exc a e discussed in he ollowing sec ion. 2.2 Exchange-co ela ion ene gy unc ionals App oxima ions o he xc ene gy unc ional can oughly be di ided in o wo g oups: explici and implici densi y unc ionals. Commonly used unc ional app oxima ions o he i s kind a e he local densi y app oxima ion (LDA) and he gene alized g adien app oxima ion (GGA). Func ionals o he second g oup do no explici ly depend on he densi y bu on he o bi als, i.e., hey possess an implici dependence on he elec on densi y by i ue o he Hohenbe g-Kohn heo em. P ominen examples a e so-called hyb id unc ionals and ange-sepa a ed hyb id (RSH) unc ionals, me a-GGAs as well as he sel -in e ac ion co ec ion (SIC). A ho ough discussion o he p ope ies and limi s o all unc ionals is beyond he scope o his hesis. A comp ehensi e o e iew can be ound in Re . [18] and e e ences he ein. Rele an app oxima ions o he p esen wo k a e desc ibed in he ollowing. 8 Chap e 2 — Theo e ical and echnical amewo k The LDA 1 is one o he mos o en used app oxima ions o Exc and was in oduced al eady by Hohenbe g and Kohn [15]. I is based on he simple idea ha in a sys em in which he elec on densi y a ies only slowly in space, exc , he xc ene gy pe elec on, is app oxima ely equal o ehom xc in he homogeneous elec on gas. One hen ob ains Exc by in eg a ion o exc o e space. The exchange con ibu ion is exac ly known o be ELDA x[n] = −3e2 43 π1/3Zn( )4/3d3 . (2.10) The co ela ion pa has been compu ed wi h high accu acy using Mon e-Ca lo me hods [19]. Many widely used LDA ene gy unc ionals, e.g., he ones by Vosko, Wilke and Nusai [20], Pe dew and Zunge [21] and Pe dew and Wang [22], a e based on pa ame iza ions o hese ea ly Mon e-Ca lo calcula ions ex ended by known limi s and scaling laws de i ed om he exac Ehom c . The la ge numbe o sys ems o which he LDA leads o quali a i ely easonable esul s 2 is su p ising aking in o accoun ha o mos sys ems he assump ion o a nea ly homogeneous elec on densi y is an o e simpli ica ion. This can be a ionalized by in oducing he concep o he xc hole nxc ( , 0 ) = ρ2 ( , 0 ) /n ( ) −n ( 0 ) wi h ρ2 ( , 0 ) ep esen ing he elec on pai densi y. The xc hole has o obey he sum ule Znxc( , 0)d3 =−1.(2.11) The LDA’s xc hole sa is ies Eq. (2.11) e en in a he inhomogeneous si ua ions [23]. This leads o a sub le e o cancella ion be ween he exchange and he co ela ion pa o ELDA xc . A sys ema ic way o imp o e on he LDA is o include no only in o ma ion abou he elec on densi y a each poin , bu also on he a e o i s spa ial a ia ion. Such GGA unc ionals ha e he o m EGGA xc [n] = Z (n( ),∇n( ))d3 . (2.12) The unc ion ( n ( ) ,∇n ( )) is cons uc ed ei he by ying o sa is y as many exac cons ain s o he exac xc ene gy unc ional as possible (leading, e.g., o one o he mos popula GGAs o Pe dew, Bu ke and E nze ho (PBE) [24] o by i ing o la ge se s o es molecules (e.g., BLYP, consis ing o he exchange unc ional o Becke [25] and he co ela ion unc ional o Lee, Yang and Pa [26]). Al hough he LDA and he GGA pe o m sa is ac o ily in many p ac ical calcula ions, hei ailu e in o he s is d ama ic. This quali a i ely inco ec beha io can in mos cases be aced back o one common sou ce, he sel -in e ac ion e o (SIE) [21], which is a p oblem ha was ecognized al eady in Thomas-Fe mi heo y [27, 28], an ances o o DFT. The SIE is i ial o de ine in an one-elec on sys em wi h densi y n1 , in which he e is no elec on-elec on in e ac ion. He e, he exac xc ene gy (po en ial) and he Ha ee ene gy 1And he local spin densi y app oxima ion (LSDA) o spin-pola ized sys ems. 2 Al hough being a om chemical accu acy, which would equi e an e o o less han 0.04336 eV pe pa icle as compa ed o he expe imen . 2.2. Exchange-co ela ion ene gy unc ionals 9 (po en ial) ha e o cancel each o he EH[n1] + Exc[n1] = 0 (2.13) H[n1]( ) + xc[n1]( )=0 by de ini ion. An app oxima e unc ional mus he e o e sa is y Eq. (2.13) in o de o be ee o one-elec on sel -in e ac ion. This no ion can be ex ended o many-elec on sys ems by iden i ying single elec ons wi h KS o bi al densi ies niσ = iσ|ϕiσ|2 . Acco ding o Pe dew and Zunge [21, 29] one can hen de ine he one-elec on SIE in a many-elec on sys em as eiσ =EH[niσ] + Exc[niσ,0].(2.14) I X σ=↑,↓ Nσ X i=1 eiσ = 0 (2.15) he app oxima e unc ional is conside ed ee o one-elec on sel -in e ac ion. This de ini ion sugges s a s aigh o wa d way o co ec ing Eapp ox xc o sel -in e ac ion by simply sub ac ing he e oneous e ms om he app oxima e xc unc ional ESIC xc [n↑, n↓] = Eapp ox xc [n↑, n↓]−X σ=↑,↓ Nσ X i=1 (EH[niσ] + Eapp ox xc [niσ,0]).(2.16) This so-called sel -in e ac ion co ec ion (SIC) an example o an o bi al-dependen , implici densi y unc ional. Two concep ual in icacies a e associa ed wi h such unc ionals. Fi s , he unc ional de i a i e δExc/δnσ canno be e alua ed as s aigh o wa dly as o explici densi y unc ionals. One way o deal wi h his p oblem and a he same ime o s ay wi hin he concep ual ealm o KS DFT, is o use a unc ional de i a i e chain ule OEP xc,σ =δExc[{ϕjτ }] δnσ( )(2.17) =X α=↑,↓ Nα X i=1 ZδExc[{ϕjτ }] δϕiα( 0) δϕiα( 0) δnσ( )d3 0+ c.c. (2.18) =X α,β=↑,↓ Nα X i=1 Z Z δExc[{ϕjτ }] δϕiα( 0) δϕiα( 0) δ KS β( 00) δ KS β( 00) δnσ( )d3 0d3 00 + c.c.,(2.19) whe e α, β, σ and τ deno e he spin pola iza ion and c.c. he complex conjuga e. F om his exp ession one ob ains he op imized e ec i e po en ial (OEP) equa ion Nσ X i=1 iσ Zϕ∗ iσ( 0)[ OEP xc,σ ( 0)−uxc,iσ( 0)]GKS iσ ( 0, )ϕiσ( )d3 0+ c.c. = 0 (2.20) in which GKS iσ ( 0, ) is he KS G een’s unc ion GKS iσ ( 0, ) = ∞ X j=1 j6=i ϕjσ( 0)ϕ∗ jσ( ) εiσ −εjσ (2.21) 10 Chap e 2 — Theo e ical and echnical amewo k and uxc,iσ( ) an o bi al speci ic po en ial uxc,iσ( ) = 1 iσϕ∗ iσ( ) δExc[{ϕjτ }] δϕiσ( ).(2.22) The xc,σ ha sol es Eq. (2.20) is called he op imized e ec i e po en ial, as i yields he KS o bi als ha minimize he o al ene gy. Fo p ac ical applica ions, a ull OEP calcula ion is o en compu a ionally oo demanding, as Eq. (2.20) is an in eg al equa ion and con ains he comple e se o occupied and unoccupied KS eigen alues. The app oxima ion o K iege , Li and Ia a e (KLI) [30] KLI xc,σ( ) = 1 2nσ( ) Nσ X i=1 |ϕiσ( )|2[uxc,iσ( ) + (¯ KLI xc,iσ −¯uxc,iσ)] + c.c. (2.23) wi h ¯ KLI xc,iσ =Zϕ∗ iσ( 0) KLI xc,σ( 0)ϕiσ( 0)d3 0(2.24) ¯uxc,iσ =Zϕ∗ iσ( 0)uxc,iσ( 0)ϕiσ( 0)d3 0(2.25) p o ides an al e na i e ha educes he compu a ional e o , bu can in many cases yield esul s e y close o hose om a ull OEP calcula ion [18]. The second in icacy o o bi al-dependen unc ionals such as he SIC, is in ima ely ela ed o he quan um mechanical na u e (indis inguishabili y) o he in e ac ing many- elec on p oblem i sel : One-elec on o bi als a e an a i icial concep in oduced h ough he KS ansa z and he KS o bi als a e in no way unique among o he o bi al se s ha also sum up o he co ec g ound s a e densi y. This uni a y in a iance p oblem means ha di e en OEP can be cons uc ed depending on he chosen o bi al se . A gene aliza ion o he OEP me hod can be employed o deal wi h he uni a y in a iance p oblem ia a densi y-conse ing uni a y ans o ma ion o he KS o bi als [31]. A de ailed discussion o he OEP and app oxima ions o i can be ound in Re . [17] and [18]. The ambigui y o he one-elec on o bi al se s also di ec ly a ec s he de ini ion o a SIE in many-elec on sys ems. A emp s ha e been made o de ine a many-elec on SIE. The issue will be discussed in Sec. 4.2. Eq. (2.14) is use ul e en wi hou ca ying ou an ac ual SIC calcula ion. I p o ides a simple c i e ion o whe he he KS eigen alue spec um esul ing om a gi en app oxima e xc unc ional is physically eliable (see discussion in Sec. 2.3 and a p ac ical example in Sec. 4.3) [32, 33]. A second class o o bi al-dependen unc ionals a e me a-GGAs. They con ain he KS kine ic ene gy densi y τσ( ) = ¯h2 2m Nσ X i=1 iσ|∇ϕiσ( )|2(2.26) (and some imes e ms ∇2nσ ) and imp o e in some espec s upon semilocal unc ionals, e.g., hey in p inciple can achie e absence o he sel -co ela ion e o . Howe e , as comple e sel -in e ac ion absence is no gua an eed by me a-GGAs, hey sha e concep ual di icul ies 2.2. Exchange-co ela ion ene gy unc ionals 11 wi h he LDA and GGA. An example o a me a-GGA ele an o he co ec desc ip ion o he 2d-3d ansi ion in small Au-clus e anions [34] is he one by Tao, Pe dew, S a o e o and Scuse ia [35]. Hyb id unc ionals a e among he mos o en used xc unc ional app oxima ions and ha e become a a o i e ool o quan um chemis y. Such unc ionals a e cons uc ed o a ixed ac ion o exac exchange Eex x as well as semilocal exchange Eapp ox x and co ela ion Eapp ox c3 . The app oach was i s in oduced in 1993 by Becke [36], who p oposed a unc ional o m Ehyb xc =bEex x+ (1 −b)Eapp ox x+Eapp ox c,(2.27) in which Eex xdeno es he Fock in eg al Eex x=−e2 2X σ=↑,↓ Nσ X i,j=1 iσ jσ Z Z ϕ∗ iσ( )ϕ∗ jσ( 0)ϕjσ( )ϕiσ( 0) | − 0|d3 d3 0.(2.28) The pa ame e b is de e mined ei he by i ing o ex ensi e es se s o molecules o a ionalized by i ue o he adiaba ic connec ion o malism [37, 38]. An example o he la e app oach is he one-pa ame e hyb id PBE0, based on he PBE GGA [24] in which b = 0 . 25 [39]. Howe e , mos o oday’s hyb id unc ionals employ e en mo e pa ame e s, he mos p ominen example being he 3-pa ame e B3LYP unc ional [40, 41], ha con ains ac ions o he semilocal Becke unc ional [25] as Eapp ox x , he GGA by Lee, Yang and Pa [26] as Eapp ox cand he LDA pa ame iza ion by Vosko, Wilke and Nusai [20]. Finally, I wan o men ion he ange-sepa a ed hyb ids (RSHs). The unde lying concep o hese unc ionals is o sepa a e he elec on-elec on in e ac ion in o a long- ange and a sho - ange pa ia 1 | − 0|=e (ω| − 0|) | − 0| | {z } long ange +1−e (ω| − 0|) | − 0| | {z } sho ange .(2.29) The sc eening unc ion is o nume ical easons o en chosen as he e o unc ion and ω is an adjus able pa ame e ha de e mines a which leng h scale he sho - ange pa o Eq. (2.29) decays o ze o and he long- ange pa becomes dominan . The long- ange pa is gi en by he sc eened Fock in eg al, so ha one p ese es he ad an ages o employing semilocal exchange a sho ange, while inco po a ing 100% o he non-local exac exchange a long ange and hus ob ains he co ec asymp o ic beha io o he xc po en ial. Hence, RSHs a e pa icula ly use ul o he desc ip ion o cha ge ans e exci a ions, bu hey can also imp o e on g ound s a e p ope ies [42]. An example o such a unc ional is ω PBE, which is a RSH ha models he exchange hole o he PBE GGA a sho ange [43]. The ange-sepa a ion pa ame e can be ob ained in di e en ways: One can empi ically de e mine he alue o ω ha desc ibes he mochemis y o cha ge ans e exci a ions bes . Howe e , e idence sugges s ha in ac conside ably 3 P io o hei o mal jus i ica ion ia he gene alized KS amewo k, hese unc ionals had been seen as ”hyb ids” be ween DFT and Ha ee-Fock. 12 Chap e 2 — Theo e ical and echnical amewo k di e en ange-sepa a ion pa ame e s migh be necessa y depending on he pa icula sys em and p ope ies o in e es . Ano he app oach is he e o e o une ω o each sys em in such a way ha ce ain p ope ies o he exac unc ional (such as Eq. (2.33) in Sec. 2.3) a e ul illed [44]. O bi al-dependen unc ionals can be used in conjunc ion wi h he OEP me hod as explained abo e. The widely used hyb ids (and RSHs) a e p ac ically always implemen ed in a way which lea es he g ounds o he KS amewo k. This second app oach is jus i ied by he obse a ion ha i is possible o o mula e a so-called gene alized KS app oach in which he in e ac ing N -elec on sys em is mapped on o ano he in e ac ing auxilia y sys em, which can, howe e , s ill be ep esen ed by a single Sla e -de e minan [45]. 2.3 In e p e a ion o (gene alized) Kohn-Sham eigen alues Gi en hei s a us as me ely auxilia y quan i ies wi hin he KS amewo k, i migh be su p ising ha he KS eigen alues should ca y any physical meaning a all. Howe e , he success o DFT pa ly elies on he ac ha in an o e whelmingly la ge numbe o sys ems he eigen alue spec a ag ee, apa om a shi o he comple e spec um, ema kably well wi h expe imen ally ob ained quasipa icle ene gies. In he ea ly days o DFT his ag eemen was ega ded acciden al. La e , G¨o ling could show ha KS eigen alue di e ences ha e in ac a well-de ined physical meaning as exci a ion ene gies o ze o h o de in he elec on-elec on in e ac ion [46]. The de i a ion is based on linking he in e ac ing ull Hamil onian o he KS Hamil onian h ough [ˆ T+αˆ W+ˆ Vα]|ψα ni=Eα n|ψα ni.(2.30) The coupling cons an α c ea es a con inuous (adiaba ic) connec ion be ween he ully in e ac ing N -elec on sys em wi h α = 1 and ˆ Vα=1 = ˆ Vex and he non-in e ac ing KS sys em wi h α = 0 and ˆ Vα=0 = KS [37, 38]. Using G¨o ling-Le y pe u ba ion heo y [47] one can hen expand Eα n in a Taylo se ies in α and show ha exci a ion ene gies can igo ously be gained om g ound s a e DFT. In pa icula , he ze o h o de e m connec s he quasipa icle exci a ion ene gies o KS eigen alue di e ences. The p ac ical use ulness o G¨o ling’s inding o cou se depends on he quali y o his ze o h o de app oxima ion. Chong e al. could show ha he ene ge ically highes lying occupied KS le els can be in e p e ed as app oxima e, bu a he accu a e, elaxed e ical ioniza ion po en ials, p o ided ha hey a e compu ed using a high-quali y xc po en ial [48]. Al eady ea lie i was obse ed by Janak [49] ha ∂E ∂ iσ =εiσ,(2.31) whe e iσ is he ( ac ional) occupa ion o he i - h KS o bi al. A igo ous physical meaning, howe e , can only be assigned o he highes occupied KS eigen alue εHOMO , which is 2.3. In e p e a ion o (gene alized) Kohn-Sham eigen alues 13 iden ical o minus he ioniza ion po en ial I ( N ) o he ully in e ac ing physical sys em. This was p o ed by Almbladh and on Ba h [50] by de i ing di e en ial equa ions o he quasipa icle ampli udes Fn,σ( ) := hψN−1 n|ˆ Ψσ( )|ψN 0i,(2.32) whe e ˆ Ψσ ( ) is he elec on- ield ope a o which annihila es an elec on o spin σ a and ψN n a e he N -elec on eigens a es o he ull many-body Hamil onian. A close look a he asymp o ic decay o his quasipa icle ampli ude shows ha he asymp o ically leading ampli ude is ob ained o n = 0 and ha he densi y i sel is domina ed by F0,σ ( ) o | |→∞ . By compa ing his esul o he asymp o ic o m o he KS densi y, which is domina ed by he mos weakly decaying KS o bi al, one a i es a he iden i y εHOMO(N) = −I(N) = E(N)−E(N−1).(2.33) Simila ly, one can show o he elec on a ini y A ( N ), i.e., he ene gy gained by b inging in a pa icle om in ini y ha εHOMO(N+ 1) = −A(N) = E(N+ 1) −E(N).(2.34) E ( N− 1) , E ( N ) and E ( N + 1) a e he ene gies o he N− 1, N and N + 1 elec on sys em, espec i ely. In p ac ical calcula ions one o en aces he p oblem o no knowing whe he he used xc unc ional desc ibes he sys em o in e es accu a ely o no . As a ule o humb may coun , ha in sys ems in which he uppe occupied KS o bi als a e localized on a simila leng h scale, s anda d xc unc ionals esul in eigen alue spec a ha a e physically eliable [21]. The SIE in such sys ems, ha can be e alua ed using Eq. (2.14) , a ec s all ele an o bi als in he same way and he e o e only amoun s o a common shi o he eigen alue spec um [32]. In sys ems in which he uppe o bi als di e in hei deg ee o localiza ion, he spec um will be dis o ed as a esul o he SIE a ec ing he o bi als di e en ly. In his sense, e alua ion o Eq. (2.14) can se e as a wa ning o sys ems ha a e s ongly a ec ed by he SIE. No e, howe e , ha he ela ion be ween o bi al localiza ion and he SIE is no i ial [51]. The close connec ion be ween he SIE and he eliabili y o an eigen alue spec um sugges s ha sel -in e ac ion ee app oaches (in he sense o Eq. (2.15) ) should yield good ag eemen wi h expe imen in cases in which semilocal unc ionals ail and indeed his has been shown o a numbe o sys ems (see e.g. Re . [32]). Howe e , hese app oaches a e nume ically expensi e, especially in cases in which inding he ene ge ically mos a o able posi ion o he nuclei poses addi ional p oblems (see Sec. 2.5). The commonly used hyb id unc ionals can a leas pa ly cancel he SIE by employing a ac ion o exac exchange. As men ioned ea lie , hyb id unc ionals a e ypically implemen ed in a GKS amewo k. The GKS equa ions −¯h2 2m∇2+ ex ( ) + H([n], ) + c,σ([n], b, ) +bˆ ex x,σ[n] + (1 −b) x,σ([n], )ϕiσ( ) = εGKS iσ ϕiσ( ), (2.35) 14 Chap e 2 — Theo e ical and echnical amewo k in which ˆ ex x,σ[n]ϕiσ( ) = −e2X σ=↑,↓ Nσ X j=1 Zϕjσ( )ϕ∗ jσ( 0) | − 0|ϕiσ( 0)d3 0(2.36) deno es he non-local Fock ope a o , a e by cons uc ion exac , jus as he KS equa ions. The esul ing eigen alues εGKS iσ di e om he exac KS eigen alues by b∆ x,σ,i =bhϕiσ( )|ˆ ex x,σ[n]− x,σ([n], )|ϕiσ( )i,(2.37) i one neglec s di e ences in he KS and GKS o bi als and he co ela ion po en ial [45]. As ∆ x,N = 0 o he highes occupied GKS eigen alue, he ela ion εGKS HOMO = −I ( N ) holds also o he GKS case. The e is again no such equali y o all he o he GKS eigen alues. Howe e , one can show ha including a ac ion o he non-local exchange mimics a pa ial sel -in e ac ion co ec ion [52]. Consequen ly, GKS eigen alues ag ee well wi h expe imen al esul s o many cases in which semilocal unc ionals ail. 2.4 Pseudopo en ials The basic idea o eplacing he s ong Coulomb po en ial o he nucleus and he sc eening e ec o he igh ly bound co e elec ons by an e ec i e co e po en ial (commonly e e ed o as pseudopo en ial), has i s oo s in he simple no ion ha only he alence elec ons o an a om de e mine i s chemical p ope ies. The app oach was o iginally in oduced by Fe mi when he s udied low ene gy elec on sca e ing om a oms [53]. In ac , he aim o he pseudopo en ial cons uc ion is o ind an e ec i e po en ial ha mimics he sca e ing p ope ies o nucleus and co e elec ons eliably o e a ce ain ene gy ange. The deg ees o eedom o pseudopo en ial gene a ion hen allow o de ise hem in such a way as o minimize he compu a ional e o . Fi s ly, by educing he numbe o elec ons ha ha e o be conside ed explici ly in he DFT calcula ion. Secondly, because pseudopo en ials allow he nume ical desc ip ion o po en ials and o bi als on much coa se g ids o wi h ewe plane wa e componen s han would be necessa y i he highly oscilla o y s uc u e o he o bi als in he co e- egion had o be ep esen ed comple ely. In he p esen hesis pseudopo en ials en e he s age a ye ano he poin . When one deals wi h hea y-elemen ansi ion me al compounds such as Au and P , ela i is ic e ec s such as spin-o bi coupling and he ela i is ic mass inc ease o elec ons in he co e egion a e c ucial o he accu a e desc ip ion o he elec onic, and hus also he geome ic s uc u e o he compounds [54]. Pseudopo en ials o e ways in which hese e ec s can be accoun ed o implici ly. An impo an concep unde lying mos pseudopo en ial gene a ion schemes is he ozen-co e app oxima ion: The co e s a es, e alua ed in an all-elec on a omic e e ence calcula ion, a e assumed no o change i in a di e en en i onmen , i.e., in a molecule o solid. Fo his eason he adius o he co e egion c is a c ucial pa ame e de e mining ans e abili y and accu acy o he pseudopo en ial. 2.4. Pseudopo en ials 15 2.4.1 No m-conse ing pseudopo en ials The spi i o no m-conse a ion is o s a he cons uc ion o he pseudopo en ial om he all-elec on o bi als ha s em om a sel -consis en solu ion o he adial KS equa ion 4 ( o a spin-unpola ized case) −¯h2 2m d2 d 2+¯h2l(l+ 1) 2m 2+ KS([n], )! Ril( ) = εil Ril( ),(2.38) in which Ril = ϕil and l deno es he angula momen um quan um numbe . The pseudopo- en ial has o ul ill ou p ope ies o be conside ed no m-conse ing: Fi s , all-elec on (AE) and pseudo (PP) alence eigen alues mus ag ee o a chosen a omic e e ence con igu a ion. Second, all-elec on and pseudo o bi als ag ee beyond a chosen co e adius c . Thi d, he in eg a ed cha ge inside c o he all-elec on and he pseudo cha ge densi ies ag ee o each alence s a e Z c 0|RPP il ( )|2 2d =Z c 0|RAE il |2 2d , (2.39) as his ensu es ha he o al cha ge in he co e egion is conse ed. And las , he loga i hmic de i a i es o he all-elec on and he pseudo o bi als and hei i s ene gy de i a i es ag ee o > c [56]. The eedom ha is le in cons uc ing he pseudopo en ial can hen be used o make i as smoo h as possible a he same ime ensu ing ans e abili y o as many di e en chemical en i onmen s as possible. Di e en schemes o pseudopo en ial gene a ion we e p oposed, e.g., by Bachele , Hamann and Schl¨u e [57] and by T oullie and Ma ins [58]. By in e ing he adial KS equa ions one hen ob ains he sc eened pseudopo en ial om he pseudo wa e unc ion PP sc eened,l =εl−¯h2 2m l(l+ 1) 2−1 RPP l( ) d2 d 2[ RPP l( )]!.(2.40) In he las s ep he sc eening o he alence elec ons has o be emo ed o ob ain an unsc eened (”ba e”) ionic pseudopo en ial PP ionic,l( ) = PP sc eened,l( )− PP H[n al]( )− PP xc [n al( )] (2.41) The esul ing pseudopo en ial di e s o e e y angula momen um componen l , i.e., he e ec i e ex e nal po en ial in he KS equa ions is no local anymo e. A u he ans o ma- ion sugges ed by Kleinman and Bylande b ings Eq. (2.41) in a sepa able non-local o m ha educes compu a ion ime and s o age space. T oullie -Ma ins pseudopo en ials in Kleinman-Bylande o m a e used h oughou his wo k o calcula ions on eal-space g ids using a local e sion o he PARSEC p og am package [59], e.g., o he calcula ion o he SIE o Mn-doped Si clus e s in Sec. 4.3. 4 I ela i is ic e ec s a e o be included one has o use Di ac’s o mula ion o he kine ic ene gy. The Sch ¨odinge -like Eq. (2.38) is hen eplaced by a pai o coupled equa ions o mino and majo o bi al componen s, which ou side he co e adius educe app oxima ely o a Sch ¨odinge - ype equa ion o he majo o bi al componen [55]. 3 The bime allic e ec in Au-P and Au-Pd nanoalloys 3.1 Au-P and Au-Pd nanoalloys in ca alysis Alloying o wo (o mo e) elemen s is a p omising ou e o enhance he ca aly ic ac i i y o ansi ion me al NP and imp o e hei elemen -speci ic selec i i y [78]. Loosely based on A is o le’s bon mo ha “ he whole is g ea e han he sum o i s pa s”, hese e ec s a e equen ly dubbed syne gis ic. Au-P nanoalloys a e a p ominen example o a sys em which exhibi s inc eased u no e a es in a a ie y o oxida ion eac ions [79–81] as well as en icing p ope ies in elec oca aly ic applica ions [82–90] compa ed o hei pu e coun e pa s. Au-P nanoalloys a e bene icial because hey educe he well-known p oblem o ”poisoning” o P ca alys anodes used in di ec me hanol uel cells wi h s ongly adso bed in e media e CO-species. Fu he mo e, inc eased u no e a es allow o ca y ou eac ions a oom empe a u e and wi hou possibly oxic sol en s, hus causing less impac on he en i onmen . Finally, by changing he Au-P composi ion one can une hei ca aly ic p ope ies which makes Au-P nanoalloys po en ially use ul o many di e en ca aly ic eac ions. When looking deepe in o hese p oblems, se e al puzzles appea . The i s puzzle is he miscibili y o Au and P on he nanoscale. I is well known ha he bulk phase diag am o Au-P alloys exhibi s a la ge miscibili y gap o a wide ange o Au-P composi ions. In going o he nanoscale i has been epo ed ha Au-P NP can o m ue solid solu ions. Such Au-P nanoalloys ha e, e.g., been syn hesized in he mally e apo a ed a y amine ilms, whe e hey we e alloyed a low empe a u e [91], in sphe ical polyelec oly e b ushes (SPBs) [80] and in ionic liquids by a spu e deposi ion echnique 24 Chap e 3 — The bime allic e ec in Au-P and Au-Pd nanoalloys [92]. O he g oups ha e ob ained s able Au-P nanoalloys by capping hem wi h hiola es [82, 93] o by suppo ing hem on silica [79]. co e-shell andom laye ed Figu e 3.1: Th ee possible miscibili y pa e ns in bime allic NP. F om le o igh : co e-shell, andomly mixed and lay- e ed. O he pa e ns such as se e al lay- e s/shells o an o de ed solid solu ion a e possible. Theo y, in con as , consis en ly p edic s a Au shell -P co e mixing pa e n as ene ge ically mos a o able [94–97]. An illus a ion o di e en mixing pa e ns in NP is gi en in Fig. 3.1. Theo e ical s udies o clus e s in he expe i- men ally ele an size ange o se e al hund ed up o housands o a oms ha e been based on semi- empi ical MD simula ions and we e mos ly con- ce ned wi h he mel ing beha iou o he Au-P NP [98–101]. A global geome y op imiza ion using a gene ic algo i hm wi h a semiempi ical Gup a- many-body po en ial o Au-P clus e s wi h 2–100 a oms has been pe o med by Logsdail e al. [102]. Fi s -p inciples DFT s udies exis ed, up o he beginning o his wo k, only o e y small clus e s wi h 2–13 a oms [103, 104]. Howe e , mo e ecen s udies on la ge clus e s (including one ha will b ie ly be p esen ed in Sec. 3.2), ha e con i med ha Au-P NP p e e a co e-shell mixing pa e n [94–97]. This seeming disc epancy o heo y and expe imen is subjec o Sec. 3.3. The second Au-P nanoalloy puzzle is hei enhanced ca aly ic ac i i y as compa ed o hei pu e coun e pa s. Conside ing he known bulk elec onic p ope ies o P and Au 1 i is no pe se clea , why a combina ion o P and Au should lead o enhanced ca aly ic ac i i y a all (see also Sec. 3.4.1). Addi ionally, while i is known how he elec onic s uc u e o Au changes upon going om he bulk o small NP [2], simila insigh s in o he elec onic s uc u e o Au-P nanoalloys a e a e. Explana ions o he special p ope ies o Au-P nanoalloys ha e been a emp ed in se e al s udies, mainly concen a ing on he local elec onic s uc u e o P [85, 95, 104, 106]. A de ailed unde s anding o he concu ence o Au and P leading o a o able ca aly ic p ope ies in a a ie y o di e en eac ions, i.e., independen o he ype o suppo o ma ix he NP a e immobilized in, possible sol en s and di e en eac ion empe a u es, is lacking. Puzzle numbe wo will be subjec o Sec. 3.4. Al hough elemen s om he P -g oup a e chemically simila , he abo e ma e s a e sligh ly di e en o Au-Pd nanoalloys. The s uc u al cha ac e iza ion o Au-Pd NP is much easie as compa ed o Au-P NP, as Au and Pd’s a omic numbe s a e su icien ly di e en o allow o he use o high- esolu ion echniques, e.g., high-angle-annula -da k- ield ansmission elec on mic oscopy [107]. Syne gis ic e ec s ha e been obse ed o Au-Pd NP ca alys s in a a ie y o eac ions. Two ecen examples a e Au-Pd nanoalloys used o elec oca aly ic H 2 O 2 p oduc ion [108] and o sol en - ee oxida ion o p ima y ca bon- hyd ogen bonds [107]. As he ollowing sec ions will ocus p ima ily on Au-P nanoalloys, I e e he eade o Re . [78] and e e ences he ein, o a summa y o s udies in which a syne gis ic e ec o Au-Pd ca alys s was obse ed. Insigh s in o he elec onic s uc u e o Au-Pd nanoalloys and hei impac on Au-Pd ca alysis a e discussed in Sec. 3.4.1. 1P esen ed in a nu shell in “Why gold is he nobles o all me als” by Hamme e al. in Re . [105]. 3.2. Geome ic s uc u e and mixing pa e ns o Au-P nanoalloys 25 3.2 Geome ic s uc u e and mixing pa e ns o Au-P nanoalloys This sec ion gi es a sho summa y o esul s ha a e discussed in de ail in Re . [109]. Mos o he nanoalloy p ope ies ha will be iden i ied in he ollowing sec ions as being a o able o ca aly ic ac i i y a e o a la ge ex end independen o he exac geome ic s uc u e o he clus e s. S ill, ene ge ically low-lying Au-P isome s a e a well-chosen s a ing poin o he s udy o hese p ope ies. In combined e o s o heo y and expe imen , he g ound s a e s uc u e o many pu e neu al and cha ged Au clus e s has been de e mined. A p ominen example a e Au 20 and Au − 20 , whose g ound s a e s uc u e is a egula e ahed on as e ealed by gas phase in a ed spec oscopy [110] and pho oelec on spec oscopy [68]. Au 20 is an excellen oy model o he p esen s udy as i ep esen s a cu ou o he cc la ice in which bo h bulk Au and P c ys allize. Fu he mo e, many o he expe imen ally obse ed Au-P nanoalloys ha e been shown o be la gely ace ed and bulk-like (e.g. [80]). Figu e 3.2: The e a e h ee symme y-inequi alen ways how o eplace one Au a om in he Au 20 e ahed on by a P a om. By sys ema ically eplacing Au by P a oms in Au 20 one inds ha he ene ge ically lowes lying homo op is always ha in which P is as highly coo dina ed as possible. In going om Au 20 o Au 19 P 1 , as illus a ed in Fig. 3.2, he ene ge ically lowes lying homo op is he one in which P occupies a posi ion in he middle o one o he ou ace s. As a esul o his g ow h pa e n a Au shell P co e s uc u e eme ges na u ally. Tha he co e-shell mixing pa e n is no a me e a i ac was es ed by compa ing homo ops and isome s wi h ixed Au-P composi ion especially o s uc u al mo i s o he han he a he special e ahed al s uc u e as well as o la ge clus e s. The s abili y o a a ie y o s uc u es was es ed using simula ed annealing (see Sec. 2.5). In hese simula ions he clus e s we e hea ed o a empe a u e o 600 K and subsequen ly slowly annealed. Impo an ly, he co e-shell mixing pa e n emains s able h oughou he en i e simula ion and can he e o e be ega ded as being s able e en a ele a ed empe a u es. Fig. 3.3 shows, as an example, di e en Au20P 20 isome s om A o F wi h inc easing o al ene gy. No e ha e en a comple e seg ega ion o he Au and he P componen (C and D) is ene ge ically mo e a o able han he andom mixing pa e n (E and F). Compu a ional de ails on he geome y op imiza ion and he accu acy o he used xc unc ional and basis se s as well as s uc u es o e ahed al and amo phous Au n P 20−n clus e s can be ound in Re . [109] and [94]. In his wo k, Au-P NP wi h ≈ 1000 alence elec ons, i.e., 60-a om clus e s, we e he la ges sys ems o which ull local geome y op imiza ions we e pe o med. The Gup a po en ial based Au 30 P 30 s uc u e [102] was used as a s a ing poin o hese calcula ions. Howe e , conside ing ha expe imen ally ele an nanoca alys s ange be ween sizes o 26 Chap e 3 — The bime allic e ec in Au-P and Au-Pd nanoalloys 1 and 4 nm, we need e en la ge sys ems o exclude ha he obse ed e ec s a e only due o he compa ably small size o he clus e s. A B C D E F Figu e 3.3: S uc u es a e named a e he geome y ha se ed as a s a ing poin o he geom y op imiza ion ei he “Gup a” [102] o “Su on Chen” [111]. A: Gup a co e-shell, B: Su on-Chen co e-shell, C: Su on-Chen laye ed, D: Gup a laye ed, E: Gup a andom, F: Su on-Chen andom. Two s a egies we e used o o e come he com- pu a ional limi a ions ha i s p inciples DFT cal- cula ions ace o sys ems wi h signi ican ly mo e han 1000 alence elec ons. Fi s ly, we cons uc ed unca ed oc ahed al clus e s wi h closed a omic shells, i.e., wi h 38, 201, 586 and 1289 a oms as illus- a ed in Fig. 3.4 and op imized hem in empi ical MD simula ions. These clus e s a e, analogously o Au 20 , cu ou s o he cc la ice and a e used in he s udies discussed in Sec. 3.3 and 3.4. Clus e s wi h egula polyhed al s uc u es and closed a omic shells a e some imes called “magic”, in analogy o he magic elec on numbe s ha we e disco e ed by Knigh e al. in 1984 in sodium clus e s [112]. This seminal disco e y igge ed he de elopmen o clus e physics much beyond he ealm o he simple sphe ical jellium model which was used o explain he abundance o sodium clus e s con aining 2, 8, 20, 40,... a oms. Elec onic shell closu e e ec s wi hin he jellium model a e esponsible o he s abili y o hese clus e sizes in he simple alkali me als and e en in Au clus e s, as hei 6ss a es o m a good ee elec on gas [2]. 1 nm 2 nm 3 nm 4 nm Figu e 3.4: T unca ed oc ahed a wi h ( om le o igh ) 38, 201, 586 and 1289 a oms. These a omic numbe s a e called magic, because hey co espond o clus e s ha ing only closed shells o a oms. The app oxima e diame e s o hese clus e s a e 1, 2, 3 and 4 nm. Fo la ge clus e s he elec onic shell closu e e ec is supe imposed by he closu e o a omic shells, a p ominen example again being sodium clus e s ha exhibi icosahed al shell closu es [113]. In sodium clus e s wi h a ew ens o a oms, elec- onic and ionic shell closu e e ec s wo k oge he in de e mining he s uc u e o he clus e s, while o la ge clus e s, s a ing wi h Na 55 , ionic shell closu e e ec s win o e elec onic ones esul ing in nea -ly sphe ical clus e s [114]. Howe e , I wan o s ess ha unca ed oc ahed al s uc u es migh no be a s uc u al mo i ound in smalle Au clus- e s. The impo an poin is ha hey a e ace ed and cc-bulk-like, which makes hem a aluable model o he ca alys s obse ed expe imen- ally. A second app oach o ca alysis on Au-P NP is o model hem as pe iodic su ace slabs. The mo i a ion o his is wo old: Fi s ly, much o he ca aly ic ac i i y o NP can be a ibu ed o hei high su ace o olume a io. Secondly, a syne gis ic e ec o Au and P has also been obse ed o bime allic su aces [115]. Au-P su aces a e s udied in Sec. 3.4. In summa y, he ollowing Au-P s uc u es a e used in he p esen hesis o de e mine s uc u al and elec onic p ope ies o Au-P sys ems: 20-a om e ahed al and amo phous low-ene gy isome s, 40-a om and 60-a om s uc u es based on empi ical po en ials and ully 3.3. Au-P alloys and Vega d’s law on he nanoscale 27 op imized using DFT, unca ed oc ahed al model s uc u es wi h 38, 201, 586 and 1289 a oms o which di e en a angemen s o Au and P , e.g., in co e-shell and andom mixing pa e ns can be cons uc ed, and inally bime allic su ace slabs. 3.3 Au-P alloys and Vega d’s law on he nanoscale All esul s o his sec ion a e published in Re . [116]. In he p e ious wo sec ions I discussed ha i s -p inciples DFT calcula ions con i m ha low-ene gy s uc u es o ee Au-P clus e s indeed exhibi an Au shell P co e mixing pa e n and hus con i m he esul s o semi-empi ical MD simula ions men ioned in Sec. 3.1. The seeming disc epancy be ween heo y (co e-shell) and expe imen (solid solu ion) mus hus be sea ched o elsewhe e. Figu e 3.5 : A ske ch o he i s wo B agg peaks as ypically obse ed in XRD measu emen s on Au-P NP. F om hese di ac ion pa e ns i can p esumably be in e ed ha he sys- em unde s udy (blue line) is a ue solid solu ion. The a ows indica e he posi ion o he i s B agg peak. pu e Au pu e P nanoalloy (111) (200) B agg angle [2 ] 30 35 40 45 50 0 10 20 30 in ensi y [a b. uni s] The nex s ep is o unde s and he assump ions ha unde lie he expe imen al cha ac- e iza ion o Au-P NP. A p ominen cha ac e iza ion me hod is he use o x- ay di ac ion (XRD) echniques. The esul ing di ac ion pa e ns a e ypically analyzed as ske ched in Fig. 3.5. The o ange and black lines indica e he i s wo B agg peaks o pu e Au and pu e P NP as obse ed in small angle x- ay sca e ing expe imen s. These peaks can be indexed in o an cc- ype la ice. Fo each se o Mille indices ( hkl ) one can hen use B agg’s equa ion a = λ√h2+k2+l2 2 sin θ o de e mine a la ice pa ame e a om he sca e ing angle θ. He e, λis he wa e leng h o he incoming x- ays. The same can be done o he di ac ion pa e n o he bime allic NP. Addi ionally, depending on he expe imen al esolu ion, one can use he peak shape o de e mine, whe he he sys em o in e es shows any sign o demixing o Au and P (in which case one would expec dis inc peaks o he Au- and he P -phase) o whe he i o ms a eal alloy (only one clea peak). The blue line in Fig. 3.5 would, ollowing his easoning, belong o a andomly mixed Au-P NP. Bulk alloys o en obey Vega d’s law [117], which s a es ha he la ice pa ame e o a wo-componen alloy can be de e mined by linea ly in e pola ing be ween he la ice pa ame e s o he wo componen s ha o m he alloy. Hence in an alloy ha consis s o 50% o bo h elemen s, he a e age la ice pa ame e would be jus he a e age o he la ice pa ame e s o bo h pu e c ys als. XRD cha ac e iza ion now e eals ha he la ice 28 Chap e 3 — The bime allic e ec in Au-P and Au-Pd nanoalloys pa ame e o Au-P NP goes linea ly wi h he Au-P a io as well. This inding is aken as e idence ha Au-P NP o m ue alloys (see, e.g., Re . [80, 93]). The ”Vega d-analysis” hus assumes i s ly ha a phase sepa a ion mus be isible in he di ac ion pa e n, secondly ha Vega d’s law is alid and can be used o de e mine he composi ion o he alloy and hi dly ha Vega d’s law is only alid o andomly mixed alloys. By simula ing he XRD pa e ns o pu e Au and P , as well as o andomly mixed and co e-shell NP one can hus es he alidi y o hese assump ions. The sca e ing in ensi y Io an ensemble o a oms is gi en by Debye’s equa ion I=X mX n m n sin kRmn kRmn ,(3.1) whe e n and m a e he o m ac o s o a oms n and m , Rmn = |Rm−Rn| is he dis ance o wo a oms a posi ions Rmand Rnand k= 4πsin θ/λ is he wa e numbe . Rmn 2θ Figu e 3.6: A ske ch o he sca - e ing o ligh a an ensemble o andomly o ien ed a oms, whe e he di e ence ec o Rmn = Rm−Rn is allowed o ake all o ien a ions in space, i.e., all po- si ions on he black ci cle wi h equal p obabili y (adap ed om [118]). The gene al sca e ing equa ion (3.1) o a oms which a e andomly o ien ed in space is easily de i ed by conside ing ha one ob ains he sca e ed x- ay in ensi y by summing o e he sca e ing ampli udes o x- ays sca e ed by di e en a oms I=X m me(2πi/λ)(s−s0)RmX n ne(−2πi/λ)(s−s0)Rn(3.2) =X mX n m ne(2πi/λ)(s−s0)Rmn , whe e s and s0 a e he di ec ions o incoming and ou going x- ays, as ske ched in Fig. 3.6. Eq. (3.2) is o cou se alid o sca e ing on any ensemble o a oms. We now assume ha his ensemble akes all o ien a ions in space wi h equal p obabili y, i.e., ha Rmn ends a all poin s o he sphe e ske ched in Fig. 3.6 (in 2d) wi h equal p obabili y. The spa ial a e age o he exponen ial e m he2πi/λ(s−s0)Rmn i can wi h k = 4 πsin θ/λ and ( s− s0)Rmn = 2 sin θRmn cos φbe w i en as heikRmn cos φi=1 4πR2 mn ZdAeikRmn cos φ =1 4πR2 mn Z2π 0 dϕ Zπ 0 dφR2 mn sin φeikRmn cos φ =sin kRmn kRmn , and leads o Debye’s sca e ing equa ion 3.1. Addi ionally, empe a u e e ec s ha cause a omic ib a ions independen om each o he could be aken in o accoun by including a Debye-Walle ac o [119, 120] in he Debye 3.3. Au-P alloys and Vega d’s law on he nanoscale 29 Table 3.1 : A e age bond leng hs and hei s anda d de ia ions o co e-shell and andom Au 22 P 38 (37% Au) as ob ained om DFT and MD op imiza- ion. DFT Au-Au [˚ A] Au-P [˚ A] P -P [˚ A] co e-shell 2.88 ±0.03 2.85 ±0.03 2.76 ±0.05 andom 2.90 ±0.05 2.82 ±0.06 2.72 ±0.06 MD Au-Au [˚ A] Au-P [˚ A] P -P [˚ A] co e-shell 2.74 ±0.03 2.70 ±0.04 2.69±0.04 andom 2.73 ±0.04 2.70 ±0.04 2.67 ±0.04 equa ion (3.1) . We do no include his ac o he e, as we will explici ly ake empe a u e in o accoun by pe o ming MD simula ions a ele a ed empe a u es. A easonable simula ion o XRD pa e ns equi es wo impo an condi ions o be me . Fi s ly, he size o he NP needs o be be ween 3 and 4 nm o con o m wi h he expe imen al size ange. Secondly, he in e a omic bond leng hs need o be ealis ic, as hey a e he decisi e inpu o Eq. (3.1) . As al eady men ioned, hese equi emen s pose se e e di icul ies o DFT, as compu a ional e o o bids o ea NP o such size, i.e., wi h mo e han 1000 Au and P a oms. On he o he hand, he eliabili y o bond leng h es ima es is bes i a i s -p inciples me hod is used o geome y op imiza ion. We assess his p oblem in he ollowing way. The 1289-a om unca ed oc ahed on as shown on he igh hand side o Fig. 3.4 is used as a s a ing poin o geome y op imiza ion. A pu e Au and P , a Au shell P co e wi h one and wo Au-shells (co esponding o he NP Au 484 P 805 and Au 830 P 459 espec i ely) and inally a andomly mixed NP wi h exac ly he same Au-P a io as he co e-shell pa icle Au484P 805 we e cons uc ed based on he unca ed oc ahed al geome y. The geome y op imiza ion o hese sys ems was ca ied ou using MD simula ions, in which he po en ial ene gy was modelled by he many-body Su on-Chen po en ial [121], ha has been employed success ully in simula ions o a a ie y o nanoalloys [100, 122–124]. The Su on-Chen pa ame e s o he Au-P in e ac ion we e ob ained om he pa ame e s o he Au-Au and P -P in e ac ion using a combina ion ule [100]. Each sys em was i s elaxed a 1 K and subsequen ly p opaga ed o 500 ps a a empe a u e o 300 K in a canonical ensemble using E ans’ he mos a [125] and acuum bounda y condi ions. E e y 20 ps a s uc u e ”snapsho ”, yielding a se o a omic posi ions {Rm} , was aken. These s uc u es we e hen op imized wi h a conjuga e g adien p ocedu e. Fo each sys em he op imized geome y wi h he lowes ene gy was aken as he inal s uc u e o u he analysis. All MD simula ions in his and Sec. 3.4.3 we e ca ied ou by Rod igo Q. Albuque que using he DL-POLY p og am package [126]. The second pa o ou s a egy consis s o de e mining he Au-Au, P -P and Au-P bond leng hs o DFT-op imized co e-shell and andom geome ies. We do no expec he bond leng hs esuling om MD simula ions o equal he DFT bond leng h, because he Su on-Chen pa ame e s a e i ed o bulk p ope ies and bulk bond leng hs a e ypically sligh ly di e en om NP bond leng hs. Howe e , wha is decisi e he e is, ha he ela i e di e ences be ween he Au-Au, P -P and Au-P bond leng hs esul ing om MD and DFT op imiza ion, espec i ely, a e simila . As one o he la ge sys ems whe e s uc u e op imiza ion wi h DFT is s ill possible we chose he 60-a om clus e Au 22 P 38 . I s Au con en o abou 37% is simila o ha o he 4 nm clus e Au 459 P 830 ha is he main ocus o ou s udy. We gene a ed s a ing 30 Chap e 3 — The bime allic e ec in Au-P and Au-Pd nanoalloys Au con en me hod Au-Au [˚ A] Au-P [˚ A] P -P [˚ A] 87 % DFT 2.87 ±0.02 2.83 ±0.03 2.72 ±0.02 MD 2.75 ±0.01 2.71 ±0.02 2.73 ±0.06 50 % DFT 2.87 ±0.08 2.83 ±0.05 2.65 ±0.05 MD 2.73 ±0.06 2.68 ±0.05 2.65 ±0.03 13 % DFT none 2.84 ±0.04 2.72 ±0.04 MD none 2.68 ±0.04 2.65 ±0.02 Table 3.2 : A e age bond leng hs and s anda d de- ia ions ob ained om DFT and MD op imiza- ion o andomly mixed 38-a omic clus e s wi h di e en Au-P mixing a ios. s uc u es by aking he coo dina es om he 60-a omic clus e in Re . [102] and dis ibu ed Au and P a oms once andomly and once in a P -co e Au-shell ashion o hese posi ions. The s uc u es we e hen elaxed wi h DFT and MD sepa a ely. In he DFT calcula ions he elaxa ion was ca ied ou wi hou any symme y cons ain s using a spli alence basis se augmen ed by one pola iza ion unc ion (SVP). The use o la ge basis se s changes he bond leng h by less han 2% [94] and does no a ec he ela i e dis ibu ion o he leng hs o Au-Au, Au-P and P -P bonds. Fo he op imized s uc u es we calcula ed he dis ance o e e y a om o i s i e nea es neighbo s, a he same ime de e mining whe he hese bonds we e Au-Au, Au-P o P -P bonds. The a e age Au-Au bond leng h was hen de ined by he a i hme ic mean o all esul ing Au-Au dis ances. In he same way, Au-P and P -P bond leng hs we e calcula ed. The esul s s ay quali a i ely he same i one somewha changes he a e aging p ocedu e, e.g., using ou ins ead o i e nea es neighbo s. Tab. 3.1 shows he a e age bond leng hs hus ob ained om DFT and MD o co e-shell and o andomly mixed a ian s o Au 22 P 38 . The MD bond leng hs a e ound o be consis en ly smalle han he DFT bond leng hs. Howe e , he ends wi h espec o bond leng h di e ences a e simila . Fo bo h DFT and MD calcula ions he di e ences be ween he a e age leng h o Au-Au bonds, Au-P bonds and P -P bonds ha one inds in andomly mixed and co e-shell pa icles a e small. In all cases Au-Au bonds a e longes and P -P bonds a e sho es , wi h Au-P alling in be ween. In o de o also es he MD bond leng h dis ibu ion o a ace ed pa icle we chose he smalles unca ed oc ahed on buil om 38 a oms. Alloy pa icles wi h Au con en s o 87 % (Au 33 P 5 ), 50 % (Au 19 P 19 ) and 13 % (Au 5 P 33 ) we e gene a ed by dis ibu ing Au and P a oms andomly o e he clus e . The s uc u es we e ene ge ically op imized as desc ibed p e iously. Tab. 3.2 shows he esul ing a e age bond leng hs. We obse e again ha he MD and DFT a e age bond leng hs a e no iden ical, wi h he MD bonds gene ally being sho e han he DFT ones. Ye , again he ends a e he same in bo h DFT and MD: Au-Au bonds a e he longes and P -P bonds a e he sho es , wi h Au-P bonds alling in be ween. The one excep ion ha we ound is shown in Tab. 3.2 o 87 % Au, whe e Au-P and P -P bonds in he MD a e qui e simila . We a ibu e his o he low P concen a ion and small pa icle size. We also pe o med he same ype o analysis o he 20-a om e ahed al and amo phous s uc u es ha a e desc ibed in Re . [94]. The esul s a e in line wi h he conclusions d awn o he 38-a om and 60-a om s uc u es shown abo e and a e he e o e no epo ed in de ail he e. We can now analyze he a e age bond leng hs o he Au 459 P 830 co e-shell and andomly 3.3. Au-P alloys and Vega d’s law on he nanoscale 31 mixed unca ed oc ahed on. In he co e-shell clus e he la ge P co e domina es he bond leng h dis ibu ion, wi h an a e age P -P dis ance o 2.77 ˚ A. The mean Au-Au dis ance in he single Au shell is he same as he mean P -P bond leng h. This inding is eminiscen o he la ice de o ma ion and ma ching ha one encoun e s when g owing single laye s o one ma e ial on a subs a e wi h a di e en la ice pa ame e . The Au-P dis ance o 2.78 ˚ A is sligh ly la ge , allowing he shell o w ap a ound he co e. The bond leng h mos o en encoun e ed in he andomly mixed NP is one ha alls in be ween he Au-Au and he P -P bond leng h, wi h Au-P bonds on a e age being 2.76 ˚ A, Au-Au bonds 2.79 ˚ A, and P -P bonds 2.74 ˚ A. a omic o m ac o gold pla inum 00.2 0.4 0.6 0.8 1.0 1.2 20 40 60 80 100 120 140 160 Figu e 3.7: A omic o m ac o s o Au and P aken om Re . [127] and polynomial leas squa e i s o he da a. A e hese p elimina y conside a ions we u n o simula ing he XRD pa e ns o he NP using Eq. (3.1) wi h he incoming x- ays ha ing a wa e leng h o λ =1.5 ˚ A. The o m ac o s o Au and P a e abula ed in Re . [127] and i ed using a polynomial leas squa es i o allow con inuous ep esen a ion o he da a (see Fig. 3.7). The op le panel in Fig. 3.8 shows he pa - e n ha one ob ains om pu e Au o P NP wi h 1289 a oms in o ange and black, espec- i ely. The ed line shows he x- ay pa e n o he co e-shell NP. As laid ou abo e he expec a ion so a has been o ind sepa a e peaks o his ype o s uc u e, as he Au and P componen a e sepa a ed. Ye , no double-peak s uc u e is obse ed a all; he peak shapes o he di e en NP a e nea ly iden ical. Ou i s impo an conclusion he e o e is ha a co e-shell s uc u e in NP does no necessa ily lead o sepa a e x- ay peaks o he di e en componen s. The lowe le panel shows he second impo an esul : The x- ay pa e n o he co e-shell NP (as abo e) and he one o he andomly mixed NP a e e y simila . Co e-shell s uc u es may hus be ha d o iden i y ia x- ay sca e ing. Figu e 3.8 : Le : Calcu- la ed XRD pa e n o pu e Au, P , and co e-shell NP ( op) and co e-shell and an- domly mixed NP (bo om). Righ : View o he co e- shell ( op) and andomly mixed (bo om) NP (ou - side and c oss sec ion). in ensi y [a b.uni s] pu e Au pu e P co e-shell co e-shell andom 30 35 40 45 50 50 100 150 250 200 50 100 150 250 200 co e-shell andom B agg angle [2 ] Howe e , we now ake he analysis one s ep u he and go h ough he Vega d’s-law- based analysis o he x- ay da a as explained abo e. Fo pu e NP he x- ay analysis leads 38 Chap e 3 — The bime allic e ec in Au-P and Au-Pd nanoalloys 0 2 4 6 8 10 0 20 40 60 80 100 DOS a Fe mi le el/elec on [a b. uni s] Au con en [%] 20 e a 20 amo phous 38 oc a 60 Gup a 40 Su on-Chen Figu e 3.16: The DOS a he Fe mi le el (in eg a ion h eshold de ined as -1 eV) as a unc ion o he Au con en o 20-a om e ahed al and amo phous, 38- a om unca ed oc ahed al and 40- and 60-a om Gup a-po en ial clus e s. We can now analyze he DOS a he Fe mi le el o a a ie y o Au-P NP wi h di e en o e all geome ies and mixing pa e ns. Fig. 3.16 shows ha i inc eases wi h inc easing P con en o 20-a om e ahed al 5 and amo phous clus e s (co e- shell), o 38-a om unca ed oc ahed a ( andom) and o 40- and 60-a om Gup a- po en ial clus e s (co e-shell). Impo an - ly, his end is independen o he speci ic way in which Au and P a oms a e a - anged in he clus e , i.e., he DOS a he Fe mi le el inc eases bo h o co e-shell and andomly mixed Au-P nanoalloys wi h inc easing P con en . This inding sugges s ha he binding ene gy be ween Au-P nanoalloys and an adso ba e should inc ease wi h inc easing P con en and ha a pu e P NP should bind adso ba es mos s ongly o i s su ace. F om his pe spec i e, a ce ain Au-P a io could co espond o an op imal binding ene gy in a speci ic eac ion in e ms o Saba ie ’s p inciple. co e-shell laye ed 20 a oms 40 a oms Figu e 3.17: Plo s o he highes occupied molecula o bi al a ± 0.04 a−3/2 0 . Top: o wo jus sligh ly di e en Au 10 P 10 clus e s. Bo om: Au 20 P 20 wi h co e- shell and laye ed mixing pa e n, espec- i ely. A close look a he local elec onic s uc u e e eals ha he addi ion o P is bene icial om ye ano he pe spec i e. Fig. 3.17 shows isosu ace plo s o he highes occupied KS o bi al (HOMO) o se e al Au-P NP. The HOMO has physical el- e ance as i asymp o ically domina es he densi y and i s exponen ial decay is go e ned by he i s ioniza ion po en ial, i.e., i ep esen s he spa ial dis ibu ion o he ene ge ically highes lying pa o he densi y. Fig. 3.17 e eals ha he spa ial loca ion o he HOMO is closely ela ed o ha o he P a oms. The wo Au 10 P 10 clus e s a he op o Fig. 3.17 di e only in he posi ion o one P a om ha si s wi hin he bo om ace on he le side and is mo ed o he op co ne on he igh side o Fig. 3.17. This sligh ea angemen esul s in a ma ked change o he spa ial dis ibu ion o he HOMO densi y. The same e ec can be seen o he wo Au 20 P 20 homo ops a he bo om o Fig. 3.17 and all o he sys ems ha we s udied. This has wo impo an consequences: Fi s ly, as he P con en inc eases, mo e P a oms a e loca ed a he su ace o he clus e , so ha he HOMO can ex end away 5 No e, ha he Au 20 e ahed on is he only excep ion ha we ound o his obse a ion. We a ibu e his de ia ion o he unusual elec onic s uc u e o Au20 [68]. 3.4. The in e play be ween luxionali y and elec onic s uc u e 39 om he su ace. Secondly, he known impo ance o he su ace o ca aly ic p ocesses is enhanced in Au-P nanoalloys, no only due o he inc eased su ace- o- olume a io on he nanoscale, bu also because o he p esence o P , he many possible dis ibu ions o Au and P on he su ace and he esul ing implica ions o he local elec onic s uc u e o he sys em. 3.4.2 Elec onic s uc u e o Au-Pd nanoalloys Au-Pd nanoalloys show an elec onic s uc u e e y simila o ha o Au-P nanoalloys. The e is bu one poin equi ing special a en ion in small Au-Pd nanoalloys ha can be neglec ed in Au-P NP: hei spin pola iza ion. Bulk Au and Pd a e no magne ic, bu high magne ic momen s ha e been epo ed o small Pd clus e s [140]. We s a om he bilaye ed g ound s a e o Pd 13 aken om Re . [140], s ep-by-s ep eplacing lowly- coo dina ed Pd a oms by Au as sugges ed in Re . [141, 142]. The esul ing Au n Pd 13−n (n=0-13) s uc u es we e hen locally elaxed. 0 2 4 6 8 10 12 0 20 40 60 80 100 DOS a Fe mi le el/elec on [a b. uni s] Au con en [%] 13 bilaye 20 e a 38 ype 1 38 ype 2 Figu e 3.18: The DOS close o he Fe mi le el as a unc ion o he Au con en o Au/Pd pa icles. Fo all h ee sizes (13-a om, 20-a om and 38-a om) he DOS dec eases wi h inc easing Au con en . Fo he 38 a omic clus e s he DOS o wo di e en s uc u e ypes is shown. The end is he same o bo h ypes. We allowed and checked o spin po- la iza ion, e alua ing he o al ene gy o di e en mul iplici ies. We ound ha o he pu poses o his s udy he magne i- za ion is no decisi e, as he DOS a he Fe mi le el depends only e y li le on he spin pola iza ion in he cases ha we s udied. The in eg a ed DOS ypically changes by less han 1% o mos s uc- u es as a unc ion o magne iza ion, and a ew pe cen a mos . Fo he sake o comple eness, some magne iza ions a e, howe e , epo ed in he ollowing. The esul ing magne ic momen s a e 6 µB o Pd 13 , 3 µB o Pd 12 Au 1 , 4 µB o Pd 11 Au 2 and 2 µB o Pd 9 Au 4 . Clus e s wi h la ge Au con en s a e ei he single s o iple s depending on whe he hei elec on num- be is odd o e en. The DOS a he Fe mi le el is hen de ined as o Au-P NP and shown in Fig. 3.18 as a unc ion o he pa icles’ composi ion ( ed diamonds). The ed inse depic s he s uc u e o Au 3 Pd 10 as an example. F om he igu e one can clea ly d aw he conclusion ha again he DOS a he Fe mi edge dec eases wi h inc easing Au amoun . In o de o check whe he his beha io o he DOS is gene al we epea ed he p ocedu e o la ge clus e s. Ou choice o u he es sys ems was mo i a ed by simila conside a ions as in he case o Au-P NP. The expe imen al s udies a e conce ned wi h ela i ely la ge pa icles ha show a bulk-like s uc u e [132]. The e o e, we also heo e ically ocus on in es iga ing sys ems whe e a bulk-like geome y is a easonable s a ing poin . Fu he mo e, as bulk Pd and Au a e no magne ic, and as he expe imen ally obse ed pa icles a e 40 Chap e 3 — The bime allic e ec in Au-P and Au-Pd nanoalloys bulk-like in hei s uc u e, i is a plausible assump ion ha magne ic e ec s a e no impo an o explaining he ca aly ic p ope ies. In addi ion, ou s udy o he 13-a om clus e s has shown ha he o al DOS is no e y sensi i e o he magne iza ion. The e o e, we choose o ou ollowing in es iga ions always he elec onic con igu a ion ha has he lowes spin pola iza ion (unde he condi ion ha he au bau p inciple is espec ed). F om he jus desc ibed poin s o iew he Au 20 e ahed on is again a good s a ing poin o he heo e ical in es iga ion. We eplaced Au by Pd as explained in Sec. 3.2 wi h P , again obse ing he ule ha high-coo dina ion numbe si es a e ene ge ically a o able o Pd. The o ange inse o Fig. 3.18 shows he s uc u e o Au 16 Pd 4 as an example and he DOS a he Fe mi le el as a unc ion o he Au con en . Also in his case an inc ease wi h inc easing amoun o Pd is obse ed o all mixed Au/Pd pa icles. This obse a ion does no depend on he speci ic s uc u al mo i . −3−2−1 0 densi y o s a es [a b. uni s] [eV] (a) (b) (c) (d) (e) ( ) Figu e 3.19: Righ : geome ic s uc u e and iso- su ace plo o he HOMO o 38-a om pa icles. F om op o bo om: (a) Au 19 Pd 19 , (b)-(d) 3 di e en s uc u es o Au 28 Pd 10 , (e) Au 30 Pd 8 , ( ) Au 32 Pd 6 . Whi e and blue a eas ep esen nega i e and posi i e isosu ace alues o 0.04 a−3/2 0. Le : he DOS o each o he clus e s. To demons a e his explici ly and o check ano he clus e size we epea ed he p ocedu e o he unca ed oc ahed al 38-a om clus e s. We s udied wo lines o s uc u es wi h 38 a oms. In bo h o hem 6 Pd a oms o m he co e (see black inse o Fig. 3.18), as his is an ene ge ically a o ed a angemen . How- e e , he wo lines di e in he posi ion o he emaining “su ace” Pd a oms. The s uc u es labeled “ ype 1” in Fig. 3.18 we e ob ained by andomly eplacing su ace Au by Pd a oms. In he s uc u es labeled “ ype 2” he su ace Pd a oms we e placed such ha hey occupy he in e io pa s o he ace s. In he li e a- u e [141] hese la e con igu a ions ha e been epo ed as being ene ge ically a o able. Ou calcula ions con i m his inding as s uc u es o ype 2 a e lowe by 0.08 eV o Au 30 Pd 8 and 0.83 eV o Au 28 Pd 10 han he co esponding s uc u es o ype 1. We obse e ha all s uc- u es ha e a he small HOMO-LUMO gaps, which may be an indica ion o u he symme- y b eaking being ene ge ically a o able. The impo an inding is ha also in he 38-a om ace ed cases bo h ypes o s uc u es show he same end: he DOS a he Fe mi le el inc eases wi h inc easing Pd con en . Again his inding could imply ha , in e ms o Saba ie ’s p inciple, a ce ain Au-Pd a io migh co espond o op imal adso ba e binding ene gies in a ce ain eac ion. In he nex s ep, ollowing he same easoning as be o e, we in es iga e he ela ion be ween he clus e s’ geome y and hei elec onic s uc u e. To his end we ake a close 3.4. The in e play be ween luxionali y and elec onic s uc u e 41 look a se e al o he ace ed 38-a om pa icles. The igh side o Fig. 3.19 shows om op o bo om s uc u es and HOMO densi ies o Au 19 Pd 19 , h ee di e en Au 28 Pd 10 homo ops, Au 30 Pd 8 and Au 32 Pd 6 . The le column depic s he occupied DOS wi h he espec i e Fe mi le els aligned a 0 eV. The e ical line a -1 eV indica es he p e iously men ioned in eg a ion ange. The conclusions ha can be d awn om Fig. 3.19 a e a he simila o he Au-P NP case: Fi s , he HOMO in each case is closely associa ed wi h he Pd a oms. Second, an inc easing amoun o Pd b ings an inc easing numbe o Pd a oms o he su ace and hus leads o a HOMO ha is dominan ly loca ed a he su ace o he clus e . This is clea ly seen, e.g., when compa ing he HOMO ep esen a ion o Au 32 Pd 6 o he one o Au 19 Pd 19 and i indica es ha Pd a oms a he su ace a e likely o be a o able in e ms o he pa icles’ ac i i y. Thi d, we ca e ully inspec h ee homo ops o Au 28 Pd 10 . S uc u e (b) as sugges ed in he li e a u e [141] has he lowes o al ene gy, (d) is highe by 0.83 eV and (c) by 1.64 eV. Independen o hese ene ge ic di e ences, he e is an impo an obse a ion o be made in compa ing s uc u es (b)-(d): al hough he h ee geome ies a e o e all a he simila , he di e ences be ween he HOMOs a e qui e no iceable. A mode a e change in he geome y leads o a signi ican change o he ene ge ically high-lying pa o he densi y. Thus, a speci ic a angemen o Au and Pd a oms a he su ace can lead o special elec onic and hus chemical p ope ies. 3.4.3 The ole o Au in Au-P nanoalloys In he spi i o Re . [131] and as explained in he in oduc o y pa ag aphs o Sec. 3.4 one could specula e ha a syne gis ic e ec in Au-P nanoalloys o a speci ic eac ion could in p inciple be explained by linking a ca aly ic desc ip o o his eac ion o he DOS a he Fe mi le el. One could hen a gue ha he binding ene gy be ween Au-P NP and he speci ic eac an s mus nei he be oo s ong no oo weak o he eac ion o ake place, in compliance wi h Saba ie ’s p inciple. An op imal Au-P mixing a io would hen eme ge om hese conside a ions. E en hough his poin o iew is pe ec ly legi ima e and has led o aluable insigh s in o gas-phase ca alysis on su aces unde well-con olled condi ions, i is no easily applicable in ou case as men ioned be o e. Howe e , as a syne gis ic e ec is obse ed in many di e en eac ions wi h di e en ma ices and sol en s, we sugges ha undamen al p ope ies o Au-P NP ha a e independen o he eac ion lead o he special p ope ies o Au-P NP. Wi h he DOS a he Fe mi le el and he spa ial a angemen o he HOMO densi y being de e mined by P as discussed ea lie , we now wan o ask he ques ion which undamen al ca alysis- a o able p ope y is associa ed o Au. He e, I wan o a gue ha his p ope y is he mobili y o a oms in he Au-P sys ems, o he NP luxionali y. Fluxionali y has been ecognized o be o c ucial impo ance o he ca aly ic ac i i y o clus e s and su aces [143]. I can be a p ime o he abili y o s uc u ally adjus o eac an s, o m ac i e si es [144] and di ec ly in luences u no e a es by a ec ing he ca alys s’ abili y o egene a e a e he eac ion. In o de o de e mine how easy i is o a oms o ea ange in Au-P sys ems we pe o med wo ypes o s udies. In he i s one we ake he pe spec i e ha wi h espec o ca aly ic ac i i y, he su ace o he NP, which o ace ed NP is simila o a bulk su ace, is likely hei mos impo an 42 Chap e 3 — The bime allic e ec in Au-P and Au-Pd nanoalloys ini fin ini finini fin ini finini fin ini fin I. P /P II. P /P +Au III. Au/P IV. Au/P +Au V. P /Au VI. Au/Au Figu e 3.20 : Ske ch o he se up used o he s udy o Au and P ada om sel -di usion om a cc (ini) o a hcp ( in) hollow posi ion. The sys ems we s udied we e: I. P ada om on pu e P (111) su ace, II. P ada om on P (111) su - ace wi h single Au a om in op su ace laye , III. Au ada om on pu e P (111), IV. Au ada om on P (111) doped wi h Au, V. P ada om on Au(111) and VI. Au ada om on Au(111). pa . When eac an s app oach he su ace, he su ace may s uc u ally eac o o m ac i e si es. The induc ion ime, i.e., he ime ha goes by be o e he eac ion s a s, seen in NP ca alysis can be an indica ion o such p ocesses [80, 132, 145]. A canonical s ep in su ace es uc u ing is he mo emen (di usion) o a oms on he su ace. The e o e, we chose he heigh o he ba ie s o he di usion o Au and P ada oms on pu e and doped P - and Au-su aces as a measu e o he su aces’ abili y o es uc u e. As an aside we no e ha his line o hinking is also suppo ed by he obse a ion ha bime allic e ec s as well play a ole in Au-P su aces [115] as men ioned be o e. sys em Eads cc Eads hcp Eba ie P /P 4.60 4.82 312 P /P +Au 4.61 4.76 287 Au/P 2.77 2.87 177 Au/P +Au 2.79 2.86 165 P /Au 4.05 4.15 143 Au/Au 2.50 2.56 115 Table 3.3 : Adso p ion ene gies (in eV) o P - and Au-ada oms in pu e and doped P - and Au su aces o he s a and he end posi ion o each NEB calcula ion. Fo a de ini ion o he di usion ba ie (in meV) see ex . Fo calcula ing hese di usion ba ie s we ely on he NEB me hod (see Sec. 2.5). I allows o de e mine he minimum ene gy pa h be ween a gi en ini ial and inal s a e. I his minimum ene gy pa h goes h ough one o se e al maxima as a unc ion o he eac ion coo dina e he la ges o hese ba ie s can be ega ded he a e-de e mining s ep o he eac ion. Calcula ions we e ca ied ou wi h VASP using he PBE GGA [24] and employing he PAW me hod o desc ibe he elec on-ion in e ac ion [64, 67] (see Sec. 2.4). Technical de ails ega ding hese calcula ions can be ound in Appendix C. The cu o -ene gy o he plane wa e basis se was se o 450 eV. Pu e and doped P and Au cc(111) su aces we e modeled by 4-laye slabs con aining 4x4 a oms in each laye and being sepa a ed by 20 ˚ A o acuum. The bo om wo laye s we e held ixed a he P (Au) bulk equilib ium dis ance o 3.977 (4.174) ˚ A, while he o he laye s we e allowed o ully elax. The B illouin-zone in eg a ions we e pe o med using Monkho s -Pack k-poin meshes o 5x5x1 size. We conside ed a di usion s ep om a cc o a hcp hollow si e o di e en combina ions o su aces and ada oms, as depic ed in Fig. 3.20. In all cases he di usion ba ie was de ined as he ene ge ic di e ence be ween he ansi ion s a e and he hcp adso p ion si e, his being he la ges ene gy di e ence and hus de e mining he a e o his pa icula di usion 3.4. The in e play be ween luxionali y and elec onic s uc u e 43 s ep. In addi ion o he di usion ba ie s we de e mined he adso p ion ene gies (see Tab. 3.3) o he ada oms a he cc and hcp hollow si e as Eads =Eslab +Eada om −Eslab+ada om (3.7) 100 120 140 160 180 200 220 240 260 280 300 320 P /P P /P +Au Au/P Au/P +Au P /Au Au/Au Sys em Diffusion ba ie [meV] Figu e 3.21: Ba ie o P - and Au- ada om di usion on P -su aces wi h and wi hou Au doping and on a pu e Au- su ace. An explana ion o he no a ion can be ound in Fig. 3.20. Fig. 3.21 shows ha di usion o a P ada om on a pu e P su ace has he highes ba ie o all conside ed sys ems. This is in line wi h he P ada om being mos s ongly adso bed o he pu e P su ace. Ye , he inse ion o one Au a om in o he P su ace lowe s he di usion ba ie by abou 25 meV. A simila e ec can be seen by compa ing he di usion ba ie o an Au ada om on a pu e P su ace and on a P su ace wi h one Au dopan . I is pa icula ly in e es ing ha he di usion ba ie dec eases e en mo e o a P ada om on a Au su - ace. In his case he adso p ion ene gy is almos as high as ha o P on P (111) leading one o expec an equally high di usion ba ie . The acil- i a ed su ace di usion on su aces wi h a highe Au con en can he e o e no only be explained by Au ada oms being mo e weakly bound o he su - ace han P ada oms. We in e p e his inding as an inc eased abili y o Au-alloyed P -su aces o unde go s uc u al changes and hus as a highe su ace luxionali y wi h inc easing Au-con en . In he second ype o s udy ha we did in o de o show ha inc easing amoun s o Au inc ease he a omic mobili y in Au-P NP we mo ed away om he su ace pe spec i e and ega ded he NP as ini e clus e s. The luxionali y o small Au clus e s has ecen ly been shown by ex ensi e scans o hei po en ial ene gy su aces [146]. This me hod, howe e , is no easible o Au-P nanoalloys in he expe imen ally ele an size ange. The e o e, we pe o med one nanosecond long MD simula ions o andomly mixed unca ed oc ahed al clus e s (38, 201 and 586 a oms) and o 60-a om and 100-a om Gup a po en ial based co e-shell clus e s o ob ain a measu e o he mobili y o he clus e a oms in e ms o hei mean squa e displacemen (MSD) hˆ 2i=*1 M M X i=1 (Ri( )−Ri( = 0))2+,(3.8) whe e M is he numbe o a oms in he clus e and Ri ( ) is he posi ion o a om i a ime . The angle b acke s indica e a ime a e age. The MD simula ions we e done using he DL POLY p og am [126] and he NVT E ans ensemble wi h he same many-body Su on-Chen po en ial as in Sec. 3.3 o desc ibe he po en ial ene gy o he sys em. Randomly mixed Au-P NP seg ega e o Au shell P co e s uc u es e en a compa ably modes empe a u es and sho simula ion imes. As ou aim was o e ain he homogenous 44 Chap e 3 — The bime allic e ec in Au-P and Au-Pd nanoalloys mixing pa e n o he unca ed oc ahed a all simula ions we e ca ied ou a 300 K. This lies well below he mel ing poin o mos o hese sys ems 6 , meaning ha he MSD a e aged o e he whole simula ion ime is app oxima ely ze o. Howe e , a sho e imescales o 1000 s a di usi e mo ion o Au and P a oms akes place. The ime a e age in Eq. 3.8 was he e o e compu ed o subsequen ime windows o 1000 s. Fo each Au-P mixing a io h ee simula ions we e ca ied ou o minimize s a is ical unce ain ies. No e ha e en hough his me hod yields only an app oxima ion o he di usion o clus e a oms, i can be used o compa e he sho - ime a omic mobili y o nanoalloys wi h a ying mixing a ios. 0.02 0.04 0.06 0.08 0.10 0.12 38 a oms 201 a oms 586 a oms 0.02 0.04 0.06 0.08 0.10 0 20 40 60 80 100 Au con en [%] 60 a oms 100 a oms MSD [ ] Figu e 3.22: Top: MSD om MD simula ions o unca ed oc ahed al clus e s ha a e solid solu ions o Au and P wi h 38, 201 and 586 a oms. Bo om: MSDs o 60-a om and 100- a om co e-shell pa icles. Fo NP in he size egime ha we a e con- ce ned wi h he high su ace- o- olume a io can ha e signi ican impac on he a e age MSD o he NP. A oms a lowly coo dina ed po- si ions, e.g., a kinks o edges can show MSDs ha a e conside ably la ge han hose o high- ly coo dina ed olume a oms o a oms in a su ace plane. Fo i egula NP like he 60- a om and he 100-a om NP his e ec is mo e p onounced han i is o he egula unca ed oc ahed a wi h hei magic a om numbe s o 38, 210 and 586. To ob ain compa able MSD alues i is he e o e necessa y o keep he ge- ome ies o NP wi h he same numbe o a oms bu di e en Au-P a ios as close as possible. Fo he i egula 60-a om and 100-a om clus- e s his makes a ull geome y op imiza ion p io o MD simula ions necessa y. Ne e he- less, he MSDs o hese NP show la ge s a is- ical a ia ions as compa ed o he MSDs o he unca ed oc ahed a. Fo his eason we wan o s ess ha we a e no conce ned wi h e alua ing he exac alue o he MSD in ou NP. In con as , we wan o s udy he end ha he MSD shows wi h inc easing Au con en . The esul ing MSDs a e depic ed in Fig. 3.22: he unca ed oc ahed a a he op and he 60-a om and 100-a om co e-shell pa icles a he bo om. I can clea ly be seen ha he composi ion-dependen end is he same o all clus e s: The MSD inc eases wi h inc easing Au-con en . This is again independen o pa icula geome ic con igu a ions o mixing pa e ns. Thus, al hough being o e y di e en na u e, he MD s udy con i ms he conclusions d awn om he su ace di usion ba ie calcula ions: s uc u al lexibili y inc eases wi h an inc easing amoun o Au. 6 The mel ing poin o he 60-a om clus e s lies below 300 K. The e o e in his case MD simula ions we e ca ied ou a a empe a u e o 200 K. 3.4. The in e play be ween luxionali y and elec onic s uc u e 45 3.4.4 Combining elec onic s uc u e and luxionali y 0.0 0.2 0.4 0.6 0.8 1.0 0 20 40 60 80 100 MSDxDOS a FL [a b. uni s] Au con en [%] 38 a oms 60 a oms Figu e 3.23: P oduc o a polynomial leas squa es i o he DOS a he Fe mi le el (FL) and MSD o 38-a om and 60-a om Au-P clus e s as a unc ion o Au con en . Quan i ies a e scaled so ha o 0 and 100% Au con en he cu es in e sec he y-axis a 0 and ha e a maximum alue o 1. We can now pu oge he all he indings. The DOS o Au-P NP de elops p ope ies ha one na u ally associa es wi h being a o able o ca al- ysis wi h inc easing P con en , whe eas he s uc- u al lexibili y inc eases wi h inc easing Au con en . The e o e, o each eac ion he e can be an ideal a io o Au and P ha yields o e all op imal p op- e ies. This is isualized in Fig. 3.23. I shows he olcano-like plo ha one ob ains by mul iply- ing a polynomial leas squa es i o he DOS a he Fe mi le el and he MSD. This has he e been done o he 38-a om and 60-a om clus e s, as o hese sys ems we could access bo h quan i ies (see Fig. 3.24). An ideal Au-P mixing a io eme ges as a con- sequence o he in e play be ween s uc u al lux- ionali y p o ided by Au and a a o able elec onic s uc u e con ibu ed by P . O cou se wha his ideal mixing a io is will sensi i ely depend on he eac ion, expe imen al condi ions, as well as on he exac s uc u e and mixing pa e n o he ca alys s. Ye , he poin ha Fig. 3.23 emphasizes is ha syne gis ic e ec s Figu e 3.24 : Le : MSD and linea i o MSD o 38- a om (black iangles) and 60-a om ( ed iangles) clus- e s. Righ : DOS a he Fe mi le el o 38-a om and 60-a om clus e s. Fo con- s uc ing he ” olcano” plo he da a was i ed wi h a cubic polynomial. 0.00 0.02 0.04 0.06 0.08 0.10 0.12 0 20 40 60 80 100 MSD [Å2] Au con en [%] 38 a oms 60 a oms DOS a FL/elec on 0 1 2 3 4 5 6 7 8 0 20 40 60 80 100 Au con en [%] 38 a oms 60 a oms in Au-P NP a e expec ed on a he gene al g ounds and ega dless o he speci ic chemical eac ion and en i onmen . Addi ionally, Fig. 3.23 illus a es ano he poin . We do no expec ” he whole o be g ea e han he sum o i s pa ”. The syne gis ic o a he bime al- lic e ec in Au-P nanoalloys eme ges as a consequence o adding up ca alysis- a o able undamen al physical p ope ies o Au and P . The e ec o he combined p ope ies migh be ampli ied by he high su ace o olume a io in NP, he p esence o s eps and kinks ha ac as ac i e si es in ca alysis [144] and he elonga ed a e age bond leng hs esul ing om he mal expansion (see Sec. 3.3) o NP-ma ix in e ac ions (see Appendix A). Di e en o wha he buzzwo d ”syne gy” sugges s, Au-P nanoalloys migh no possess special ca aly ic p ope ies ha go beyond a combina ion o he p ope ies o pu e Au and P NP. 4 Magne ic- o-nonmagne ic ansi ion in Mn-doped Si-clus e s 4.1 Dilu e magne ic semiconduc o s Semiconduc o s doped wi h ew impu i ies ha ca y a magne ic momen a e called dilu e magne ic semiconduc o s. Du ing he las decades hey ha e been subjec o in ense s udy, a e i had been obse ed ha he in oduc ion o a small ac ion o magne ic impu i ies would lea e he a o able anspo and op ical p ope ies o he semiconduc o s unchanged, a he same ime being accompanied by la ge magne ic momen s and many in e es ing magne ic o de phenomena. To name only one example, i has been shown by Ohno e al., ha (Ga,Mn)As exhibi s e omagne ic o de up o empe a u es as high as 110 K [147]. La e heo e ical wo k p o ided a model o he o igin o his e omagne ic o de ing and p edic ed ma e ials wi h Cu ie empe a u es e en exceeding oom empe a u e. E en hough i is belie ed oday ha dilu e magne ic semiconduc o s wi h such high Cu ie empe a u es may no be ealizable, he ield ne e heless bea s plen y o undamen al ques ions, many o which a e s ill unanswe ed. The seemingly simples o hese ques ions is how a single magne ic impu i y in e ac s wi h he semiconduc ing hos , i.e., whe he he impu i y’s magne ic momen is quenched o su i es. Pionee ing s udies ha e been conduc ed by Ludwig and Woodbu y in he 1960s using elec on-pa amagne ic esonance spec oscopy o in es iga e magne ic p ope ies o Si doped wi h a ange o 3 d ansi ion me al impu i ies a subs i u ional and in e s i ial posi ions [148]. Fu he mo e, Ludwig and Woodbu y p o ided a simple and desc ip i e model o p edic quenching o su i al o he magne ic momen o he ansi ion me al impu i y. 54 Chap e 4 — Magne ic- o-nonmagne ic ansi ion in Mn-doped Si-clus e s MnSi7 +MnSi8 +MnSi9 +MnSi10 + MnSi11 +MnSi12 +MnSi13 +MnSi14 + Figu e 4.5: Ene ge ically mos a o able s uc u es o MnSi + n ( n = 7 − 14) as p edic ed by PBE0. Con- a y o expe imen MnSi+ 11 is exohed ally doped. Fo he simula ed annealing simula- ions di e en s a ing s uc u es we e hea- ed o empe a u es be ween 3500 and 4500 K and p opaga ed a his empe a- u e o 3 ps. Subsequen ly, he empe a- u e o he sys em was educed by a ac o o 0.9 e e y 0.2 ps un il i ell below 300 K. A his poin he s uc u e usually was apped in a s i local minimum. I is he e o e o g ea impo ance o s a sim- ula ions om a la ge numbe o di e en geome ies o scan as much o he po en- ial ene gy su ace as possible. The lowes ene gy isome s om hese simula ions we e hen locally elaxed using a quasi-New on me hod and la ge TZVPP basis se s un il he o al ene gy changed by less han 10 −8 Ha ee and he o ces ac ing upon he ionic co es we e smalle han 0.001 a.u.. The modi ied big bang app oach consis ed o andomly gene a ing di e en se s o s a ing coo dina es wi h bond leng hs comp essed by abou 10% o he equilib ium bond leng h and p opaga ing hese coo dina es a a ixed empe a u e o 300 K o a sho ime span o 1.5 ps. Again, lowes ene gy geome ies we e locally elaxed as desc ibed be o e. Figu e 4.6: S uc u e o he ene ge ically lowes lying endohe- d ally doped MnSi + 11 as p edic ed by he PBE0 xc unc ional. The ene ge ically lowes lying s uc u es ha PBE0 p edic s a e shown in Fig. 4.5. Consis en wi h expe imen clus e s wi h n = 7 − 10 a e exohed ally and clus e s wi h n = 12 − 14 a e endohed ally doped. Simila ly, in pe ec ag eemen wi h he XMCD spec a o Fig. 4.4 clus e s wi h n = 7 − 10 exhibi a la ge magne ic momen o 4 µB , while he magne ic momen o hose wi h n = 12 − 14 is comple ely quenched. MnSi + 11 is, con a y o he expe imen ally p edic ed exohed al- o- endohed al ansi ion, exohed ally doped. Fu he mo e, he calcula ed s uc u e ca ies a magne ic momen o 2 µB in con adic ion o he anishing XMCD signal o MnSi + 11 in Fig. 4.4. The ene ge ically lowes lying endohed al s uc u e, al hough 1.1 eV highe in ene gy han he PBE0 g ound s a e, (shown in Fig.4.6) would howe e ag ee wi h expe imen . As he expe imen ally obse ed s uc u es a e expec ed o be he g ound s a e s uc u es, a close look a he heo e ical esul s e eals ha he ailu e o PBE0 o MnSi+ 11 can indeed be explained in e ms o he e ec o sel -in e ac ion on he eigen alue spec um o bo h isome s. To cla i y he si ua ion we calcula ed he o bi al- SIE, Eq. (2.14) , as in oduced in Sec. 2.2. To de e mine eiσ a local de elope s e sion o he p og am package PARSEC [59] was used o calcula e he elec onic s uc u e o MnSi + n ( n = 7 − 14). We employed T oullie -Ma ins pseudopo en ials in Kleinman-Bylande o m cons uc ed wi hin he PBE GGA [24] wi h a non-linea co e co ec ion as discussed in Sec. 2.4. The pseudopo en ials we e based on he elec onic con igu a ion 4 s1.75 4 p0.25 3 d5 wi h s/p/d cu -o adii o 1.884/2.573/1.980 a.u. o Mn and 3 s2 3 p2 wi h s/p cu -o adii o 4.3. A DFT-s udy o he magne ic- o-nonmagne ic ansi ion in Mn-doped Si-clus e s 55 2.493/2.390 a.u. o Si. I was es ed ha he in luence o he used pseudopo en ials on he eigen alue spec um was negligible by compa ing hem wi h ou all-elec on TURBOMOLE calcula ions. A g id spacing o 0.3 a.u. was used. eiσ was e alua ed using he PBE GGA xc unc ional. Following Pe dew and Zunge and neglec ing he di e ence be ween sel - in e ac ion co ec ed and unco ec ed o bi als, he sel -in e ac ion co ec ed eigen alues can be es ima ed as εiσ ≈εapp ox iσ −eiσ,(4.9) whe e εapp ox iσ esul s om a sel -consis en calcula ion using app ox xc , which is PBE in ou case [21]. We can now compa e hese app oxima ely sel -in e ac ion co ec ed eigen alues wi h hose ha PBE0 yields. The smalle he di e ence ∆ εiσ = εiσ −εPBE0 iσ , he mo e eliable he PBE0 eigen alues εPBE0 iσ a e. Figu e 4.7 : Eigen alue spec um (ba s), o al DOS (do ed line), and Mn-p ojec ed local DOS (solid line) o endohed al MnSi+ 11 ( op) and he exo- hed al PBE0 g ound s a e (bo om). ∆ εi is a measu e o how much he PBE0 eigen alues a e a ec ed by he SIE. −200 0 200 −200 0 200 MnSi11 + −3 −2 −1 0 1 −3 −2 −1 0 1 −7−6−5−4−3−2−1 0 1 DOS DOS Fig. 4.7 shows he occupied eigen alue spec um in e p e ed as a DOS and he espec i e ∆ εi o bo h MnSi + 11 isome s, he endohed ally doped a he op and he exohed ally doped PBE0 g ound s a e a he bo om. While o he endohed ally doped isome ∆ εi≈ 0 o all uppe o bi als, he si ua ion is conside ably di e en o he exohed ally doped isome . He e, he PBE0 eigen alues a e s ongly a ec ed by he SIE. This is no only ue o he o bi als 4-5 eV below he Fe mi le el, ha can be associa ed wi h he Mn dopan 5 , bu also o highe lying s a es ha pa icipa e in bonding. All in all, Fig. 4.7 illus a es ha he ene ge ic o de ing o MnSi+ 11 isome s as p edic ed by PBE0 is un eliable. In a second s ep we used a comple ely di e en app oach o u he elucida e he si ua ion. Using he RSH xc unc ional ω PBE [43] (see Sec. 2.2) as implemen ed in he QChem p og am package [174], we checked he ene ge ic o de ing o he exohed al and he endohed al MnSi + 11 isome s as a unc ion o he ange-sepa a ion pa ame e ω . Fo alues o ω = 0 . 2 − 0 . 4 a.u. he ω PBE unc ional p edic s he ene ge ic o de ing ha is in ag eemen wi h expe imen . This inding highligh s he delica e balance o exchange and co ela ion e ec s ha de e mines he ene ge ic o de ing in MnSi+ 11. I will explain in he ollowing, why he SIE, al hough also plaguing he o he exohed al sys ems, only becomes decisi e o he ene ge ic o de ing in case o MnSi + 11 . To his end 5 This can be done by p ojec ing he KS eigen alues on o a omic o bi als ia a Mulliken popula ion analysis as explained in Sec.3.4.1. 56 Chap e 4 — Magne ic- o-nonmagne ic ansi ion in Mn-doped Si-clus e s Fig. 4.8 shows he o al DOS, he DOS p ojec ed on o he Mn- d -o bi als and ∆ εi o MnSi + 7 ( op) and MnSi + 14 (bo om). These wo sys ems a e exempla y o he exohed al and he endohed al size egime, espec i ely. Addi ionally, isosu ace plo s o he o bi als wi h he la ges Mn-3dcha ac e and o he HOMO a e shown. −200 0 200 −2 0 −400 0 400 −2 0 −7−6−5−4−3−2−1 0 1 MnSi7 + MnSi14 + DOS DOS Figu e 4.8 : Eigen alue spec um (ba s), o al (do ed line), and local Mn 3 d -p ojec ed (solid line) DOS wi h isosu ace plo s (a ± 0.04 a.u.) o he Mn 3 d o bi als and he highes occu- pied o bi al o MnSi + 7 and MnSi + 14 . Posi i e (nega i e) alues ep esen he spin-up (spin-down) channel. The d -like o bi als a e localized on he Mn a om in MnSi + 7 , whe eas hey a e delo- calized in MnSi + 14 . Also shown is ∆ εiσ , he e ec o he SIE on he PBE0 eigen alues. Fig. 4.8 illus a es se e al poin s. Fi s , o he endohed al size egime, as ep esen ed by MnSi + 14 , PBE0 ai h ully p edic s he elec onic s uc u e, while in he exohed al size egime his is no he case o all ele an s a es. Howe e , he eigen alues mos se e ely a ec ed by he SIE (4-5 eV below he Fe mi le el) a e ene ge ically well sepa a ed om he Si-s a es. I is o be expec ed ha he shi o hese o bi als in a ully sel -in e ac ion co ec ed calcula ion would ha dly inc ease he hyb idiza ion be ween he Mn- and he Si-s a es and he e o e he elec onic and magne ic p ope ies o hese clus e s a e expec ed o be only sligh ly a ec ed. Fu he mo e, he ela i e shi be ween delocalized and localized o bi als ha esul s om he SIE is less c i ical when s uc u es wi h localized and delocalized o bi als a e ene ge ically well sepa a ed [52]. He e, his is he case o all s uc u es excep MnSi + 11 . The la e is he only clus e o which he global geome y op imiza ion esul ed in bo h exohed al and endohed al isome s close in ene gy. Fo MnSi + n ( n = 7 − 10) exohed al isome s a e consis en ly lowe in ene gy han endohed al ones. We checked, howe e , ha he ene ge ic o de ing o exohed ally and endohed ally doped isome s emains he same o clus e sizes a he s uc u al ansi ion, i.e., o MnSi + 10 and o MnSi + 12 . To his end we calcula ed he o al ene gy o he g ound s a e geome y as p edic ed by PBE0 and he ene ge ically lowes lying endohed al (exohed al) isome o MnSi + 10 (MnSi + 12 ) using ω PBE as be o e wi h a ange sepa a ion pa ame e o ω =0.2 a.u. as his ω leads o he co ec ene ge ic o de ing o MnSi + 11 . Tab. 4.1 shows he ene ge ic di e ence be ween exohed ally and endohed ally doped isome s calcula ed wi h PBE0 and ω PBE ene gies o MnSi + n (n=10-12) and independen ly suppo s he iew ha he SIE only a ec s MnSi + 11 in such a way as o change he ene ge ic o de ing o exohed ally and endohed ally doped isome s. Fo MnSi + 10 and MnSi + 12 he ene ge ic o de ing as well as he 4.3. A DFT-s udy o he magne ic- o-nonmagne ic ansi ion in Mn-doped Si-clus e s 57 Table 4.1 : To al ene gies di e ences o MnSi + n (n=10-12) exohed al and endohed al isome s ∆ E = Eexo −Eendo calcula ed using PBE0 and ω PBE wi h a ange sepa a ion pa ame e o ω = 0 . 2 a.u. Posi i e numbe s mean ha he endohed al isome is ene ge ically mo e a o able. sys em ∆EωPBE in eV ∆EPBE0 in eV MnSi+ 10 -0.67 -1.09 MnSi+ 11 1.67 -1.13 MnSi+ 12 2.05 1.05 ela i e ene gy di e ence be ween he exohed al and he endohed al isome oughly s ay he same o PBE0 and ω PBE. Consequen ly, only MnSi + 11 needs special conside a ion and he endohed al s uc u e shown in Fig. 4.6 can in ag eemen wi h expe imen conside ed o be he g ound s a e s uc u e o MnSi+ 11. The second poin ha Fig. 4.8 emphasizes is ha he change in local elec onic s uc u e o he Mn dopan ha was obse ed in he x- ay abso p ion spec a (Fig. 4.4) is e lec ed in he occupied eigen alue spec um and also by he shape o he KS o bi als. −200 0 200 400 −7−6−5−4−3−2−1 0 1 DOS [a b. uni s] Figu e 4.9: Eigen alue spec um (ba s) and DOS o a omic Mn + wi h 4 unpai ed elec ons. No e ha he g ound s a e o Mn+has a spin momen o 6 µB. Fo MnSi+ 7 , he occupied s a es o Mn 3 d cha ac e a e mos ly isola ed a ≈ 4 − 5 eV below he Fe mi le el and a e igh ly localized a he Mn dopan , i.e., hey la gely p ese e a (pe u bed) a omic cha ac e and only weakly in e ac wi h Si s a es, as can be seen om he DOS and he isosu - ace plo s. No e also he simila i y be ween he eigen alue spec um o a omic Mn + wi h 4 µB as shown in Fig. 4.9 wi h he Mn-3 d -PDOS o MnSi + 7 . Because o his weak in e ac ion, he localized o bi als o Mn 3 d cha ac e a e quali a i ely e y simila o all exohed al MnSi+ n clus e s and lead o he nea ly iden ical x- ay abso p ion spec a in Fig. 4.4. This no ion o , a leas pa ly, a omic Mn 3 d s a es is los in he endohed al size egime, ep esen ed by MnSi+ 14 in Fig. 4.8. He e, o bi als wi h pa ial Mn 3 d cha ac e a e s ongly hyb idized wi h Si s a es and a e shi ed o ≈ 0 . 5 − 2 eV below he Fe mi le el, i.e., hey pa icipa e s ongly in bonding and a e delocalized o e he Si ame. Consequen ly, he Mn 3 d de i ed PDOS sensi i ely depends on he s uc u e o he Si cage, which is e lec ed in he a ia ion o he x- ay abso p ion spec a o endohed al MnSi+ n o di e en n ( n = 11 − 14) in Fig. 4.4. The s ong spd hyb idiza ion wi h he pa icipa ion o all Mn alence o bi als explains he comple e quenching o he magne ic momen om 4 µB in exohed ally o 0 µB in endohed ally doped clus e s, which is again in acco dance wi h he expe imen al XMCD spec a. Thi dly, Fig. 4.8 is exempla y o he well-known connec ion be ween he SIE and he o bi al localiza ion [51]. The igh ly localized, nea ly a omic 3 d -o bi als o MnSi + 7 a e mo e s ongly a ec ed by he SIE han he a he delocalized o bi als o MnSi + 14 , despi e he pa ial Mn-3 d -cha ac e o he la e . Fo a ho ough discussion o o bi al localiza ion and sel -in e ac ion, he eade is e e ed o Re . [51]. 58 Chap e 4 — Magne ic- o-nonmagne ic ansi ion in Mn-doped Si-clus e s The s uc u al change a he exohed al- o-endohed al ansi ion o MnSi+ n can be quan i ied by he coo dina ion numbe Nc o Mn, i.e., he numbe o Si a oms in he i s coo dina ion sphe e as exempli ied o MnSi+ 8 in he inse o Fig. 4.11. In exohed al clus e s, Mn adop s a minimal coo dina ion numbe o Nc = 2 − 4, while in endohed al clus e s Nc is maximized o 11−14, i.e., all Si a oms a e wi hin he i s coo dina ion sphe e o Mn. 2.35 Figu e 4.10: Lowes endohed al isome o MnSi + 10 . The Mn-Si bond leng h is comp essed o only 2.35 ˚ A in his clus e . The a e age Mn-Si bond leng h a elucida es why encap- sula ion o he Mn dopan becomes ene ge ically a o able only o n≥ 11: In he g ound s a e s uc u es, he Mn-Si nea es neighbo dis ance expands om a = 2 . 43 − 2 . 58 ˚ A in exohed al o a = 2 . 53 − 2 . 67 ˚ A in endohed al clus e s. In con as , i would be comp essed o a = 2 . 35 ˚ A in he ene ge ically mos a o able endohed al isome o MnSi+ 10 (as illus a ed in Fig 4.10). E en hough Mn a o s high coo dina ion in Si [175], his s ain, which becomes e en mo e p onounced in smalle clus e s, p ecludes endohed al g ound s a es o n≤10. The ab up change in coo dina ion a he s uc u al ansi ion is in e ela ed wi h he quenching o he magne ic momen as illus a ed in Fig. 4.11: He e, he calcula ed magne ic momen s o MnSi+ n a e plo ed e sus he weigh ed coo dina ion numbe d0Nc/a , which akes in o accoun bo h he numbe o Si nea es neighbo s Nc as well as hei a e age dis ance a o he Mn a om, no malized o he nea es neighbo dis ance d0 in bulk Si. Low-coo dina ed exohed al clus e s wi h d0Nc/a = 1 . 9 − 3 . 7 ( Nc = 2 − 4) ca y a magne ic momen o 4 µB , which is quenched o 0 µB in high-coo dina ed species wi h d0Nc/a = 10 . 2 − 12 . 3 ( Nc = 11 − 14). This ela ion o magne ic momen and weigh ed coo dina ion also holds o highe -ene gy isome s ha a e included in Fig. 4.11 and ma k he ansi ion om magne ic o nonmagne ic impu i ies a ound d0Nc/a ≈4. # o bonds MnSi8 + 2 3 4 5 121086420 magne ic momen 0 1 2 3 4 5 6 14 Mn-Si dis ance 6 0 1 2 3 weigh ed coo dina ion numbe d0Nc/a Figu e 4.11 : Magne ic momen e - sus weigh ed coo dina ion d0Nc/a o g ound s a e (open ci cles) and highe ene gy (solid ci cles) isome s o MnSi+ n ( n = 7 − 14), isola ed neu- al Mn impu i ies in bulk ([151], solid squa es) and amo phous Si ([154], open iangles), and in a Si nanoc ys- al ([156], solid diamond). Inse : bond leng h dis ibu ion o MnSi+ 8 wi h i s coo dina ion sphe e (do ed line). Fig. 4.11 shows ha his obse a ion can be gene alized o ex ended sys ems, i.e., o a neu al Mn impu i y a a subs i u ional si e in c ys alline Si [151] o in hyd ogen-passi a ed 4.3. A DFT-s udy o he magne ic- o-nonmagne ic ansi ion in Mn-doped Si-clus e s 59 Si nanoc ys als [156]. I also applies o e y low concen a ions o Mn in amo phous Si, o which a possible ela ion be ween magne ic momen and coo dina ion has been poin ed ou [154]. Ye , al hough Nc is he leading e m, i does no accoun o he dependence o he local magne ic momen on he nea es -neighbo dis ance [151] ha becomes impo an a ound Nc = 4 and is included in d0Nc/a . As can be seen in Fig. 4.11, Mn-doped bulk Si is jus a he ansi ion om high-spin o low-spin s a es and he e o e eac s e y sensi i ely o changes in d0Nc/a . This migh explain he la ge sca e in expe imen al esul s on Mn-doped Si and indica es ha high-spin s a es could be s abilized by an app op ia e expansion o he la ice pa ame e , e.g., in ul a- hin ilms o passi a ed nanoc ys als. In summa y, he magne ic momen o Mn-doped Si has been in es iga ed o e a wide ange o s uc u al pa ame e s, including ex eme coo dina ion numbe s om 2 - 14. The s udy o singly doped, size-selec ed MnSi+ n clus e s a oids impu i y-band o ma ion o in e ac ion be ween impu i ies ha migh be p esen in expe imen s on bulk samples, bu also in calcula ions wi h pe iodic bounda y condi ions. Thus he obse ed quenching o he magne ic momen is no a esul o impu i y band o ma ion bu o he elec onic in e ac ion wi h he Si hos . While he SIE s ongly a ec s igh ly localized o bi als wi h Mn-3 d cha ac e and is hus pe sis en in exohed ally doped clus e s in which he dopan only weakly in e ac s wi h he Si hos , i only pe u bs he ene ge ic o de ing o exohed al and endohed al isome s o MnSi + 11 . Finally, we ound an uni e sal co ela ion o he magne ic momen and he weigh ed coo dina ion numbe . Bulk Si doped wi h Mn lies a he ansi ion om high o low magne ic momen s and he e o e is pa icula ly sensi i e o he nea es neighbo bond leng hs o he Mn dopan . 5 Summa y and ou look 5.1 Summa y In he p esen hesis s uc u al and elec onic p ope ies o Au-P , Au-Pd nanoalloys and Mn-doped Si clus e s we e s udied using non-empi ical DFT and MD simula ions employing empi ical Su on-Chen po en ials. Au-P and Au-Pd nanoalloys show a bime allic e ec , i.e., inc eased ac i i y and selec i i y in many di e en ca aly ic eac ions. This wo k con ibu es o he long-s anding deba e be ween heo y and expe imen on he mixing pa e n o Au-P nanoalloys. We we e able o show ha a p ominen cha ac e iza ion echnique based on a combina ion o XRD and Vega d’s law elies on ques ionable assump ions. Au shell P co e and homogeneously mixed NP in he expe imen ally ele an size ange o ≈ 4 nm a e indis inguishable in a ypical XRD expe imen . Fu he mo e, i was shown ha Vega d’s law is alid o bo h ypes o mixing pa e ns. Theo y consis en ly p edic s co e-shell s uc u es as ene ge ically mos a o able, bu homogeneously mixed Au-P nanoalloys a e obse ed in expe imen using elemen -speci ic cha ac e iza ion echniques. Ou indings show ha he o ma ion o mixing pa e ns o he han he co e-shell one mus be linked o e ec s such as in e ac ions wi h he suppo ma e ial o he syn hesis p ocedu e. The s udy o he elec onic s uc u e o Au-P and Au-P nanoalloys showed ha he DOS a he Fe mi le el depends on he P and Pd con en o he NP, espec i ely. The DOS a he Fe mi le el is ela ed o he binding ene gy be ween he ca alys and possible adso ba es and i inc eases wi h inc easing P (Pd) con en . Addi ionally, he spa ial dis ibu ion o he HOMO is connec ed o he posi ion o he P (Pd) a oms. Empi ical MD simula ions we e used o ex ac he MSD o he a oms in an Au-P NP. The MSD is ela ed o he a omic mobili y o s uc u al lexibili y o he NP. The la e can be bene icial in ha i allows he ca alys o s uc u ally adjus o eac an s, o m elemen -speci ic binding si es 62 Chap e 5 — Summa y and ou look and egene a e a e a eac ion has aken place. I was shown ha he MSD inc eases wi h inc easing Au con en . This esul could be suppo ed by modeling he NP as su ace slabs and s udying he di usion o a single Au o P ada om on hese su aces. The ac i a ion ba ie o ada om di usion, he simples ype o su ace econs uc ion, was shown o dec ease wi h inc easing Au con en . A combina ion o he undamen al p ope ies DOS a he Fe mi le el and a omic mobili y migh explain why a bime allic e ec is obse ed o Au-P nanoalloys unde e y a ying expe imen al condi ions. A speci ic Au-P a io could amoun o an op imal in e ac ion be ween ca alys and eac an s media ed by an op imal DOS a he Fe mi le el and an op imal amoun o s uc u al lexibili y. In he second pa o his hesis he magne ic p ope ies o Mn-doped Si clus e s we e s udied. These sys ems can be seen as well-de ined models o a semiconduc ing hos ha in e ac s wi h a single magne ic impu i y. A global geome y op imiza ion was ca ied ou o iden i y he g ound s a e s uc u es o MnSi + n (n=7-14). In ag eemen wi h expe imen , i could be shown ha a s uc u al ansi ion om exohed al o endohed al doping om MnSi + 10 o MnSi + 11 is accompanied by comple e quenching o he magne ic momen . The size o he magne ic momen is co ela ed wi h he coo dina ion numbe o he Mn dopan weigh ed wi h i s a e age nea es neighbo dis ance. This ela ion holds also o Mn in amo phous and c ys alline Si and sugges s ways o s abilize he magne ic momen o a Mn impu i y by app op ia e la ice expansion. Sys ems o which he in e ac ion be ween s ongly localized and delocalized o bi als plays a majo ole pose se ious di icul ies o exis ing app oxima ions o DFT because o he well-known sel -in e ac ion p oblem. By e alua ing i s o bi al-SIE we ound ha MnSi + 11 su e s pa icula ly se e ely om sel -in e ac ion. The one-pa ame e hyb id xc unc ional PBE0 which co ec ly p edic ed s uc u es and magne ic momen s o all o he clus e s he e o e yields a w ong, exohed ally doped g ound s a e s uc u e in his case. By using a RSH unc ional we saw ha he co ec ene ge ic o de ing o he endohed al and he exohed al isome is ob ained o ce ain alues o he ange sepa a ion pa ame e . Toge he wi h he o bi al-SIE his inding highligh s how delica ely exchange and co ela ion e ec s de e mine he ene ge ic o de ing in MnSi+ 11. 5.2 An ou look o Ni-Pd nanoalloys The sea ch o bime allic ca alys s ha pe o m e icien ly in la ge-scale ca alysis is equen ly guided by he simple p inciple o ind he combina ion o ma e ials ha yields he highes u no e a es and is as cheap as possible. Ni-Pd nanoalloys a e in his sense op imal. Addi ionally, hey cons i u e a sys em co e ing a a ie y o en icing undamen al ques ions. Fi s ly, small Ni and Pd clus e s a e known o exhibi high magne ic momen s [11, 140]. The size- and composi ion-dependen magne ic momen o Ni-Pd nanoalloys is he e o e expec ed o exhibi in e es ing ea u es. Howe e , Chap. 4.3 showed ha he co ec desc ip ion o 5.2. An ou look o Ni-Pd nanoalloys 63 Table 5.1 : Adso p ion ene gies (in eV) o Pd- and Ni-ada oms on pu e and doped Pd- and Ni su aces o he s a (ini) and he end ( in) po- si ion o each NEB calcula ion. The di usion ba ie (in meV) is de ined as he ene ge ic di e - ence be ween he ini al posi ion and he saddle poin o he minimum ene gy pa h. sys em Eads ini Eads in Eba ie Pd/Pd 2.70 2.70 70 Pd/Pd+Ni 2.72 2.69 76 Ni/Pd 3.48 3.38 296 Pd/Ni 3.14 3.14 65 Ni/Ni+Pd 3.63 3.69 380 Ni/Ni 3.63 3.63 80 he geome ical and elec onic s uc u e e en o allegedly simple magne ic ansi ion me al compounds is a huge challenge o heo y. DFT s udies o small Ni-Pd clus e s ha e up o now only been ca ied ou using s anda d semilocal xc unc ionals [176, 177] and global geome y op imiza ions a e a e [178] o hese sys ems. A emp s o cons uc a eliable pseudopo en ial o Ni ha could be used in conjunc ion wi h o bi al-dependen unc ionals ha e so a p o ed challenging as well [179]. ini fin ini fin (a) (b) Figu e 5.1: Se up o Pd/Ni ada om di usion on Ni/Pd su ace (a) wi hou and (b) wi h a single Pd/Ni a om in he uppe su ace laye . The di usion s ep akes place om one hcp hollow o ano he hcp hollow si e. Secondly, a bime allic e ec in ca aly- sis has ecen ly been obse ed o Ni-Pd nanoalloys immobilized in so-called me- al-o ganic amewo ks [180] as well as in o he eac ions [78]. A close lying ques ion is he e o e whe he he gene al concep s ha apply o Au-P nanoalloys, a e also sui ed o explain he special p ope ies o Ni-Pd pa icles. This would again equi e a ai h ul desc ip ion o hei elec onic s uc u e, a leas close o he Fe mi le el. Fu he mo e, he a omic mobili y o luxionali y o Ni-Pd sys ems has o be es ed. Ini ial s eps in o his di ec ion ha e been aken ollowing a simila line o a gumen as desc ibed in Sec. 3.4.3. Pu e and doped Ni- and Pd-su aces ha e been modeled by 4x4x4-laye slabs wi h equilib ium la ice pa ame e s o 3.520 ˚ A o Ni and 3.939 ˚ A o Pd. A su ace di usion s ep be ween wo equi alen hcp hollow si es as depic ed in Fig. 5.1 has been conside ed. CI-NEB calcula ions we e ca ied ou in o de o de e mine he ac i a ion ene gy o his di usion s ep (see Sec. 2.5). Fu he mo e, he adso p ion ene gies o he ada om a he ini al and he inal posi ion ha e been de e mined using Eq. (3.7) . The esul ing ba ie s and adso p ion ene gies a e lis ed in Tab. 5.1. No e ha bo h he adso p ion ene gies and he ba ie s a e conside ably smalle han o Au-P su aces (c . Tab. 3.3). This inding is in line wi h he ecen obse a ion ha la ge Ni-Pd clus e s (wi h diame e s o ≈ 4 nm) p e e ably o m solid solu ions and ha hei su ace is co uga ed due o he la ge la ice misma ch o Ni and Pd [180]. The s ong endency o seg ega ion o he wo componen s ound in Au-P NP hus seems no o be p esen in Ni-Pd NP. Fu he mo e, no speci ic end o he di usion ba ie s can be seen in Tab. 5.1. Excep ions o he o e all small ba ie s o be ween 65 and 80 meV a e ound o he di usion o a Ni ada om on Pd wi h a ba ie o almos 300 meV and di usion o a Ni ada om on a Ni su ace doped wi h a 70 Appendix A — Tempe a u e and suppo e ec s Figu e A.4: DOS a he Fe mi le el wi h inc easing la ice pa ame e o bulk Au (le ) and P ( igh ). Di e en in eg a ion h esholds a e shown, ha demons a e ha he Au bulk DOS inc eases mo e apidly, as soon as he 5d-de i ed DOS is aken in o accoun . Au P 3.90 3.95 4.00 4.05 4.10 4.15 4.20 la ice pa ame e [Å] up o -1 eV up o -2 eV 0 1 2 3 4 5 4.00 4.05 4.10 4.15 4.20 DOS a Fe mi le el [a b. uni s] up o -1 eV up o -2 eV up o -3 eV subs i u ion o one Au a om in, e.g., Au 38 by a P a om (upon which he mean in e a omic dis ance e en dec eases, as shown in Sec. 3.3), leads o a ela i e inc ease o he DOS a he Fe mi le el o nea ly 100%, while he posi ion o he Fe mi le el emains app oxima ely cons an . The e ec o he mal expansion on he elec onic s uc u e o Au-P NP is he e o e negligibly small as compa ed o he inc ease ha is obse ed o alloying Au clus e s wi h P a oms. A.2 In luence o he nanopa icle suppo The p ac ical use ulness o ca aly ic NP is o en dic a ed by how success ully hey can be s abilized agains coagula ion, by hei handling and hei e-usabili y. S abiliza ion by end g oups ha s ongly in e ac wi h he NP’s su ace can se e ely al e and e en limi he ca aly ic ac i i y. Recen ly, i has been shown ha highly ca aly ically ac i e Au-P nanoalloys wi h a diame e o ≈ 3-4 nm can be syn hesized and immobilized in so-called sphe ical polyelec oly e b ushes (SPBs) [80]. SPBs consis o a solid polys y ene co e on o which long ca ionic polyelec oly e chains a e densely g a ed as ske ched in Fig. A.5. The polyelec oly e is 2-amino-e hyl me hac yla e (AEMH) which is posi i ely cha ged in aqueous solu ion. Cl − anions ep esen he coun e ions which a e mos ly con ined o he su ace laye o he SPB, ha is indica ed in ligh g ay in Fig. A.5. The Cl − coun e ions can be eplaced by me al ions like [AuCl 4 ] − and [P Cl 6 ] −2 . These a e hen educed o elemen al Au and P using NaBH 2 . Du ing he p ocess o Au-P nanoalloy o ma ion he su ace laye conside ably dec eases om abou 70 nm o only 20 nm. This can be a ibu ed o he nanoalloys ca ying a nega i e su ace cha ge and he e o e s ongly in e ac ing wi h he ca ionic polyelec oly e chains. Howe e , his in e ac ion seems o be weak enough no o obs uc he ca aly ic ac i i y o he Au-P nanoalloys [80]. A.2. In luence o he nanopa icle suppo 71 Figu e A.5 : Au-P nanoalloys can be syn hesized wi hin he dense polyelec oly e laye o a sphe ical polyelec oly e b ush. C H CH3 O CH2 NH3 C C H O CH2 n + polys y ene 70nm Cl- Cl- Cl- Cl- Cl- polys y ene 20nm [AuCl4]- [P Cl6]2- NaBH4 + C H CH3 O CH2 NH3 C C H O CH2 n + C H CH3 O CH2 NH3 C C H O CH2 n + C H CH3 O CH2 NH3 C C H O CH2 n + C H CH3 O CH2 NH3 C C H O CH2 n + polyelec oly e laye wi h NP In his sec ion, I wan o p esen i s esul s on how he in e ac ion be ween he NP and he SPB suppo a ec s he undamen al s uc u al and elec onic p ope ies o he Au-P nanoalloys ha ha e been es ablished in he main pa o his hesis. As he simples possible model se up we assume ha he me al NP in e ac wi h monome ic AEMH uni s and ha he combined sys em consis ing o NP and AEMH uni s is neu al. The polys y ene co e is assumed o be ine and will no be conside ed he e. Fu he mo e, we dis ega d he aqueous solu ion in which he eac ion akes place, as we wan o concen a e on he e ec o he NP-AEMH in e ac ion. pu e Au andom co e-shell Figu e A.6: Two AEMH monome uni s a e placed close o he su ace o an Au 20 , a andomly mixed Au10P 10 and a co e-shell Au10P 10 clus e . Ou i s model se up is shown in Fig. A.6. Two AEMH monome uni s a e plac- ed close o he su ace o an Au 20 , a an- domly mixed Au 10 P 10 and a co e-shell like Au 10 P 10 clus e 1 . The dynamical e olu ion o hese sys ems is simula ed us- ing cons an empe a u e MD simula ions ia a Nos´e-Hoo e he mos a o 300 K empe a u e. In he o iginal o mula ion o Nos´e, a canonical ensemble is achie ed by aug- men ing he classical Hamil onian by a ime scale a iable s , i s conjuga e momen um ps and a pa ame e Q HNos´e = M X i=1 p2 i 2mis2+U(q1, ..., qM) + ps 2Q+ (N + 1)kBTln s. (A.4) The ime scale a iable s luc ua es which amoun s o a coupling o a hea ba h o empe a u e T. N is he numbe o deg ees o eedom. I can be shown ha he mic ocanonical dis ibu ion in he augmen ed se o a iables is equi alen o a canonical dis ibu ion o he a iables {q1, ..., qM} , {p0 1, ..., p0 M} , whe e p0 i = pi/s [182]. As i is incon enien o wo k wi h luc ua ing ime in e als in p ac ical simula ions, one o en uses he o mula ion o Hoo e , in which ps/Q is eplaced by a he modynamic ic ion 1 Simila o Sec. 3.2, he label ”co e-shell” indica es ha all P a oms a e as highly coo dina ed and all Au a oms a e as lowly coo dina ed as possible. 72 Appendix A — Tempe a u e and suppo e ec s coe icien η, which can be exp essed ia a phenomenological elaxa ion ime τ[183] η=ps N kBTτ2,(A.5) which is he e chosen ou imes as la ge as he simula ion ime s ep ∆ = 80 a.u. b= 1.9 s. All sys ems we e p opaga ed o abou 16 ps. The elec onic s uc u e in each imes ep was calcula ed using he PBE GGA and a SVP basis se augmen ed as implemen ed in he TURBOMOLE p og am package [184]. As an-de -Waals o ces a e expec ed o play a ole in he in e ac ion be ween he Au-P nanoalloys and he AEMH molecules, he empi ical dispe sion co ec ion o G imme was employed [185], which adds a e m Edisp =−s6 M−1 X i=1 M X j=i+1 Cij 6 R6 ij damp(Rij) (A.6) o he o al ene gy o he sys em. He e, s6 is a global scaling pa ame e , which depends on he xc unc ional employed, Cij 6 a e empi ical dispe sion coe icien s o a om pai i and j , Rij = |Ri−Rj| , damp is a damping unc ion ha a oids singula i ies a small in e a omic dis ances Rij and M deno es he numbe o a oms. To di e en ia e be ween he e ec s o empe a u e and hose o he AEMH monome uni s, a second se o MD simula ions o jus he ba e sys ems wi hou AEMH molecules was ca ied ou . andom pu e Au co e-shell Figu e A.7: S uc u e snapsho aken a he end o each simula ion, showing ha conside - able s uc u al changes ha e se e ely dis o ed he e ahed al geome y o all h ee sys ems. Fig. A.7 p o ides a isual imp ession o he i s impo an esul o hese simula ions. I shows an exempla y s uc u e snapsho aken a he end o he simula ion. The geome y o all h ee sys ems is conside ably di e en om he e ahed al s uc u e we s a ed wi h. Tha his is no only an e ec o empe a u e can be seen by compa ison wi h he s uc u es o he ba e Au 20 and Au 10 P 10 sys ems ha o e all e ain hei e ahed al shape. The s uc u e o he ba e sys em is e en s able up o empe a u es o 600 K, as men ioned in Sec. 3.2 and Re . [94]. In ac , du ing he whole simula ion s ong s uc u al changes ake place in he combined NP-AEMH sys em, while in he ba e sys ems mainly ib a ions o he a oms abou hei equilib ium posi ions can be seen. To quan i y hese e ec s we calcula ed he a e age nea es neighbo bond leng h o he NP wi h and wi hou AEMH monome s o e e y imes ep. Fig. A.8 shows he nea es neighbo bond leng h o Au 20 (le ), andomly mixed Au 10 P 10 (middle) and co e-shell Au 10 P 10 ( igh ). The nea es neighbo bond leng h was de ined as he a e age dis ance o e e y a om o i s ou nea es neighbo s. No e ha he elaxed co e-shell Au 10 P 10 was aken as he ini ial geome y o all NP. Fo Au 20 all P a oms in his NP we e hen eplaced by Au a oms. Fo he andomly mixed clus e , he P a oms we e edis ibu ed o e he clus e as depic ed in Fig. A.6. This means, ha one has o compa e he nea es A.2. In luence o he nanopa icle suppo 73 Figu e A.8: A e age nea es neighbo bond leng h o e e y ime s ep o he simula ion o he Au 20 (le ), he Au 10 P 10 andomly mixed (middle) and he Au 10 P 10 co e-shell sys em wi h ( op) and wi hou (bo om) AEMH monome s. The yellow line, a mo ing window a e age, is a guide o he eye. 2.75 2.80 2.85 2.90 2.95 0 2000 4000 6000 8000 10000 12000 14000 16000 nea es neighbo dis ance [Å] Au20 2.75 2.80 2.85 2.90 2.95 0 2000 4000 6000 8000 10000 12000 14000 16000 Au20 wi h AEMH wi hou AEMH simula ion ime [ s] 2.70 2.75 2.80 2.85 2.90 0 2000 4000 6000 8000 10000 12000 14000 16000 andom 2.70 2.75 2.80 2.85 2.90 0 2000 4000 6000 8000 10000 12000 14000 16000 andom 2.70 2.75 2.80 2.85 2.90 0 2000 4000 6000 8000 10000 12000 14000 16000 co e-shell 2.70 2.75 2.80 2.85 2.90 0 2000 4000 6000 8000 10000 12000 14000 16000 co e-shell neighbo bond leng h o he NP-AEMH sys em wi h he equilib ium bond leng h o he ba e NP, a e all ansien p ocesses ha e decayed. On a e age, he nea es neighbo bond leng h inc eases om 2.84 ˚ A o 2.85 ˚ A in Au 20 , om 2.78 ˚ A o 2.79 ˚ A in he andomly mixed Au 10 P 10 clus e and om 2.78 ˚ A o 2.80 ˚ A in he co e-shell Au 10 P 10 clus e . Mo e signi ican han his sligh inc ease o he a e age bond leng h by less han 1% is ha he nea es neighbo bond leng h luc ua es much mo e s ongly in he NP-AEMH sys ems han in he ba e clus e s. This is consis en wi h he la ge s uc u al changes in he o me . Addi ionally, we coun ed he numbe o Au-Au, Au-P and P -P bonds o he wo Au 10 P 10 sys ems in each simula ion s ep. A bond is he e de ined as e e y in e a omic dis ance ≤ 2.9 ˚ A. Fo he ba e NP hese numbe s a e cons an du ing he whole simula ion. Fo he combined NP-AEMH sys ems one has o dis inguish wo ime domains, simila ly as o he nea es neighbo bond leng h. The ansien ime domain om 0 o ≈ 6000 s and he s eady s a e un il he end o he simula ion. O cou se, ansien e ec s a e also p esen o he ba e clus e s, bu hey do no a ec he numbe o Au-Au, Au-P and P -P bonds he e. Tab. A.1 shows he a e age s eady s a e numbe o bonds and he espec i e s anda d de ia ions o Au 10 P 10 wi h and wi hou AEMH monome s. Again, we compa e he a e age numbe o bonds o he combined NP-AEMH sys em wi h he numbe o bonds o he ba e NP. Fo bo h Au 10 P 10 +AEMH sys ems he numbe o Au-Au bonds dec eases as compa ed o hei ba e coun e pa s. Fo he andomly mixed clus e also he numbe o Au-P bonds dec eases, while he numbe o P -P bonds inc eases. Fo he co e-shell clus e he numbe o Au-P bonds sligh ly inc eases and he numbe o P -P bonds dec eases. I wan o s ess ha he o e all numbe o Au-Au, Au-P and P -P bonds is oo small o allow conclusi e 74 Appendix A — Tempe a u e and suppo e ec s wi hou AEMH andom co e-shell Au-Au 4 ±1 8 ±1 Au-P 22 ±2 13 ±2 P -P 12 ±1 19 ±1 wi h AEMH andom co e-shell Au-Au 2 ±1 5 ±2 Au-P 20 ±2 14 ±2 P -P 14 ±1 16 ±1 Table A.1 : Numbe o Au-Au, Au-P and P - P bonds ≤ 2.9 ˚ A o andomly mixed and co e- shell clus e s wi hou and wi h wo AEMH monome s. The a e ages o he ba e sys em is aken o e he comple e simula ion ime. Fo he combined sys em he a e age was compu ed om 6000 o 16 000 s. s a emen s he e. A i s cau ious in e p e a ion o hese esul s, howe e , indica es ha he co e-shell clus e pa ly looses i s clea P -co e/Au-shell cha ac e is ics, while he andomly mixed clus e unde goes s uc u al changes making i mo e co e-shell like. In o he wo ds, in he combined NP-AEMH sys em he di e en ia ion be ween andomly mixed and co e-shell clus e s is no meaning ul anymo e. Fi s ly, because on a e age clus e s will exhibi an in e media e mixing pa e n in be ween co e-shell and a andom dis ibu ion o Au and P . Secondly, a ealis ic expe imen al condi ions, i.e., a ini e empe a u e and embedded in a suppo ing ma ix, Au-P nanoalloys canno be desc ibed in a s a ic, monos uc u e ashion anymo e [146]. Indeed, in ou case a 300 K in e ac ion wi h he AEMH monome s seems o be he dominan ac o ha induces s uc u al luxionali y. Figu e A.9: To al ene gy o andomly mixed and co e-shell Au 10 P 10 in he MD simula ion o he combined NP-AEMH sys em and he ba e NP sys em. −93394 −93392 −93390 −93388 −93386 −93384 −93382 −93380 0 2000 4000 6000 8000 10000 12000 14000 16000 o al ene gy [eV] simula ion ime [ s] andom co e-shell −69464 −69462 −69460 −69458 −69456 −69454 −69452 −69450 0 2000 4000 6000 8000 10000 12000 14000 16000 simula ion ime [ s] andom co e-shell wi h AEMH wi hou AEMH These conclusions a e u he suppo ed by conside ing he o al ene gy o he wo Au 10 P 10 sys ems wi h and wi hou AEMH suppo as depic ed in Fig. A.9. Fo he ba e sys em he o al ene gy o he co e-shell clus e is consis en ly lowe han ha o he andomly mixed pa icles. This esul is no su p ising conside ing ha s uc u e and mixing pa e n in hese simula ions emain la gely undis u bed. The si ua ion is comple ely di e en o he combined NP-AEMH sys em. He e he co e-shell clus e -AEMH sys em is ene ge ically mo e a o able a he beginning o he simula ion. F om ≈ 6000 s on he o al ene gies o he wo sys ems a e nea ly iden ical. Finally, we will ake a close look a he elec onic s uc u e o ou model sys em. To A.2. In luence o he nanopa icle suppo 75 his end, we ook 11 s uc u e snapsho s o each sys em, being 194 s apa om each o he ( b= 100 MD s eps) s a ing a 13 545 s. The elec onic s uc u e o hese sys ems was calcula ed using a TZVPP basis se . No s uc u al elaxa ion was ca ied ou , as we a e in e es ed in he elec onic s uc u e o each pa icula snapsho he e. Fo each s uc u e snapsho we calcula ed he o al DOS and he me al DOS o he combined NP-AEMH sys em (in he ollowing called me al-PDOS), which is ob ained by p ojec ing he o al DOS on o he Au and P a oms. Fig. A.10 shows he o al DOS o he ba e Au 20 and Au 10 P 10 sys ems and he espec i e me al-PDOS o he combined NP-AEMH sys ems. The depic ed DOS’s ep esen he a i hme ic mean o he DOS’s o each o he 11 s uc u e snapsho s o each sys em. The me al-PDOS o he combined sys em is in e es ing, because we assume ha possible eac an s in a ca aly ic p ocess a e adso bed o he NP and eac a i s su ace. Thus, ollowing he conside a ions o Sec. 3.4.1, he in e ac ion s eng h be ween he ca alys s and he adso ba es is go e ned by he me al-PDOS a he Fe mi le el. Figu e A.10: To al DOS o ba e NP (black) and DOS p ojec ed on o he me al a oms o he combined NP-AEMH sys em (yellow) a e aged o e 11 s uc u e snapsho s each, as explained in he ex . 0 500 1000 1500 2000 2500 3000 −10 −8−6−4−2 0 densi y o s a es NP +AEMH PDOS ba e NP DOS −10 −8−6−4−2 0−10 −8−6−4−2 0 pu e Au andom co e-shell The le panel o Fig. A.10 again demons a es he special elec onic s uc u e o he ba e Au 20 e ahed on [68]. I ’s DOS exhibi s disc e e ea u es and is high close o he Fe mi le el as compa ed o o he Au clus e s (see Sec. 3.4.1). In con as , he DOS o he ba e Au 10 P 10 clus e s, is mo e con inuous and band-like. No e ha con a y o he isual imp ession he DOS a he Fe mi le el o Au 10 P 10 is signi ican ly highe han ha o Au 20 in acco dance wi h he esul s o Sec. 3.4.1. No iceably, Fig. A.10 shows ha he me al-PDOS a he Fe mi le el is conside ably educed as compa ed o he ba e NP case. This is mos p onounced o Au 20 . He e, he me al-PDOS is mo e con inuous han ha o he ba e NP. Mo e impo an ly, he me al-PDOS a he Fe mi le el is educed by 60% as compa ed o he o al DOS o he ba e NP. The same e ec , hough less dis inc , is seen o Au 10 P 10 . The me al-PDOS a he Fe mi le el is educed by 16% o he andomly mixed and by 30% o he co e-shell clus e . The la ge de ia ions be ween he DOS educ ion o he wo Au 10 P 10 sys ems can be asc ibed o he o e all small numbe o s uc u al snapsho s o which he DOS was e alua ed. A educ ion o he DOS a he Fe mi le el is expec ed o lead o weake in e ac ions wi h possible adso ba es. Whe he i esul s in an inc eased o dec eased ca aly ic ac i i y, howe e , depends on he speci ic eac ion one is in e es ed in. Qui e gene ally he esul s o his sec ion indica e ha he in e ac ion o 76 Appendix A — Tempe a u e and suppo e ec s ou 20-a om Au and Au-P NP wi h AEMH monome uni s leads o a ma ked inc ease o he luxionali y o he clus e s on he one hand. On he o he hand he DOS a he Fe mi le el dec eases in he combined NP-AEMH sys em. In e ms o he olcano pic u e in oduced in Sec. 3.4.4, his migh co espond o a shi o he olcano peak as compa ed o he ba e NP case. Ou indings suppo he no ion ha liquid-phase ca alysis canno be modeled using he same concep s ha ha e success ully been applied o gas-phase ca alysis on su aces [131]. Unde ealis ic expe imen al condi ions he desc ip ion o suppo ed Au-P nanoalloys as s a ic en i ies being go e ned by hei ( acuum and ze o- empe a u e) g ound s a e p ope ies is likely o be lawed. Ye , e en hough he esul s p esen ed in his sec ion allow aluable i s insigh s in o he in e ac ion be ween suppo molecules and Au-P NP u he simula ions ha e o be ca ied ou o achie e conclusi e esul s and o judge he signi icance o he indings p esen ed abo e. Fi s , simula ions ha e o be done o la ge NP, e.g., 40-a om Au-P nanoalloys, possibly wi h a less ”special” geome y han he e ahed on, which has only su ace and no olume a oms and is he e o e expec ed o be pa icula ly eac i e. Second, i has o be cla i ied how signi ican he ype o NP suppo in e ms o he epo ed e ec s is. Typical suppo s ha could be modeled wi h easonable compu a ional e o a e, e.g., ca bon [87] o silica [79] suppo s. B Op ical p ope ies o Au-P nanoalloys The use o inely dispe sed Au pa icles in uby glass and he colo ul s ained-glass windows o Eu opean’s medie al ca hed als shows ha hei special op ical p ope ies ha e been known o cen u ies. The i s scien i ic examina ion o he subjec is a ibu ed o Michael Fa aday, who s udied he in e ac ion o ligh wi h colloidal Au pa icles, as well as hin Au ilms and lea es as ea ly as 1857 [186]. Today i is well unde s ood ha he beau i ul colo o suspensions o colloidal Au is a esul o collec i e exci a ions o he conduc ion elec ons ha domina e he abso p ion spec um, accumula ing in a so called Mie esonance o su ace plasma esonance 1 . The posi ion o he Mie esonance depends on he size o he Au colloids, bu he op ical ma e ial unc ions ha desc ibe he linea esponse o he clus e s o elec omagne ic wa es, a e la gely size-independen and simila o he bulk alues. Wi h hese op ical ma e ial unc ions one can de e mine he op ical esponse o ( a he la ge) NP using classical elec odynamics and he so called Mie heo y [188]. Fo NP smalle han ≈ 10 nm, ha a e s ill oo la ge o assess hem wi h quan um chemical me hods such as ime-dependen DFT, he si ua ion is mo e di icul . The op ical esponse unc ions become size dependen and can de ia e subs an ially om he bulk. A comp ehensi e accoun on op ical p ope ies o me al clus e s is gi en in Re . [187] and pe inen e e ences he ein. He e, he ocus will be on sphe ical clus e s o 10 nm diame e ( R = 5 nm) in e ac ing wi h isible ligh o wa eleng h λ , i.e., Rλ . The op ical esponse o clus e s in his size ange can be de e mined by i ue o Mie heo y. Fo Rλ (quasi-s a ic case) phase e a da ion e ec s and highe mul ipole o de s can be neglec ed and he abso p ion c oss sec ion in he dipola limi is σ(ω) = 4πω c√εm=[α(ω)],(B.1) 1 The e m su ace plasma esonance e e s o he su ace pola iza ion being he main es o a i e o ce in clus e s. This su ace pola iza ion is due o cha ges wi hin he elec onic sc eening leng h [187]. 78 Appendix B — Op ical p ope ies o Au-P nanoalloys whe e = [ α ( ω )] is he imagina y pa o he pola izabili y o he clus e , which o a homogeneous sphe e is αhomogeneous(ω) = ε−εm ε+ 2εm εmR3.(B.2) −10 −5 0 5 10 ε1 ε2 −100 −50 0 50 100 0 1 2 3 4 5 dielec ic unc ion ene gy [eV] ε1 ε2 dielec ic unc ion Gold Pla inum Figu e B.1: Real and imagina y pa o he dielec ic unc ion o bulk Au and P , aken om Re . [189] and [190]. In Eq. (B.1) and (B.2) , ε is he (complex and ω -dependen ) dielec ic unc ion o he clus e . We will he e assume ha ε is simila o i s bulk alue. The alidi y o his assump ion will be discussed below. The eal and imagina y pa o he dielec ic unc ion ε = ε1 + iε2 desc ibe pola iza ion and ene gy dissipa ion o ma e , espec i ely, and a e ela ed by he amous K ame s-K onig ela ions. Fig. B.1 shows ε1 and ε2 o bulk Au and P as abula ed in Re . [189] and [190]. Fu he mo e, Eq.(B.1) and B.2 con ain he ela i e pe mi i i y εm o he ma ix in which he pa icle is embedded. Fo all ollowing calcula ions εm = 2 . 7 was used, which co esponds o he ela i e pe mi i i y o polys y ene, one o he majo ing edien s o he polyelec oly e ma ices used o Au-P nanoalloys s abiliza ion in Re . [80]. Finally, c is he acuum speed o ligh . Fig. B.2 depic s he op ical abso p ion spec a o a pu e Au and a pu e P NP o 10 nm diame e . The Mie esonance o he Au NP can clea ly be seen a a ound 2.25 eV. The P NP exhibi s no Mie esonance in he isible ene gy ange. 0 200 400 600 800 1000 1200 1400 1600 1800 0 1 2 3 4 5 abso p ion c oss sec ion [nm2] ene gy [eV] pu e Au pu e P Figu e B.2 : Op ical abso p ion spec- um calcula ed using Mie heo y in he dipola limi o sphe ical me al sphe es o 10 nm diame e . Con a y o P , Au NP show a p onounced Mie esonance. 79 A his poin he ques ion a ises how he op ical esponse o an Au-P nanoalloy will e ol e as a unc ion o he Au/P a io and how he mixing pa e n o Au and P in luences he op ical abso p ion spec um o he NP. The op ical p ope ies o Au-P NP ha e been s udied expe imen ally [191, 192] and o co e-shell clus e s also heo e ically using Mie- heo y as will be explained in he ollowing [192]. Ou app oach ollows a s udy o Ni-Ag clus e s by Gaud y e al. [193]. Mie heo y can be ex ended o he sca e ing o an elec omagne ic wa e by wo concen ic sphe es [194]. In he dipola limi one hen ob ains o he pola izabili y αco e-shell(ω) = (εshell −εm)(εco e + 2εshell) + (εco e −εshell)(εm+ 2εshell) (εshell + 2εm)(εco e + 2εshell)+2 (εshell −εm)(εco e −εshell)εmR3,(B.3) whe e εco e and εshell a e he dielec ic unc ions o he co e and he shell, espec i ely. He e, = Rc/R and Rc is he adius o he co e, ha one can es ima e using he Au-P a io (1 −x)/x and he a omic adii o Au and P o a AushellP co e clus e Vco e =4 3πR3 c(B.4) Vshell =4 3π(R3−R3 c).(B.5) Assuming he clus e o consis o MAu Au and MP P a oms MAu =Vshell VAu (B.6) MP =Vco e VP ,(B.7) whe e VAu,P =4 3π 3 Au,P is he olume occupied by a single Au o P a om, one ob ains MAu MP =1−x x= 3 P 3 Au R3−R3 c R3 c .(B.8) Wi h his, he co e adius is de e mined o be app oxima ely Rc=R3 sx 3 P (1 −x) 3 Au +x 3 P .(B.9) Fo εco e and εshell we ake he bulk dielec ic unc ions o P and Au. This migh no be jus i ied, no ably o he shell egion and only small pe cen ages o Au. Howe e , o he case o Ni-Ag NP (R=1–2.5 nm) his app oxima ion ep oduced he expe imen ally obse ed Mie esonance and i s blueshi wi h inc easing Ni-con en quali a i ely well [193]. Fo a Au-P solid solu ion, he op ical abso p ion can be de e mined using Eq. (B.1) and (B.2) wi h an a e aged dielec ic unc ion [187] ε=εP x0Au1−x0(ω) = x0εP (ω) + (1 −x0)εAu(ω),(B.10) 86 Appendix C — Su ace slab calcula ions wi h VASP The selec i e dynamics lag allows o de e mine which a omic posi ions a e supposed o be kep ixed du ing a geome y op imiza ion o a ansi ion s a e sea ch using he NEB me hod (see Sec. 2.5). Fixing, e.g., he bo om ew laye s o he su ace slab a he bulk equilib ium la ice cons an can educe compu a ional cos conside ably and simul aneously imp o e he con e gence pe o mance. This, howe e , has o be es ed ca e ully, as will be explained in he ollowing sec ion. Simila ly, he numbe o a oms pe su ace laye depends on he speci ics o he p oblem one is in e es ed in. In he p esen wo k 4x4 a oms pe laye we e necessa y o a oid in e ac ions be ween ada oms in adjacen uni cells. C.3 Slab calcula ions P io o any slab calcula ion one has o de e mine he numbe o k-poin s ha yield accu a e esul s a easonable compu a ional cos . In he di ec ion no mal o he su ace one k-poin is usually su icien 1 . In he wo o he di ec ions one can s a wi h he bulk alues and hen es con e gence o he ee ene gy when dec easing he alue along hese wo di ec ions. Fo he 4x4x4 a oms su ace slabs a 5x5x1 k-poin mesh was used h oughou all calcula ions. 620 625 630 635 640 645 650 655 660 2 3 4 5 6 7 8 9 su ace ene gy/a om [meV] numbe o laye s un elaxed elaxed Figu e C.3: Su ace ene gy pe a om o a 2x2 a oms pe laye P (111) slab as a unc ion o he numbe o laye s. One hen has o de e mine how many laye s o bulk a e necessa y o ge con- e ged esul s o he su ace ene gy σ=1 2(Eslab −MEbulk),(C.3) whe e Eslab is he ee ene gy o he slab, Ebulk is he ee ene gy pe a om o he bulk sys em and M is he numbe o a oms used o cons uc ion o he su ace slab. The ac o 1 / 2 a ises as a consequence o he su ace slab ha ing wo su aces. Fig. C.3 depic s σ pe a om as a unc ion o he numbe o laye s (using 4 a oms pe laye ) o a P (111) su ace slab. As 16 a oms pe laye we e used o he inal calcula ions, a minimum numbe o 4 laye s is chosen. No e, ha e en his seemingly small se up yields uni cells con aining 64 a oms which in he case o a pu e P slab, using he PAW me hod o desc ibe he elec on-ion in e ac ion (see Sec. 2.4), co esponds o a o al o 640 elec ons. The compu a ional cos can be educed by ixing he bo om wo laye s a he equilib ium la ice pa ame e o he bulk using VASP’s selec i e dynamics lag. I was es ed ha eac ion ba ie s change only insigni ican ly as compa ed o a ull elaxa ion o he slab. 1 Using mo e han one k-poin in his di ec ion only imp o es he desc ip ion o he in e ac ion be ween adjacen uni cells ha has o be minimized anyway by choosing an adequa e amoun o acuum space in be ween neighbo ing slabs. C.3. Slab calcula ions 87 -164.5175 -164.5170 -164.5165 -164.5160 -164.5155 -164.5150 5 10 15 20 25 30 35 ene gy [eV] acuum heigh [Å] ee Figu e C.4: F ee ene gy o a 2x2x7 P (111) su ace slab as a unc ion o he acuum be ween adjacen uni cells. Finally, he con e gence wi h espec o he acuum heigh in be ween neighbo ing slabs has o be es ed. Fig. C.4 shows he ee ene gy o a 2x2x7 laye P (111) slab as a unc ion o acuum heigh . Fo he inal calcula ions a acuum heigh o 20 ˚ A was chosen. The ini ial and inal con igu a ion be- ween which he NEB algo i hm is supposed o ind a ansi ion s a e, ha e o be elaxed p io o se ing up he NEB calcula ion. The ini ial guess o he elas ic band can be a linea in e pola ion be ween s a and end con igu a ion. I one is in e es ed in he en- e gy o he saddle poin i is ecommended o use he CI-NEB oge he wi h a mini- mum amoun o elas ic band images. Fo he simple di usion s ep conside ed in case o he Au-P su aces one single image was su icien o de e mine he eac ion ba ie . Using h ee ins ead o one image did no change he heigh o he ba ie signi ican ly. Howe e , i a u h ul map o he MEP is equi ed, mo e images migh be necessa y depending on he pa icula s o he sys em one is in e es ed in. A ypical inpu ile used o he NEB calcula ions p esen ed in Sec. 3.4 is SYSTEM = P /P (111) #inse sys em name ENCUT = 450 #plane wa e cu o ISTART = 0 #s a calcula ion om sc a ch ISPIN = 2 #allow spin pola ized solu ion PREC = accu a e #con ols se e al p ecision pa ame e s # elax NSW = 100 #maximum o 100 ionic s eps IBRION = 2 #conjuga e g adien algo i hm EDIFFG = -0.02 # elax ole ance SPRING = -5 # u ns on NEB IMAGES = 1 #numbe o images LCLIMB = .TRUE. # u n on CI-NEB IALGO=48 #algo i hm o elec onic elaxa ion LREAL=au o #p ojec ion ope a o s e alua ed in eal space LCHARG=.TRUE. #w i e elec onic densi y LWAVE=.TRUE. #w i e wa e unc ions # hese can be u ned o i memo y is an issue Lis o abb e ia ions AEMH 2-amino-e hyl-me hac yla e Au Gold (CI)-NEB (Climbing-image)-Nudged Elas ic Band DFT Densi y Func ional Theo y GGA Gene alized g adien app oxima ion (G)KS (Gene alized) Kohn-Sham HOMO Highes occupied (gene alized) Kohn-Sham o bi al LDA Local densi y app oxima ion LUMO Lowes unoccupied (gene alized) Kohn-Sham o bi al MEP Minimum ene gy pa h Mn Manganese MD Molecula dynamics Ni Nickel NP Nanopa icle(s) OEP Op imized e ec i e po en ial PAW P ojec o augmen ed wa es PBE xc unc ional o Pe dew, Bu ke and E nze ho Pd Palladium P Pla inum RSH Range-sepa a ed hyb id SIC Sel -in e ac ion co ec ion SIE Sel -in e ac ion e o Si Silicon SVP Spli alence basis se TZVPP T iple-ζbasis se VASP Vienna-Ab-ini io-Simula ion Package XAS X- ay abso p ion spec oscopy xc exchange-co ela ion XMCD X- ay magne ic ci cula dich oism XRD X- ay di ac ion Lis o Publica ions 1. 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[194] A. L. Aden and M. Ke ke , “Sca e ing o elec omagne ic wa es om wo concen ic sphe es”, J. Appl. Phys. 22, 1242 (1951). Acknowledgmen I hank he ollowing people o hei help and suppo wi h his hesis: S ephan K¨ ummel . I ha e i on good au ho i y, ha ≈ 8.23 ou o 10 o my sen- ences du ing an a e age week in he las couple o yea s s a ed wi h ”S ephan says ha ...”. I hank him o he mos inspi ing quan um mechanics lec u e o all ime (in my second yea o unde g adua e s udy), o con incing me o s ay jus a li le bi longe in Bay eu h (in my hi d and i h yea ) and o all he ad ice, discussions and pa ience du ing he las yea s ha we e o ines imable alue o me. Rod igo Albuque que , who has been inc edibly pa ien in pe o ming ye ano he molecula dynamics un. I hank him o inspi ing discussions abou he bime allic e ec in Au-P nanoalloys, aluable insigh in o molecula dynamics simula ions and his indes uc ible en husiasm o ou p ojec . Adam Fos e , o always lending a hand when i came o Nudged Elas ic Band calcula ions. I enjoyed his and his g oup’s hospi ali y in Tampe e, Finland, o six win e ly weeks and lea ned a lo abou VASP, ansi ion s a e sea ch and (no so) Finnish bee . Tobias Lau and Vicen e Zamudio-Baye , o coun less physics discussions, he mos amusing science gossip, a desk in Tobias’ o ice whene e I a eled o Be lin, o mo e suppo han I e e expec ed o ecei e and o le ing me con ince hem ha DFT can ac ually be a lo o un. And eas Ka olewski has been my bes iend o he las 8 yea s and he bes o ice is-`a- is one could wish o . F om physics o poli ics and especially ”beyond”; he e is li le ha And eas does no know abou . He always imp essed me wi h his g ea cu iosi y and s amina. Besides, he’s jus p e y cool. Ma hias Dau h , Tobias Schmid and all he o he membe s and alumni o he K¨ ummel g oup . I hank you o making hese las yea s mos exci ing and ins uc i e bu i s and o emos o making hem an aw ul lo o un. Fu he mo e, I hank And eas, Ma hias and Tobias o hei ho ough p oo - eading o pa s o his hesis. Ma kus Hil , Be nha d Winkle and Monika Bi kelbach o echnical and ad- minis a i e suppo whene e i was needed. My amily, bu especially Ka ena and J¨ o g , he second physicis in he amily, who a oused an ea ly in e es o physics in me. Kons an in Hi sch , o pa ien p oo - eading o his hesis and o many help ul commen s, ad ice and discussion. Mos o all, I wan o hank him o p o ing an easy and a di icul hing o me, ha changed my li e in coun less ways: 1. Only a physicis can be a physicis . 2. All can be well. Fo inancial and in ellec ual suppo I am g a e ul o he SFB 840 o he DFG, he Wilhelm und Else He aeus S i ung and he Eli e S udy P og amme Mac o- molecula Science. E kl¨a ung Hie mi e kl¨a e ich, dass ich die o liegende A bei selbs s ¨andig e ass und keine ande en als die angegebenen Quellen und Hil smi el e wende habe. Die A bei wu de wede in gleiche noch in ¨ahnliche Fo m bei ande en P ü ungsbehö den zu E langung eines akademischen G ades o geleg . Ich e kl¨a e, dass ich keine Hil e on gewe blichen P omo ionsbe a e n bzw. - e mi le n ode ¨ahnlichen Diens leis e n in Ansp uch genommen habe und auch nich beabsich ige diese zuk¨un ig in Ansp uch zu nehmen. Wei e hin e kl¨a e ich, dass ich bishe keinen ande wei igen P omo ions e such un e nommen habe. Bay eu h, den 29. Mai 2013 Linn Leppe