Structural and electronic properties of transition metal nanoalloys and magnetic compounds

Structural and electronic properties of transition metal nanoalloys and magnetic compounds Genehmigte Abhandlung zur Erlangung des akademischen Grades eines Doktors der Naturwissenschaften (Dr. rer. nat.) im Promotionsprogramm Physik Weicher Materie, Nichtlineare Dynamik und Festk¨orperphysik der Bayreuther Graduiertenschule f¨ur Mathematik und Naturwissenschaften von Linn Leppert geboren in Chemnitz 1. Gutachter: Prof. Dr. Stephan K¨ummel 2. Gutachter: Prof. Dr. Matthias Schmidt 3. Gutachter: Prof. Dr. Paul-Gerhard Reinhard Tag der Einreichung: 28. Mai 2013 Tag des Kolloqiums: 5. September 2013 I might be wrong I could have sworn I saw a light coming on (Radiohead) Abstract In transition metal clusters, potentially profitable technological applications and fascinating fundamental questions are closely connected. Bimetallic nanoalloys, e.g., have become increasingly popular as their performance in catalysis is often superior to their pure counterparts. Exemplary for this are gold-platinum (Au-Pt) nanoalloys that have been used as highly potent catalysts in electrocatalysis and in a variety of oxidation reactions. However, the mere existence of Au-Pt nanoalloys is astonishing, as Au and Pt cannot be mixed in bulk over a wide range of compositions. Furthermore, how a combination of Au and Pt in nanoalloys results in their special properties has not yet been determined conclusively. It has been shown in empirical simulations and first-principles density functional theory (DFT) calculations that Au-Pt nanoalloys preferably arrange in a core-shell mixing pattern with Au forming a shell around a Pt core. This is in contradiction to many experimental studies that report the formation of solid solutions of Au and Pt. In the present work, this seeming discrepancy is addressed by simulating x-ray diffraction patterns that are experimentally used to characterize nanoalloys. It is shown that the interpretation of the diffraction patterns relies on questionable assumptions and therefore does not suffice as a definite characterization tool for Au-Pt nanoalloys. To shed light on the special catalytic properties of Au-Pt nanoalloys under rather different experimental conditions, a thorough investigation of their electronic and structural properties has been carried out. It is found that features favorable for catalysis in Au-Pt nanoalloys emerge as a consequence of combining two fundamental properties: Pt contributes a high density of states close to the Fermi level, which promotes chemical activity. Au increases the structural flexibility of the Au-Pt system, which might be beneficial for the formation of active and element-specific binding sites as well as regeneration of the catalyst after the reaction. Although DFT offers an attractive compromise between computational effort and accuracy for a theoretical description of Au-Pt nanoalloys, other transition metal compounds severely challenge existing DFT approximations. Manganese (Mn) doped silicon (Si) clusters represent an ideal model system to study the interaction of a single magnetic impurity with a semiconducting host both experimentally and theoretically. The transition from exohedral (lowly coordinated) to endohedral (highly coordinated) doping that occurs for Si clusters with more than ten atoms, is accompanied by complete quenching of the magnetic moment of Mn. We show that MnSi + 11 , the smallest endohedral cluster found in experiment, suffers strongly from a well-known general problem of most DFT approximations: the self-interaction error. Finally, a universal correlation between magnetic moment and the coordination of the Mn dopant is established that can be generalized to extended systems and suggests a route to stabilize the magnetic moment of bulk Mn-Si compounds. Kurzdarstellung In ¨ Ubergangsmetallclustern liegen gewinnbringende technische Anwendungen und fundamentale Fragen oftmals nah beieinander. Dimetallische Nanolegierungen erfreuen sich beispielsweise großer Beliebtheit, weil sie in diversen katalytischen Reaktionen den entsprech-enden reinen Metallen ¨uberlegen sind. Gold-Platin (Au-Pt) Nanolegierungen wurden als ¨außerst wirksame Katalysatoren in der Elektrokatalyse und f¨ur eine Reihe von Oxidationsreaktionen identifiziert. Genau genommen ist jedoch die bloße Existenz dieser Nanolegierungen erstaunlich, da Au und Pt in ausgedehnten Systemen kaum mischbar sind. Bislang ist zudem nicht klar, auf welche Weise eine Kombination von Au und Pt in einer Nanolegierung zu speziellen Eigenschaften f¨uhrt. In empirischen Simulationen und nicht-empirischen Dichtefunktionaltheorie (DFT) Rechnungen konnte gezeigt werden, dass Au-Pt Nanolegierungen ein Au Schale Pt Kern Mischungsmuster bevorzugen. Dies steht im Widerspruch zu experimentellen Studien, in denen homogen gemischte Cluster beobachtet wurden. In der vorliegenden Arbeit wird diese scheinbare Diskrepanz durch Simulation von R¨ontgenbeugungsmustern untersucht. Es wird gezeigt, dass die ¨ubliche Auswertung dieser Beugungsmuster auf mehrdeutigen Annahmen beruht und somit nicht zur alleinigen Charakterisierung von Au-Pt Nanolegierungen ausreicht. Um Aufschluss ¨uber die hohe katalytische Aktivit¨at von Au-Pt Nanolegierungen zu erlangen, wurden ihre elektronischen und strukturellen Eigenschaften untersucht. Es wird gezeigt, dass sich die g¨unstigen Eigenschaften von Au-Pt Nanolegierungen als Konsequenz der Kombination zweier fundamentaler Eigenschaften ergeben k¨onnen: Pt tr¨agt zu einer hohen Zustandsdichte am Ferminiveau bei, welche f¨orderlich f¨ur die chemische Aktivit¨at sein kann. Mit steigendem Au-Anteil steigt wiederum die strukturelle Flexibilit¨at der Au-Pt Systeme. Dies kann f¨ur die Bildung aktiver und elementspezifischer Katalysezentren sowie zur Regeneration des Katalysators nach der Reaktion von Nutzen sein. DFT bietet f¨ur die Behandlung von Au-Pt Nanolegierungen einen guten Kompromiss zwischen rechnerischem Aufwand und Genauigkeit. Im zweiten Teil dieser Arbeit geht es jedoch um eine andere ¨ Ubergangsmetallverbindung, welche existierende N¨aherungen der DFT auf eine harte Probe stellt. Siliziumcluster (Si) dotiert mit einem einzelnen Manganatom (Mn) sind ein ideales Modellsystem um die Frage zu untersuchen, wie eine magnetische Verunreinigung mit einem halbleitenden Wirtsmaterial wechselwirkt. Der ¨ Ubergang von exohedraler (niedrige Koordination) zu endohedraler (hohe Koordination) Dotierung findet f¨ur Cluster mit mehr als zehn Si-Atomen statt und wird von einer kompletten Ausl¨oschung des magnetischen Moments begleitet. In dieser Arbeit wird gezeigt, dass MnSi + 11 , der kleinste experimentell identifizierte endohedrale Cluster, besonders stark unter einem wohlbekannten Problem vieler DFT N¨aherungen leidet: dem Selbstwechselwirkungsfehler. Abschließend wird ein universeller Zusammenhang zwischen dem magnetischen Moment und der Koordination des Mn-Dotieratoms gezeigt, der auch auf Festk¨orpersysteme ¨ubertragen werden kann. Dieser Zusammenhang er¨offnet die M¨oglichkeit das magnetische Moment von ausgedehnten Mn-Si Verbindungen zu stabilisieren. Contents 1 Introduction 1 2 Theoretical and technical framework 5 2.1 Foundations of Density Functional Theory . . . . . . . . . . . . . . . . . . . 5 2.2 Exchange-correlation energy functionals . . . . . . . . . . . . . . . . . . . . 7 2.3 Interpretation of (generalized) Kohn-Sham eigenvalues . . . . . . . . . . . . 12 2.4 Pseudopotentials ................................. 14 2.4.1 Norm-conserving pseudopotentials . . . . . . . . . . . . . . . . . . . 15 2.4.2 Energy-adjusted pseudopotentials . . . . . . . . . . . . . . . . . . . . 16 2.4.3 Projector augmented waves . . . . . . . . . . . . . . . . . . . . . . . 17 2.5 Methods of geometry optimization . . . . . . . . . . . . . . . . . . . . . . . 18 2.5.1 Simulated annealing . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 2.5.2 The ”Big Bang” search . . . . . . . . . . . . . . . . . . . . . . . . . 19 2.5.3 The Nudged Elastic Band method . . . . . . . . . . . . . . . . . . . 20 3 The bimetallic effect in Au-Pt and Au-Pd nanoalloys 23 3.1 Au-Pt and Au-Pd nanoalloys in catalysis . . . . . . . . . . . . . . . . . . . . 23 3.2 Geometric structure and mixing patterns of Au-Pt nanoalloys . . . . . . . . 25 3.3 Au-Pt alloys and Vegard’s law on the nanoscale . . . . . . . . . . . . . . . . 27 3.4 The interplay between fluxionality and electronic structure . . . . . . . . . 34 3.4.1 Electronic structure of Au-Pt nanoalloys . . . . . . . . . . . . . . . . 35 3.4.2 Electronic structure of Au-Pd nanoalloys . . . . . . . . . . . . . . . 39 3.4.3 The role of Au in Au-Pt nanoalloys . . . . . . . . . . . . . . . . . . . 41 3.4.4 Combining electronic structure and fluxionality . . . . . . . . . . . . 45 4 Magnetic-to-nonmagnetic transition in Mn-doped Si-clusters 47 4.1 Dilute magnetic semiconductors . . . . . . . . . . . . . . . . . . . . . . . . . 47 4.2 Strong correlation in Density Functional Theory . . . . . . . . . . . . . . . 49 4.3 A DFT-study of the magnetic-to-nonmagnetic transition in Mn-doped Siclusters ...................................... 52 5 Summary and outlook 61 5.1 Summary ..................................... 61 6 Chapter 2 — Theoretical and technical framework the interaction of the electrons with all external potentials and ˆ W = Pi<j w ( ˆri,ˆrj ) is the electron-electron interaction. The Hohenberg-Kohn theorem can be subsumed in two statements: First, for a given electron-electron interaction ˆ W there is a one-to-one mapping between the external potential vext (up to a physically irrelevant constant), the (non-degenerate) ground state |ψ0i resulting from Schr¨odinger’s equation ˆ H|ψ0i = E0|ψ0i and the ground state density n ( r ). |ψ0i is a unique functional of the ground state density, hence every ground state observable, and in particular the ground state energy, is a density functional, too. Second, using the Raleigh-Ritz variational principle one can obtain the exact ground state density and energy corresponding to vext by minimizing the energy functional E[n] = F[n] + Zvext(r)n(r)d3r. (2.1) F [ n ] = hψ [ n ] |ˆ T + ˆ W|ψ [ n ] i is a universal functional, i.e., it is independent of vext ( r ). This provides a simple and exact reformulation of Schr¨odinger’s equation. However, as Hohenberg and Kohn noted already in 1964 [15]: “The major part of the complexities of the manyelectron problem are associated with the determination of the universal functional F[n].” The most successful approach to determine F [ n ] was proposed in 1965 by Kohn and Sham [16], who reformulated the energy functional of a system of N interacting electrons as E[n] = Ts[n] + EH[n] + Eext[n] + Exc[n],(2.2) where Ts [ n ] is the kinetic energy of a system of N non-interacting electrons. EH [ n ] is the classical electrostatic Hartree energy EH=e2 2Z Z n(r)n(r0) |r−r0|d3rd3r0(2.3) and Eext[n] describes the interaction between the electrons and the external potential Eext[n] = Zvext(r)n(r)d3r. (2.4) Exc [ n ], the exchange-correlation (xc) energy functional, is defined as consisting of every contribution not treated by EHand Ts, i.e., Exc =T−Ts+W−EH.(2.5) Here, T is the full kinetic energy of the interacting electron system and W is the electronelectron interaction energy. Minimization of Eq. (2.2) with respect to nσ , the total spin density, and comparison with the corresponding term for a system of N non-interacting electrons leads to the same ground state density if the N non-interacting electrons are subject to an effective, local, multiplicative potential vKS σ(r) = vext(r) + vH(r) + vxc,σ(r),(2.6) in which vH is the Hartree potential vH = e2Rn ( r0 ) /|r−r0|d3r0 and vxc,σ = δExc/δnσ the xc potential. Eq. (2.6) defines the so-called KS potential. Regarding the question of 2.2. Exchange-correlation energy functionals 7 whether such a potential exists for all densities (v-representability problem) the reader is referred to Ref. [17] and references therein. In most practically relevant cases the density is v-representable. This means that one can calculate the ground state density n ( r ) of the many-electron system in an external potential vext ( r ) by solving the Schr¨odinger-like one-particle equations −¯h2 2m∇2+vKS σ(r)!ϕiσ(r) = εiσϕiσ(r),(2.7) and summing over all occupied KS orbitals, ϕiσ(r), n(r) = X σ=↑,↓ nσ(r) = X σ=↑,↓ Nσ X i=1 fiσ|ϕiσ(r)|2.(2.8) The sum over all occupation numbers fiσ yields the total number of electrons X σ=↑,↓ Nσ X i=1 fiσ =N. (2.9) Equations (2.6) – (2.8) constitute the KS equations, that have to be solved self-consistently in most practical applications of DFT. The KS equations are in principle an exact reformulation of the full interacting manybody problem of quantum mechanics. However, all exchange and correlation effects that go beyond the Hartree energy EHand the non-interacting kinetic energy Tsare by definition (Eq. (2.5) ) included in the xc energy Exc , which is in general not known exactly. The most important approximations to Exc are discussed in the following section. 2.2 Exchange-correlation energy functionals Approximations to the xc energy functional can roughly be divided into two groups: explicit and implicit density functionals. Commonly used functional approximations of the first kind are the local density approximation (LDA) and the generalized gradient approximation (GGA). Functionals of the second group do not explicitly depend on the density but on the orbitals, i.e., they possess an implicit dependence on the electron density by virtue of the Hohenberg-Kohn theorem. Prominent examples are so-called hybrid functionals and range-separated hybrid (RSH) functionals, meta-GGAs as well as the self-interaction correction (SIC). A thorough discussion of the properties and limits of all functionals is beyond the scope of this thesis. A comprehensive overview can be found in Ref. [18] and references therein. Relevant approximations for the present work are described in the following. 8 Chapter 2 — Theoretical and technical framework The LDA 1 is one of the most often used approximations to Exc and was introduced already by Hohenberg and Kohn [15]. It is based on the simple idea that in a system in which the electron density varies only slowly in space, exc , the xc energy per electron, is approximately equal to ehom xc in the homogeneous electron gas. One then obtains Exc by integration of exc over space. The exchange contribution is exactly known to be ELDA x[n] = −3e2 43 π1/3Zn(r)4/3d3r. (2.10) The correlation part has been computed with high accuracy using Monte-Carlo methods [19]. Many widely used LDA energy functionals, e.g., the ones by Vosko, Wilke and Nusair [20], Perdew and Zunger [21] and Perdew and Wang [22], are based on parametrizations of these early Monte-Carlo calculations extended by known limits and scaling laws derived from the exact Ehom c . The large number of systems for which the LDA leads to qualitatively reasonable results 2 is surprising taking into account that for most systems the assumption of a nearly homogeneous electron density is an oversimplification. This can be rationalized by introducing the concept of the xc hole nxc ( r,r0 ) = ρ2 ( r,r0 ) /n ( r ) −n ( r0 ) with ρ2 ( r,r0 ) representing the electron pair density. The xc hole has to obey the sum rule Znxc(r,r0)d3r=−1.(2.11) The LDA’s xc hole satisfies Eq. (2.11) even in rather inhomogeneous situations [23]. This leads to a subtle error cancellation between the exchange and the correlation part of ELDA xc . A systematic way to improve on the LDA is to include not only information about the electron density at each point r , but also on the rate of its spatial variation. Such GGA functionals have the form EGGA xc [n] = Zf(n(r),∇n(r))d3r. (2.12) The function f ( n ( r ) ,∇n ( r )) is constructed either by trying to satisfy as many exact constraints of the exact xc energy functional as possible (leading, e.g., to one of the most popular GGAs of Perdew, Burke and Ernzerhof (PBE) [24] or by fitting to large sets of test molecules (e.g., BLYP, consisting of the exchange functional of Becke [25] and the correlation functional of Lee, Yang and Parr [26]). Although the LDA and the GGA perform satisfactorily in many practical calculations, their failure in others is dramatic. This qualitatively incorrect behavior can in most cases be traced back to one common source, the self-interaction error (SIE) [21], which is a problem that was recognized already in Thomas-Fermi theory [27, 28], an ancestor of DFT. The SIE is trivial to define in an one-electron system with density n1 , in which there is no electron-electron interaction. Here, the exact xc energy (potential) and the Hartree energy 1And the local spin density approximation (LSDA) for spin-polarized systems. 2 Although being far from chemical accuracy, which would require an error of less than 0.04336 eV per particle as compared to the experiment. 2.2. Exchange-correlation energy functionals 9 (potential) have to cancel each other EH[n1] + Exc[n1] = 0 (2.13) vH[n1](r) + vxc[n1](r)=0 by definition. An approximate functional must therefore satisfy Eq. (2.13) in order to be free of one-electron self-interaction. This notion can be extended to many-electron systems by identifying single electrons with KS orbital densities niσ = fiσ|ϕiσ|2 . According to Perdew and Zunger [21, 29] one can then define the one-electron SIE in a many-electron system as eiσ =EH[niσ] + Exc[niσ,0].(2.14) If X σ=↑,↓ Nσ X i=1 eiσ = 0 (2.15) the approximate functional is considered free of one-electron self-interaction. This definition suggests a straightforward way of correcting Eapprox xc for self-interaction by simply subtracting the erroneous terms from the approximate xc functional ESIC xc [n↑, n↓] = Eapprox xc [n↑, n↓]−X σ=↑,↓ Nσ X i=1 (EH[niσ] + Eapprox xc [niσ,0]).(2.16) This so-called self-interaction correction (SIC) an example for an orbital-dependent, implicit density functional. Two conceptual intricacies are associated with such functionals. First, the functional derivative δExc/δnσ cannot be evaluated as straightforwardly as for explicit density functionals. One way to deal with this problem and at the same time to stay within the conceptual realm of KS DFT, is to use a functional derivative chain rule vOEP xc,σ =δExc[{ϕjτ }] δnσ(r)(2.17) =X α=↑,↓ Nα X i=1 ZδExc[{ϕjτ }] δϕiα(r0) δϕiα(r0) δnσ(r)d3r0+ c.c. (2.18) =X α,β=↑,↓ Nα X i=1 Z Z δExc[{ϕjτ }] δϕiα(r0) δϕiα(r0) δvKS β(r00) δvKS β(r00) δnσ(r)d3r0d3r00 + c.c.,(2.19) where α, β, σ and τ denote the spin polarization and c.c. the complex conjugate. From this expression one obtains the optimized effective potential (OEP) equation Nσ X i=1 fiσ Zϕ∗ iσ(r0)[vOEP xc,σ (r0)−uxc,iσ(r0)]GKS iσ (r0,r)ϕiσ(r)d3r0+ c.c. = 0 (2.20) in which GKS iσ (r0,r) is the KS Green’s function GKS iσ (r0,r) = ∞ X j=1 j6=i ϕjσ(r0)ϕ∗ jσ(r) εiσ −εjσ (2.21) 10 Chapter 2 — Theoretical and technical framework and uxc,iσ(r) an orbital specific potential uxc,iσ(r) = 1 fiσϕ∗ iσ(r) δExc[{ϕjτ }] δϕiσ(r).(2.22) The vxc,σ that solves Eq. (2.20) is called the optimized effective potential, as it yields the KS orbitals that minimize the total energy. For practical applications, a full OEP calculation is often computationally too demanding, as Eq. (2.20) is an integral equation and contains the complete set of occupied and unoccupied KS eigenvalues. The approximation of Krieger, Li and Iafrate (KLI) [30] vKLI xc,σ(r) = 1 2nσ(r) Nσ X i=1 |ϕiσ(r)|2[uxc,iσ(r) + (¯vKLI xc,iσ −¯uxc,iσ)] + c.c. (2.23) with ¯vKLI xc,iσ =Zϕ∗ iσ(r0)vKLI xc,σ(r0)ϕiσ(r0)d3r0(2.24) ¯uxc,iσ =Zϕ∗ iσ(r0)uxc,iσ(r0)ϕiσ(r0)d3r0(2.25) provides an alternative that reduces the computational effort, but can in many cases yield results very close to those from a full OEP calculation [18]. The second intricacy of orbital-dependent functionals such as the SIC, is intimately related to the quantum mechanical nature (indistinguishability) of the interacting manyelectron problem itself: One-electron orbitals are an artificial concept introduced through the KS ansatz and the KS orbitals are in no way unique among other orbital sets that also sum up to the correct ground state density. This unitary invariance problem means that different OEP can be constructed depending on the chosen orbital set. A generalization of the OEP method can be employed to deal with the unitary invariance problem via a density-conserving unitary transformation of the KS orbitals [31]. A detailed discussion of the OEP and approximations to it can be found in Ref. [17] and [18]. The ambiguity of the one-electron orbital sets also directly affects the definition of a SIE in many-electron systems. Attempts have been made to define a many-electron SIE. The issue will be discussed in Sec. 4.2. Eq. (2.14) is useful even without carrying out an actual SIC calculation. It provides a simple criterion for whether the KS eigenvalue spectrum resulting from a given approximate xc functional is physically reliable (see discussion in Sec. 2.3 and a practical example in Sec. 4.3) [32, 33]. A second class of orbital-dependent functionals are meta-GGAs. They contain the KS kinetic energy density τσ(r) = ¯h2 2m Nσ X i=1 fiσ|∇ϕiσ(r)|2(2.26) (and sometimes terms ∇2nσ ) and improve in some respects upon semilocal functionals, e.g., they in principle can achieve absence of the self-correlation error. However, as complete self-interaction absence is not guaranteed by meta-GGAs, they share conceptual difficulties 2.2. Exchange-correlation energy functionals 11 with the LDA and GGA. An example for a meta-GGA relevant for the correct description of the 2d-3d transition in small Au-cluster anions [34] is the one by Tao, Perdew, Staroverov and Scuseria [35]. Hybrid functionals are among the most often used xc functional approximations and have become a favorite tool of quantum chemistry. Such functionals are constructed of a fixed fraction of exact exchange Eex x as well as semilocal exchange Eapprox x and correlation Eapprox c3 . The approach was first introduced in 1993 by Becke [36], who proposed a functional form Ehyb xc =bEex x+ (1 −b)Eapprox x+Eapprox c,(2.27) in which Eex xdenotes the Fock integral Eex x=−e2 2X σ=↑,↓ Nσ X i,j=1 fiσfjσ Z Z ϕ∗ iσ(r)ϕ∗ jσ(r0)ϕjσ(r)ϕiσ(r0) |r−r0|d3rd3r0.(2.28) The parameter b is determined either by fitting to extensive test sets of molecules or rationalized by virtue of the adiabatic connection formalism [37, 38]. An example for the latter approach is the one-parameter hybrid PBE0, based on the PBE GGA [24] in which b = 0 . 25 [39]. However, most of today’s hybrid functionals employ even more parameters, the most prominent example being the 3-parameter B3LYP functional [40, 41], that contains fractions of the semilocal Becke functional [25] as Eapprox x , the GGA by Lee, Yang and Parr [26] as Eapprox cand the LDA parametrization by Vosko, Wilke and Nusair [20]. Finally, I want to mention the range-separated hybrids (RSHs). The underlying concept of these functionals is to separate the electron-electron interaction into a long-range and a short-range part via 1 |r−r0|=erf(ω|r−r0|) |r−r0| | {z } long range +1−erf(ω|r−r0|) |r−r0| | {z } short range .(2.29) The screening function is for numerical reasons often chosen as the error function and ω is an adjustable parameter that determines at which length scale the short-range part of Eq. (2.29) decays to zero and the long-range part becomes dominant. The long-range part is given by the screened Fock integral, so that one preserves the advantages of employing semilocal exchange at short range, while incorporating 100% of the non-local exact exchange at long range and thus obtains the correct asymptotic behavior of the xc potential. Hence, RSHs are particularly useful for the description of charge transfer excitations, but they can also improve on ground state properties [42]. An example for such a functional is ω PBE, which is a RSH that models the exchange hole of the PBE GGA at short range [43]. The range-separation parameter can be obtained in different ways: One can empirically determine the value of ω that describes thermochemistry or charge transfer excitations best. However, evidence suggests that in fact considerably 3 Prior to their formal justification via the generalized KS framework, these functionals had been seen as ”hybrids” between DFT and Hartree-Fock. 12 Chapter 2 — Theoretical and technical framework different range-separation parameters might be necessary depending on the particular system and properties of interest. Another approach is therefore to tune ω for each system in such a way that certain properties of the exact functional (such as Eq. (2.33) in Sec. 2.3) are fulfilled [44]. Orbital-dependent functionals can be used in conjunction with the OEP method as explained above. The widely used hybrids (and RSHs) are practically always implemented in a way which leaves the grounds of the KS framework. This second approach is justified by the observation that it is possible to formulate a so-called generalized KS approach in which the interacting N -electron system is mapped onto another interacting auxiliary system, which can, however, still be represented by a single Slater-determinant [45]. 2.3 Interpretation of (generalized) Kohn-Sham eigenvalues Given their status as merely auxiliary quantities within the KS framework, it might be surprising that the KS eigenvalues should carry any physical meaning at all. However, the success of DFT partly relies on the fact that in an overwhelmingly large number of systems the eigenvalue spectra agree, apart from a shift of the complete spectrum, remarkably well with experimentally obtained quasiparticle energies. In the early days of DFT this agreement was regarded accidental. Later, G¨orling could show that KS eigenvalue differences have in fact a well-defined physical meaning as excitation energies of zeroth order in the electron-electron interaction [46]. The derivation is based on linking the interacting full Hamiltonian to the KS Hamiltonian through [ˆ T+αˆ W+ˆ Vα]|ψα ni=Eα n|ψα ni.(2.30) The coupling constant α creates a continuous (adiabatic) connection between the fully interacting N -electron system with α = 1 and ˆ Vα=1 = ˆ Vext and the non-interacting KS system with α = 0 and ˆ Vα=0 = vKS [37, 38]. Using G¨orling-Levy perturbation theory [47] one can then expand Eα n in a Taylor series in α and show that excitation energies can rigorously be gained from ground state DFT. In particular, the zeroth order term connects the quasiparticle excitation energies to KS eigenvalue differences. The practical usefulness of G¨orling’s finding of course depends on the quality of this zeroth order approximation. Chong et al. could show that the energetically highest lying occupied KS levels can be interpreted as approximate, but rather accurate, relaxed vertical ionization potentials, provided that they are computed using a high-quality xc potential [48]. Already earlier it was observed by Janak [49] that ∂E ∂fiσ =εiσ,(2.31) where fiσ is the (fractional) occupation of the i -th KS orbital. A rigorous physical meaning, however, can only be assigned to the highest occupied KS eigenvalue εHOMO , which is 2.3. Interpretation of (generalized) Kohn-Sham eigenvalues 13 identical to minus the ionization potential I ( N ) of the fully interacting physical system. This was proved by Almbladh and von Barth [50] by deriving differential equations for the quasiparticle amplitudes Fn,σ(r) := hψN−1 n|ˆ Ψσ(r)|ψN 0i,(2.32) where ˆ Ψσ ( r ) is the electron-field operator which annihilates an electron of spin σ at r and ψN n are the N -electron eigenstates of the full many-body Hamiltonian. A closer look at the asymptotic decay of this quasiparticle amplitude shows that the asymptotically leading amplitude is obtained for n = 0 and that the density itself is dominated by F0,σ ( r ) for |r|→∞ . By comparing this result to the asymptotic form of the KS density, which is dominated by the most weakly decaying KS orbital, one arrives at the identity εHOMO(N) = −I(N) = E(N)−E(N−1).(2.33) Similarly, one can show for the electron affinity A ( N ), i.e., the energy gained by bringing in a particle from infinity that εHOMO(N+ 1) = −A(N) = E(N+ 1) −E(N).(2.34) E ( N− 1) , E ( N ) and E ( N + 1) are the energies of the N− 1, N and N + 1 electron system, respectively. In practical calculations one often faces the problem of not knowing whether the used xc functional describes the system of interest accurately or not. As a rule of thumb may count, that in systems in which the upper occupied KS orbitals are localized on a similar length scale, standard xc functionals result in eigenvalue spectra that are physically reliable [21]. The SIE in such systems, that can be evaluated using Eq. (2.14) , affects all relevant orbitals in the same way and therefore only amounts to a common shift of the eigenvalue spectrum [32]. In systems in which the upper orbitals differ in their degree of localization, the spectrum will be distorted as a result of the SIE affecting the orbitals differently. In this sense, evaluation of Eq. (2.14) can serve as a warning for systems that are strongly affected by the SIE. Note, however, that the relation between orbital localization and the SIE is not trivial [51]. The close connection between the SIE and the reliability of an eigenvalue spectrum suggests that self-interaction free approaches (in the sense of Eq. (2.15) ) should yield good agreement with experiment in cases in which semilocal functionals fail and indeed this has been shown for a number of systems (see e.g. Ref. [32]). However, these approaches are numerically expensive, especially in cases in which finding the energetically most favorable position of the nuclei poses additional problems (see Sec. 2.5). The commonly used hybrid functionals can at least partly cancel the SIE by employing a fraction of exact exchange. As mentioned earlier, hybrid functionals are typically implemented in a GKS framework. The GKS equations −¯h2 2m∇2+vext(r) + vH([n],r) + vc,σ([n], b, r) +bˆvex x,σ[n] + (1 −b)vx,σ([n],r)ϕiσ(r) = εGKS iσ ϕiσ(r), (2.35) 14 Chapter 2 — Theoretical and technical framework in which ˆvex x,σ[n]ϕiσ(r) = −e2X σ=↑,↓ Nσ X j=1 Zϕjσ(r)ϕ∗ jσ(r0) |r−r0|ϕiσ(r0)d3r0(2.36) denotes the non-local Fock operator, are by construction exact, just as the KS equations. The resulting eigenvalues εGKS iσ differ from the exact KS eigenvalues by b∆vx,σ,i =bhϕiσ(r)|ˆvex x,σ[n]−vx,σ([n],r)|ϕiσ(r)i,(2.37) if one neglects differences in the KS and GKS orbitals and the correlation potential [45]. As ∆ vx,N = 0 for the highest occupied GKS eigenvalue, the relation εGKS HOMO = −I ( N ) holds also for the GKS case. There is again no such equality for all the other GKS eigenvalues. However, one can show that including a fraction of the non-local exchange mimics a partial self-interaction correction [52]. Consequently, GKS eigenvalues agree well with experimental results for many cases in which semilocal functionals fail. 2.4 Pseudopotentials The basic idea of replacing the strong Coulomb potential of the nucleus and the screening effect of the tightly bound core electrons by an effective core potential (commonly refered to as pseudopotential), has its roots in the simple notion that only the valence electrons of an atom determine its chemical properties. The approach was originally introduced by Fermi when he studied low energy electron scattering from atoms [53]. In fact, the aim of the pseudopotential construction is to find an effective potential that mimics the scattering properties of nucleus and core electrons reliably over a certain energy range. The degrees of freedom of pseudopotential generation then allow to devise them in such a way as to minimize the computational effort. Firstly, by reducing the number of electrons that have to be considered explicitly in the DFT calculation. Secondly, because pseudopotentials allow the numerical description of potentials and orbitals on much coarser grids or with fewer plane wave components than would be necessary if the highly oscillatory structure of the orbitals in the core-region had to be represented completely. In the present thesis pseudopotentials enter the stage at yet another point. When one deals with heavy-element transition metal compounds such as Au and Pt, relativistic effects such as spin-orbit coupling and the relativistic mass increase of electrons in the core region are crucial for the accurate description of the electronic, and thus also the geometric structure of the compounds [54]. Pseudopotentials offer ways in which these effects can be accounted for implicitly. An important concept underlying most pseudopotential generation schemes is the frozen-core approximation: The core states, evaluated in an all-electron atomic reference calculation, are assumed not to change if in a different environment, i.e., in a molecule or solid. For this reason the radius of the core region rc is a crucial parameter determining transferability and accuracy of the pseudopotential. 2.4. Pseudopotentials 15 2.4.1 Norm-conserving pseudopotentials The spirit of norm-conservation is to start the construction of the pseudopotential from the all-electron orbitals that stem from a self-consistent solution of the radial KS equation 4 (for a spin-unpolarized case) −¯h2 2m d2 dr2+¯h2l(l+ 1) 2mr2+vKS([n], r)!rRil(r) = εilrRil(r),(2.38) in which rRil = ϕil and l denotes the angular momentum quantum number. The pseudopotential has to fulfill four properties to be considered norm-conserving: First, all-electron (AE) and pseudo (PP) valence eigenvalues must agree for a chosen atomic reference configuration. Second, all-electron and pseudo orbitals agree beyond a chosen core radius rc . Third, the integrated charge inside rc of the all-electron and the pseudo charge densities agree for each valence state Zrc 0|RPP il (r)|2r2dr =Zrc 0|RAE il |2r2dr, (2.39) as this ensures that the total charge in the core region is conserved. And last, the logarithmic derivatives of the all-electron and the pseudo orbitals and their first energy derivatives agree for r > rc [56]. The freedom that is left in constructing the pseudopotential can then be used to make it as smooth as possible at the same time ensuring transferability to as many different chemical environments as possible. Different schemes for pseudopotential generation were proposed, e.g., by Bachelet, Hamann and Schl¨uter [57] and by Troullier and Martins [58]. By inverting the radial KS equations one then obtains the screened pseudopotential from the pseudo wavefunction vPP screened,l =εl−¯h2 2m l(l+ 1) r2−1 rRPP l(r) d2 dr2[rRPP l(r)]!.(2.40) In the last step the screening of the valence electrons has to be removed to obtain an unscreened (”bare”) ionic pseudopotential vPP ionic,l(r) = vPP screened,l(r)−vPP H[nval](r)−vPP xc [nval(r)] (2.41) The resulting pseudopotential differs for every angular momentum component l , i.e., the effective external potential in the KS equations is not local anymore. A further transformation suggested by Kleinman and Bylander brings Eq. (2.41) in a separable non-local form that reduces computation time and storage space. Troullier-Martins pseudopotentials in Kleinman-Bylander form are used throughout this work for calculations on real-space grids using a local version of the PARSEC program package [59], e.g., for the calculation of the SIE of Mn-doped Si clusters in Sec. 4.3. 4 If relativistic effects are to be included one has to use Dirac’s formulation of the kinetic energy. The Schr¨odinger-like Eq. (2.38) is then replaced by a pair of coupled equations for minor and major orbital components, which outside the core radius reduce approximately to a Schr¨odinger-type equation for the major orbital component [55]. 3 The bimetallic effect in Au-Pt and Au-Pd nanoalloys 3.1 Au-Pt and Au-Pd nanoalloys in catalysis Alloying of two (or more) elements is a promising route to enhance the catalytic activity of transition metal NP and improve their element-specific selectivity [78]. Loosely based on Aristotle’s bon mot that “the whole is greater than the sum of its parts”, these effects are frequently dubbed synergistic. Au-Pt nanoalloys are a prominent example of a system which exhibits increased turnover rates in a variety of oxidation reactions [79–81] as well as enticing properties in electrocatalytic applications [82–90] compared to their pure counterparts. Au-Pt nanoalloys are beneficial because they reduce the well-known problem of ”poisoning” of Pt catalyst anodes used in direct methanol fuel cells with strongly adsorbed intermediate CO-species. Furthermore, increased turnover rates allow to carry out reactions at room temperature and without possibly toxic solvents, thus causing less impact on the environment. Finally, by changing the Au-Pt composition one can tune their catalytic properties which makes Au-Pt nanoalloys potentially useful for many different catalytic reactions. When looking deeper into these problems, several puzzles appear. The first puzzle is the miscibility of Au and Pt on the nanoscale. It is well known that the bulk phase diagram of Au-Pt alloys exhibits a large miscibility gap for a wide range of Au-Pt compositions. In going to the nanoscale it has been reported that Au-Pt NP can form true solid solutions. Such Au-Pt nanoalloys have, e.g., been synthesized in thermally evaporated fatty amine films, where they were alloyed at low temperature [91], in spherical polyelectrolyte brushes (SPBs) [80] and in ionic liquids by a sputter deposition technique 24 Chapter 3 — The bimetallic effect in Au-Pt and Au-Pd nanoalloys [92]. Other groups have obtained stable Au-Pt nanoalloys by capping them with thiolates [82, 93] or by supporting them on silica [79]. core-shell random layered Figure 3.1: Three possible miscibility patterns in bimetallic NP. From left to right: core-shell, randomly mixed and layered. Other patterns such as several layers/shells or an ordered solid solution are possible. Theory, in contrast, consistently predicts a Au shell -Pt core mixing pattern as energetically most favorable [94–97]. An illustration of different mixing patterns in NP is given in Fig. 3.1. Theoretical studies of clusters in the experimentally relevant size range of several hundred up to thousands of atoms have been based on semiempirical MD simulations and were mostly concerned with the melting behaviour of the Au-Pt NP [98–101]. A global geometry optimization using a genetic algorithm with a semiempirical Guptamany-body potential of Au-Pt clusters with 2–100 atoms has been performed by Logsdail et al. [102]. First-principles DFT studies existed, up to the beginning of this work, only for very small clusters with 2–13 atoms [103, 104]. However, more recent studies on larger clusters (including one that will briefly be presented in Sec. 3.2), have confirmed that Au-Pt NP prefer a core-shell mixing pattern [94–97]. This seeming discrepancy of theory and experiment is subject of Sec. 3.3. The second Au-Pt nanoalloy puzzle is their enhanced catalytic activity as compared to their pure counterparts. Considering the known bulk electronic properties of Pt and Au 1 it is not per se clear, why a combination of Pt and Au should lead to enhanced catalytic activity at all (see also Sec. 3.4.1). Additionally, while it is known how the electronic structure of Au changes upon going from the bulk to small NP [2], similar insights into the electronic structure of Au-Pt nanoalloys are rare. Explanations for the special properties of Au-Pt nanoalloys have been attempted in several studies, mainly concentrating on the local electronic structure of Pt [85, 95, 104, 106]. A detailed understanding of the concurrence of Au and Pt leading to favorable catalytic properties in a variety of different reactions, i.e., independent of the type of support or matrix the NP are immobilized in, possible solvents and different reaction temperatures, is lacking. Puzzle number two will be subject of Sec. 3.4. Although elements from the Pt-group are chemically similar, the above matters are slightly different for Au-Pd nanoalloys. The structural characterization of Au-Pd NP is much easier as compared to Au-Pt NP, as Au and Pd’s atomic numbers are sufficiently different to allow for the use of high-resolution techniques, e.g., high-angle-annular-dark-field transmission electron microscopy [107]. Synergistic effects have been observed for Au-Pd NP catalysts in a variety of reactions. Two recent examples are Au-Pd nanoalloys used for electrocatalytic H 2 O 2 production [108] and for solvent-free oxidation of primary carbonhydrogen bonds [107]. As the following sections will focus primarily on Au-Pt nanoalloys, I refer the reader to Ref. [78] and references therein, for a summary of studies in which a synergistic effect of Au-Pd catalysts was observed. Insights into the electronic structure of Au-Pd nanoalloys and their impact on Au-Pd catalysis are discussed in Sec. 3.4.1. 1Presented in a nutshell in “Why gold is the noblest of all metals” by Hammer et al. in Ref. [105]. 3.2. Geometric structure and mixing patterns of Au-Pt nanoalloys 25 3.2 Geometric structure and mixing patterns of Au-Pt nanoalloys This section gives a short summary of results that are discussed in detail in Ref. [109]. Most of the nanoalloy properties that will be identified in the following sections as being favorable for catalytic activity are to a large extend independent of the exact geometric structure of the clusters. Still, energetically low-lying Au-Pt isomers are a well-chosen starting point for the study of these properties. In combined efforts of theory and experiment, the ground state structure of many pure neutral and charged Au clusters has been determined. A prominent example are Au 20 and Au − 20 , whose ground state structure is a regular tetrahedron as revealed by gas phase infrared spectroscopy [110] and photoelectron spectroscopy [68]. Au 20 is an excellent toy model for the present study as it represents a cutout of the fcc lattice in which both bulk Au and Pt crystallize. Furthermore, many of the experimentally observed Au-Pt nanoalloys have been shown to be largely faceted and bulk-like (e.g. [80]). Figure 3.2: There are three symmetry-inequivalent ways how to replace one Au atom in the Au 20 tetrahedron by a Pt atom. By systematically replacing Au by Pt atoms in Au 20 one finds that the energetically lowest lying homotop is always that in which Pt is as highly coordinated as possible. In going from Au 20 to Au 19 Pt 1 , as illustrated in Fig. 3.2, the energetically lowest lying homotop is the one in which Pt occupies a position in the middle of one of the four facets. As a result of this growth pattern a Au shell Pt core structure emerges naturally. That the core-shell mixing pattern is not a mere artifact was tested by comparing homotops and isomers with fixed Au-Pt composition especially for structural motifs other than the rather special tetrahedral structure as well as for larger clusters. The stability of a variety of structures was tested using simulated annealing (see Sec. 2.5). In these simulations the clusters were heated to a temperature of 600 K and subsequently slowly annealed. Importantly, the core-shell mixing pattern remains stable throughout the entire simulation and can therefore be regarded as being stable even at elevated temperatures. Fig. 3.3 shows, as an example, different Au20Pt20 isomers from A to F with increasing total energy. Note that even a complete segregation of the Au and the Pt component (C and D) is energetically more favorable than the random mixing pattern (E and F). Computational details on the geometry optimization and the accuracy of the used xc functional and basis sets as well as structures of tetrahedral and amorphous Au n Pt 20−n clusters can be found in Ref. [109] and [94]. In this work, Au-Pt NP with ≈ 1000 valence electrons, i.e., 60-atom clusters, were the largest systems for which full local geometry optimizations were performed. The Gupta potential based Au 30 Pt 30 structure [102] was used as a starting point for these calculations. However, considering that experimentally relevant nanocatalysts range between sizes of 26 Chapter 3 — The bimetallic effect in Au-Pt and Au-Pd nanoalloys 1 and 4 nm, we need even larger systems to exclude that the observed effects are only due to the comparably small size of the clusters. A B C D E F Figure 3.3: Structures are named after the geometry that served as a starting point of the geomtry optimization either “Gupta” [102] or “Sutton Chen” [111]. A: Gupta core-shell, B: Sutton-Chen core-shell, C: Sutton-Chen layered, D: Gupta layered, E: Gupta random, F: Sutton-Chen random. Two strategies were used to overcome the computational limitations that first principles DFT calculations face for systems with significantly more than 1000 valence electrons. Firstly, we constructed truncated octahedral clusters with closed atomic shells, i.e., with 38, 201, 586 and 1289 atoms as illustrated in Fig. 3.4 and optimized them in empirical MD simulations. These clusters are, analogously to Au 20 , cutouts of the fcc lattice and are used in the studies discussed in Sec. 3.3 and 3.4. Clusters with regular polyhedral structures and closed atomic shells are sometimes called “magic”, in analogy to the magic electron numbers that were discovered by Knight et al. in 1984 in sodium clusters [112]. This seminal discovery triggered the development of cluster physics much beyond the realm of the simple spherical jellium model which was used to explain the abundance of sodium clusters containing 2, 8, 20, 40,... atoms. Electronic shell closure effects within the jellium model are responsible for the stability of these cluster sizes in the simple alkali metals and even in Au clusters, as their 6sstates form a good free electron gas [2]. 1 nm 2 nm 3 nm 4 nm Figure 3.4: Truncated octahedra with (from left to right) 38, 201, 586 and 1289 atoms. These atomic numbers are called magic, because they correspond to clusters having only closed shells of atoms. The approximate diameters of these clusters are 1, 2, 3 and 4 nm. For larger clusters the electronic shell closure effect is superimposed by the closure of atomic shells, a prominent example again being sodium clusters that exhibit icosahedral shell closures [113]. In sodium clusters with a few tens of atoms, electronic and ionic shell closure effects work together in determining the structure of the clusters, while for larger clusters, starting with Na 55 , ionic shell closure effects win over electronic ones resulting in near-ly spherical clusters [114]. However, I want to stress that truncated octahedral structures might not be a structural motif found in smaller Au clusters. The important point is that they are faceted and fcc-bulk-like, which makes them a valuable model for the catalysts observed experimentally. A second approach to catalysis on Au-Pt NP is to model them as periodic surface slabs. The motivation for this is twofold: Firstly, much of the catalytic activity of NP can be attributed to their high surface to volume ratio. Secondly, a synergistic effect of Au and Pt has also been observed for bimetallic surfaces [115]. Au-Pt surfaces are studied in Sec. 3.4. In summary, the following Au-Pt structures are used in the present thesis to determine structural and electronic properties of Au-Pt systems: 20-atom tetrahedral and amorphous low-energy isomers, 40-atom and 60-atom structures based on empirical potentials and fully 3.3. Au-Pt alloys and Vegard’s law on the nanoscale 27 optimized using DFT, truncated octahedral model structures with 38, 201, 586 and 1289 atoms for which different arrangements of Au and Pt, e.g., in core-shell and random mixing patterns can be constructed, and finally bimetallic surface slabs. 3.3 Au-Pt alloys and Vegard’s law on the nanoscale All results of this section are published in Ref. [116]. In the previous two sections I discussed that first-principles DFT calculations confirm that low-energy structures of free Au-Pt clusters indeed exhibit an Au shell Pt core mixing pattern and thus confirm the results of semi-empirical MD simulations mentioned in Sec. 3.1. The seeming discrepancy between theory (core-shell) and experiment (solid solution) must thus be searched for elsewhere. Figure 3.5 : A sketch of the first two Bragg peaks as typically observed in XRD measurements on Au-Pt NP. From these diffraction patterns it can presumably be inferred that the system under study (blue line) is a true solid solution. The arrows indicate the position of the first Bragg peak. pure Au pure Pt nanoalloy (111) (200) Bragg angle [2 ] 30 35 40 45 50 0 10 20 30 intensity [arb. units] The next step is to understand the assumptions that underlie the experimental characterization of Au-Pt NP. A prominent characterization method is the use of x-ray diffraction (XRD) techniques. The resulting diffraction patterns are typically analyzed as sketched in Fig. 3.5. The orange and black lines indicate the first two Bragg peaks of pure Au and pure Pt NP as observed in small angle x-ray scattering experiments. These peaks can be indexed into an fcc-type lattice. For each set of Miller indices ( hkl ) one can then use Bragg’s equation a = λ√h2+k2+l2 2 sin θ to determine a lattice parameter a from the scattering angle θ. Here, λis the wave length of the incoming x-rays. The same can be done for the diffraction pattern of the bimetallic NP. Additionally, depending on the experimental resolution, one can use the peak shape to determine, whether the system of interest shows any sign of demixing of Au and Pt (in which case one would expect distinct peaks for the Auand the Pt-phase) or whether it forms a real alloy (only one clear peak). The blue line in Fig. 3.5 would, following this reasoning, belong to a randomly mixed Au-Pt NP. Bulk alloys often obey Vegard’s law [117], which states that the lattice parameter of a two-component alloy can be determined by linearly interpolating between the lattice parameters of the two components that form the alloy. Hence in an alloy that consists of 50% of both elements, the average lattice parameter would be just the average of the lattice parameters of both pure crystals. XRD characterization now reveals that the lattice 28 Chapter 3 — The bimetallic effect in Au-Pt and Au-Pd nanoalloys parameter of Au-Pt NP goes linearly with the Au-Pt ratio as well. This finding is taken as evidence that Au-Pt NP form true alloys (see, e.g., Ref. [80, 93]). The ”Vegard-analysis” thus assumes firstly that a phase separation must be visible in the diffraction pattern, secondly that Vegard’s law is valid and can be used to determine the composition of the alloy and thirdly that Vegard’s law is only valid for randomly mixed alloys. By simulating the XRD patterns of pure Au and Pt, as well as of randomly mixed and core-shell NP one can thus test the validity of these assumptions. The scattering intensity Iof an ensemble of atoms is given by Debye’s equation I=X mX n fmfn sin kRmn kRmn ,(3.1) where fn and fm are the form factors of atoms n and m , Rmn = |Rm−Rn| is the distance of two atoms at positions Rmand Rnand k= 4πsin θ/λ is the wave number. Rmn 2θ Figure 3.6: A sketch of the scattering of light at an ensemble of randomly oriented atoms, where the difference vector Rmn = Rm−Rn is allowed to take all orientations in space, i.e., all positions on the black circle with equal probability (adapted from [118]). The general scattering equation (3.1) for atoms which are randomly oriented in space is easily derived by considering that one obtains the scattered x-ray intensity by summing over the scattering amplitudes of x-rays scattered by different atoms I=X m fme(2πi/λ)(s−s0)RmX n fne(−2πi/λ)(s−s0)Rn(3.2) =X mX n fmfne(2πi/λ)(s−s0)Rmn , where s and s0 are the directions of incoming and outgoing x-rays, as sketched in Fig. 3.6. Eq. (3.2) is of course valid for scattering on any ensemble of atoms. We now assume that this ensemble takes all orientations in space with equal probability, i.e., that Rmn ends at all points of the sphere sketched in Fig. 3.6 (in 2d) with equal probability. The spatial average of the exponential term he2πi/λ(s−s0)Rmn i can with k = 4 πsin θ/λ and ( s− s0)Rmn = 2 sin θRmn cos φbe written 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 to Debye’s scattering equation 3.1. Additionally, temperature effects that cause atomic vibrations independent from each other could be taken into account by including a Debye-Waller factor [119, 120] in the Debye 3.3. Au-Pt alloys and Vegard’s law on the nanoscale 29 Table 3.1 : Average bond lengths and their standard deviations for core-shell and random Au 22 Pt 38 (37% Au) as obtained from DFT and MD optimization. DFT Au-Au [˚ A] Au-Pt [˚ A] Pt-Pt [˚ A] core-shell 2.88 ±0.03 2.85 ±0.03 2.76 ±0.05 random 2.90 ±0.05 2.82 ±0.06 2.72 ±0.06 MD Au-Au [˚ A] Au-Pt [˚ A] Pt-Pt [˚ A] core-shell 2.74 ±0.03 2.70 ±0.04 2.69±0.04 random 2.73 ±0.04 2.70 ±0.04 2.67 ±0.04 equation (3.1) . We do not include this factor here, as we will explicitly take temperature into account by performing MD simulations at elevated temperatures. A reasonable simulation of XRD patterns requires two important conditions to be met. Firstly, the size of the NP needs to be between 3 and 4 nm to conform with the experimental size range. Secondly, the interatomic bond lengths need to be realistic, as they are the decisive input to Eq. (3.1) . As already mentioned, these requirements pose severe difficulties to DFT, as computational effort forbids to treat NP of such size, i.e., with more than 1000 Au and Pt atoms. On the other hand, the reliability of bond length estimates is best if a first-principles method is used for geometry optimization. We assess this problem in the following way. The 1289-atom truncated octahedron as shown on the right hand side of Fig. 3.4 is used as a starting point for geometry optimization. A pure Au and Pt, a Au shell Pt core with one and two Au-shells (corresponding to the NP Au 484 Pt 805 and Au 830 Pt 459 respectively) and finally a randomly mixed NP with exactly the same Au-Pt ratio as the core-shell particle Au484Pt805 were constructed based on the truncated octahedral geometry. The geometry optimization of these systems was carried out using MD simulations, in which the potential energy was modelled by the many-body Sutton-Chen potential [121], that has been employed successfully in simulations of a variety of nanoalloys [100, 122–124]. The Sutton-Chen parameters for the Au-Pt interaction were obtained from the parameters for the Au-Au and Pt-Pt interaction using a combination rule [100]. Each system was first relaxed at 1 K and subsequently propagated for 500 ps at a temperature of 300 K in a canonical ensemble using Evans’ thermostat [125] and vacuum boundary conditions. Every 20 ps a structure ”snapshot”, yielding a set of atomic positions {Rm} , was taken. These structures were then optimized with a conjugate gradient procedure. For each system the optimized geometry with the lowest energy was taken as the final structure for further analysis. All MD simulations in this and Sec. 3.4.3 were carried out by Rodrigo Q. Albuquerque using the DL-POLY program package [126]. The second part of our strategy consists of determining the Au-Au, Pt-Pt and Au-Pt bond lengths of DFT-optimized core-shell and random geometries. We do not expect the bond lengths resuling from MD simulations to equal the DFT bond length, because the Sutton-Chen parameters are fitted to bulk properties and bulk bond lengths are typically slightly different from NP bond lengths. However, what is decisive here is, that the relative differences between the Au-Au, Pt-Pt and Au-Pt bond lengths resulting from MD and DFT optimization, respectively, are similar. As one of the larger systems where structure optimization with DFT is still possible we chose the 60-atom cluster Au 22 Pt 38 . Its Au content of about 37% is similar to that of the 4 nm cluster Au 459 Pt 830 that is the main focus of our study. We generated starting 30 Chapter 3 — The bimetallic effect in Au-Pt and Au-Pd nanoalloys Au content method Au-Au [˚ A] Au-Pt [˚ A] Pt-Pt [˚ 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 : Average bond lengths and standard deviations obtained from DFT and MD optimization of randomly mixed 38-atomic clusters with different Au-Pt mixing ratios. structures by taking the coordinates from the 60-atomic cluster in Ref. [102] and distributed Au and Pt atoms once randomly and once in a Pt-core Au-shell fashion to these positions. The structures were then relaxed with DFT and MD separately. In the DFT calculations the relaxation was carried out without any symmetry constraints using a split valence basis set augmented by one polarization function (SVP). The use of larger basis sets changes the bond length by less than 2% [94] and does not affect the relative distribution of the lengths of Au-Au, Au-Pt and Pt-Pt bonds. For the optimized structures we calculated the distance of every atom to its five nearest neighbors, at the same time determining whether these bonds were Au-Au, Au-Pt or Pt-Pt bonds. The average Au-Au bond length was then defined by the arithmetic mean of all resulting Au-Au distances. In the same way, Au-Pt and Pt-Pt bond lengths were calculated. The results stay qualitatively the same if one somewhat changes the averaging procedure, e.g., using four instead of five nearest neighbors. Tab. 3.1 shows the average bond lengths thus obtained from DFT and MD for core-shell and for randomly mixed variants of Au 22 Pt 38 . The MD bond lengths are found to be consistently smaller than the DFT bond lengths. However, the trends with respect to bond length differences are similar. For both DFT and MD calculations the differences between the average length of Au-Au bonds, Au-Pt bonds and Pt-Pt bonds that one finds in randomly mixed and core-shell particles are small. In all cases Au-Au bonds are longest and Pt-Pt bonds are shortest, with Au-Pt falling in between. In order to also test the MD bond length distribution for a faceted particle we chose the smallest truncated octahedron built from 38 atoms. Alloy particles with Au contents of 87 % (Au 33 Pt 5 ), 50 % (Au 19 Pt 19 ) and 13 % (Au 5 Pt 33 ) were generated by distributing Au and Pt atoms randomly over the cluster. The structures were energetically optimized as described previously. Tab. 3.2 shows the resulting average bond lengths. We observe again that the MD and DFT average bond lengths are not identical, with the MD bonds generally being shorter than the DFT ones. Yet, again the trends are the same in both DFT and MD: Au-Au bonds are the longest and Pt-Pt bonds are the shortest, with Au-Pt bonds falling in between. The one exception that we found is shown in Tab. 3.2 for 87 % Au, where Au-Pt and Pt-Pt bonds in the MD are quite similar. We attribute this to the low Pt concentration and small particle size. We also performed the same type of analysis for the 20-atom tetrahedral and amorphous structures that are described in Ref. [94]. The results are in line with the conclusions drawn for the 38-atom and 60-atom structures shown above and are therefore not reported in detail here. We can now analyze the average bond lengths of the Au 459 Pt 830 core-shell and randomly 3.3. Au-Pt alloys and Vegard’s law on the nanoscale 31 mixed truncated octahedron. In the core-shell cluster the large Pt core dominates the bond length distribution, with an average Pt-Pt distance of 2.77 ˚ A. The mean Au-Au distance in the single Au shell is the same as the mean Pt-Pt bond length. This finding is reminiscent of the lattice deformation and matching that one encounters when growing single layers of one material on a substrate with a different lattice parameter. The Au-Pt distance of 2.78 ˚ A is slightly larger, allowing the shell to wrap around the core. The bond length most often encountered in the randomly mixed NP is one that falls in between the Au-Au and the Pt-Pt bond length, with Au-Pt bonds on average being 2.76 ˚ A, Au-Au bonds 2.79 ˚ A, and Pt-Pt bonds 2.74 ˚ A. atomic form factor gold platinum 00.2 0.4 0.6 0.8 1.0 1.2 20 40 60 80 100 120 140 160 Figure 3.7: Atomic form factors for Au and Pt taken from Ref. [127] and polynomial least square fits to the data. After these preliminary considerations we turn to simulating the XRD patterns of the NP using Eq. (3.1) with the incoming x-rays having a wave length of λ =1.5 ˚ A. The form factors of Au and Pt are tabulated in Ref. [127] and fitted using a polynomial least squares fit to allow continuous representation of the data (see Fig. 3.7). The top left panel in Fig. 3.8 shows the pattern that one obtains from pure Au or Pt NP with 1289 atoms in orange and black, respectively. The red line shows the x-ray pattern for the core-shell NP. As laid out above the expectation so far has been to find separate peaks for this type of structure, as the Au and Pt component are separated. Yet, no double-peak structure is observed at all; the peak shapes for the different NP are nearly identical. Our first important conclusion therefore is that a core-shell structure in NP does not necessarily lead to separate x-ray peaks for the different components. The lower left panel shows the second important result: The x-ray pattern for the core-shell NP (as above) and the one for the randomly mixed NP are very similar. Core-shell structures may thus be hard to identify via x-ray scattering. Figure 3.8 : Left: Calculated XRD pattern of pure Au, Pt, and core-shell NP (top) and core-shell and randomly mixed NP (bottom). Right: View of the coreshell (top) and randomly mixed (bottom) NP (outside and cross section). intensity [arb.units] pure Au pure Pt core-shell core-shell random 30 35 40 45 50 50 100 150 250 200 50 100 150 250 200 core-shell random Bragg angle [2 ] However, we now take the analysis one step further and go through the Vegard’s-lawbased analysis of the x-ray data as explained above. For pure NP the x-ray analysis leads 38 Chapter 3 — The bimetallic effect in Au-Pt and Au-Pd nanoalloys 0 2 4 6 8 10 0 20 40 60 80 100 DOS at Fermi level/electron [arb. units] Au content [%] 20 tetra 20 amorphous 38 octa 60 Gupta 40 Sutton-Chen Figure 3.16: The DOS at the Fermi level (integration threshold defined as -1 eV) as a function of the Au content of 20-atom tetrahedral and amorphous, 38atom truncated octahedral and 40and 60-atom Gupta-potential clusters. We can now analyze the DOS at the Fermi level for a variety of Au-Pt NP with different overall geometries and mixing patterns. Fig. 3.16 shows that it increases with increasing Pt content for 20-atom tetrahedral 5 and amorphous clusters (coreshell), for 38-atom truncated octahedra (random) and for 40and 60-atom Guptapotential clusters (core-shell). Importantly, this trend is independent of the specific way in which Au and Pt atoms are arranged in the cluster, i.e., the DOS at the Fermi level increases both for core-shell and randomly mixed Au-Pt nanoalloys with increasing Pt content. This finding suggests that the binding energy between Au-Pt nanoalloys and an adsorbate should increase with increasing Pt content and that a pure Pt NP should bind adsorbates most strongly to its surface. From this perspective, a certain Au-Pt ratio could correspond to an optimal binding energy in a specific reaction in terms of Sabatier’s principle. core-shell layered 20 atoms 40 atoms Figure 3.17: Plots of the highest occupied molecular orbital at ± 0.04 a−3/2 0 . Top: for two just slightly different Au 10 Pt 10 clusters. Bottom: Au 20 Pt 20 with coreshell and layered mixing pattern, respectively. A closer look at the local electronic structure reveals that the addition of Pt is beneficial from yet another perspective. Fig. 3.17 shows isosurface plots of the highest occupied KS orbital (HOMO) of several Au-Pt NP. The HOMO has physical relevance as it asymptotically dominates the density and its exponential decay is governed by the first ionization potential, i.e., it represents the spatial distribution of the energetically highest lying part of the density. Fig. 3.17 reveals that the spatial location of the HOMO is closely related to that of the Pt atoms. The two Au 10 Pt 10 clusters at the top of Fig. 3.17 differ only in the position of one Pt atom that sits within the bottom facet on the left side and is moved to the top corner on the right side of Fig. 3.17. This slight rearrangement results in a marked change of the spatial distribution of the HOMO density. The same effect can be seen for the two Au 20 Pt 20 homotops at the bottom of Fig. 3.17 and all other systems that we studied. This has two important consequences: Firstly, as the Pt content increases, more Pt atoms are located at the surface of the cluster, so that the HOMO can extend away 5 Note, that the Au 20 tetrahedron is the only exception that we found for this observation. We attribute this deviation to the unusual electronic structure of Au20 [68]. 3.4. The interplay between fluxionality and electronic structure 39 from the surface. Secondly, the known importance of the surface for catalytic processes is enhanced in Au-Pt nanoalloys, not only due to the increased surface-to-volume ratio on the nanoscale, but also because of the presence of Pt, the many possible distributions of Au and Pt on the surface and the resulting implications for the local electronic structure of the system. 3.4.2 Electronic structure of Au-Pd nanoalloys Au-Pd nanoalloys show an electronic structure very similar to that of Au-Pt nanoalloys. There is but one point requiring special attention in small Au-Pd nanoalloys that can be neglected in Au-Pt NP: their spin polarization. Bulk Au and Pd are not magnetic, but high magnetic moments have been reported for small Pd clusters [140]. We start from the bilayered ground state of Pd 13 taken from Ref. [140], step-by-step replacing lowlycoordinated Pd atoms by Au as suggested in Ref. [141, 142]. The resulting Au n Pd 13−n (n=0-13) structures were then locally relaxed. 0 2 4 6 8 10 12 0 20 40 60 80 100 DOS at Fermi level/electron [arb. units] Au content [%] 13 bilayer 20 tetra 38 type 1 38 type 2 Figure 3.18: The DOS close to the Fermi level as a function of the Au content for Au/Pd particles. For all three sizes (13-atom, 20-atom and 38-atom) the DOS decreases with increasing Au content. For the 38 atomic clusters the DOS for two different structure types is shown. The trend is the same for both types. We allowed and checked for spin polarization, evaluating the total energy for different multiplicities. We found that for the purposes of this study the magnetization is not decisive, as the DOS at the Fermi level depends only very little on the spin polarization in the cases that we studied. The integrated DOS typically changes by less than 1% for most structures as a function of magnetization, and a few percent at most. For the sake of completeness, some magnetizations are, however, reported in the following. The resulting magnetic moments are 6 µB for Pd 13 , 3 µB for Pd 12 Au 1 , 4 µB for Pd 11 Au 2 and 2 µB for Pd 9 Au 4 . Clusters with larger Au contents are either singlets or triplets depending on whether their electron number is odd or even. The DOS at the Fermi level is then defined as for Au-Pt NP and shown in Fig. 3.18 as a function of the particles’ composition (red diamonds). The red inset depicts the structure of Au 3 Pd 10 as an example. From the figure one can clearly draw the conclusion that again the DOS at the Fermi edge decreases with increasing Au amount. In order to check whether this behavior of the DOS is general we repeated the procedure for larger clusters. Our choice of further test systems was motivated by similar considerations as in the case of Au-Pt NP. The experimental studies are concerned with relatively large particles that show a bulk-like structure [132]. Therefore, we also theoretically focus on investigating systems where a bulk-like geometry is a reasonable starting point. Furthermore, as bulk Pd and Au are not magnetic, and as the experimentally observed particles are 40 Chapter 3 — The bimetallic effect in Au-Pt and Au-Pd nanoalloys bulk-like in their structure, it is a plausible assumption that magnetic effects are not important for explaining the catalytic properties. In addition, our study of the 13-atom clusters has shown that the total DOS is not very sensitive to the magnetization. Therefore, we choose for our following investigations always the electronic configuration that has the lowest spin polarization (under the condition that the aufbau principle is respected). From the just described points of view the Au 20 tetrahedron is again a good starting point for the theoretical investigation. We replaced Au by Pd as explained in Sec. 3.2 with Pt, again observing the rule that high-coordination number sites are energetically favorable for Pd. The orange inset of Fig. 3.18 shows the structure of Au 16 Pd 4 as an example and the DOS at the Fermi level as a function of the Au content. Also in this case an increase with increasing amount of Pd is observed for all mixed Au/Pd particles. This observation does not depend on the specific structural motif. −3−2−1 0 density of states [arb. units] [eV] (a) (b) (c) (d) (e) (f) Figure 3.19: Right: geometric structure and isosurface plot of the HOMO of 38-atom particles. From top to bottom: (a) Au 19 Pd 19 , (b)-(d) 3 different structures for Au 28 Pd 10 , (e) Au 30 Pd 8 , (f) Au 32 Pd 6 . White and blue areas represent negative and positive isosurface values of 0.04 a−3/2 0. Left: the DOS of each of the clusters. To demonstrate this explicitly and to check another cluster size we repeated the procedure for the truncated octahedral 38-atom clusters. We studied two lines of structures with 38 atoms. In both of them 6 Pd atoms form the core (see black inset of Fig. 3.18), as this is an energetically favored arrangement. However, the two lines differ in the position of the remaining “surface” Pd atoms. The structures labeled “type 1” in Fig. 3.18 were obtained by randomly replacing surface Au by Pd atoms. In the structures labeled “type 2” the surface Pd atoms were placed such that they occupy the interior parts of the facets. In the literature [141] these latter configurations have been reported as being energetically favorable. Our calculations confirm this finding as structures of type 2 are lower by 0.08 eV for Au 30 Pd 8 and 0.83 eV for Au 28 Pd 10 than the corresponding structures of type 1. We observe that all structures have rather small HOMO-LUMO gaps, which may be an indication for further symmetry breaking being energetically favorable. The important finding is that also in the 38-atom faceted cases both types of structures show the same trend: the DOS at the Fermi level increases with increasing Pd content. Again this finding could imply that, in terms of Sabatier’s principle, a certain Au-Pd ratio might correspond to optimal adsorbate binding energies in a certain reaction. In the next step, following the same reasoning as before, we investigate the relation between the clusters’ geometry and their electronic structure. To this end we take a closer 3.4. The interplay between fluxionality and electronic structure 41 look at several of the faceted 38-atom particles. The right side of Fig. 3.19 shows from top to bottom structures and HOMO densities of Au 19 Pd 19 , three different Au 28 Pd 10 homotops, Au 30 Pd 8 and Au 32 Pd 6 . The left column depicts the occupied DOS with the respective Fermi levels aligned at 0 eV. The vertical line at -1 eV indicates the previously mentioned integration range. The conclusions that can be drawn from Fig. 3.19 are rather similar to the Au-Pt NP case: First, the HOMO in each case is closely associated with the Pd atoms. Second, an increasing amount of Pd brings an increasing number of Pd atoms to the surface and thus leads to a HOMO that is dominantly located at the surface of the cluster. This is clearly seen, e.g., when comparing the HOMO representation of Au 32 Pd 6 to the one of Au 19 Pd 19 and it indicates that Pd atoms at the surface are likely to be favorable in terms of the particles’ activity. Third, we carefully inspect three homotops of Au 28 Pd 10 . Structure (b) as suggested in the literature [141] has the lowest total energy, (d) is higher by 0.83 eV and (c) by 1.64 eV. Independent of these energetic differences, there is an important observation to be made in comparing structures (b)-(d): although the three geometries are overall rather similar, the differences between the HOMOs are quite noticeable. A moderate change in the geometry leads to a significant change of the energetically high-lying part of the density. Thus, a specific arrangement of Au and Pd atoms at the surface can lead to special electronic and thus chemical properties. 3.4.3 The role of Au in Au-Pt nanoalloys In the spirit of Ref. [131] and as explained in the introductory paragraphs of Sec. 3.4 one could speculate that a synergistic effect in Au-Pt nanoalloys for a specific reaction could in principle be explained by linking a catalytic descriptor of this reaction to the DOS at the Fermi level. One could then argue that the binding energy between Au-Pt NP and the specific reactants must neither be too strong nor too weak for the reaction to take place, in compliance with Sabatier’s principle. An optimal Au-Pt mixing ratio would then emerge from these considerations. Even though this point of view is perfectly legitimate and has led to valuable insights into gas-phase catalysis on surfaces under well-controlled conditions, it is not easily applicable in our case as mentioned before. However, as a synergistic effect is observed in many different reactions with different matrices and solvents, we suggest that fundamental properties of Au-Pt NP that are independent of the reaction lead to the special properties of Au-Pt NP. With the DOS at the Fermi level and the spatial arrangement of the HOMO density being determined by Pt as discussed earlier, we now want to ask the question which fundamental catalysis-favorable property is associated to Au. Here, I want to argue that this property is the mobility of atoms in the Au-Pt systems, or the NP fluxionality. Fluxionality has been recognized to be of crucial importance for the catalytic activity of clusters and surfaces [143]. It can be a primer for the ability to structurally adjust to reactants, form active sites [144] and directly influences turnover rates by affecting the catalysts’ ability to regenerate after the reaction. In order to determine how easy it is for atoms to rearrange in Au-Pt systems we performed two types of studies. In the first one we take the perspective that with respect to catalytic activity, the surface of the NP, which for faceted NP is similar to a bulk surface, is likely their most important 42 Chapter 3 — The bimetallic effect in Au-Pt and Au-Pd nanoalloys ini fin ini finini fin ini finini fin ini fin I. Pt/Pt II. Pt/Pt+Au III. Au/Pt IV. Au/Pt+Au V. Pt/Au VI. Au/Au Figure 3.20 : Sketch of the setup used for the study of Au and Pt adatom self-diffusion from a fcc (ini) to a hcp (fin) hollow position. The systems we studied were: I. Pt adatom on pure Pt(111) surface, II. Pt adatom on Pt(111) surface with single Au atom in top surface layer, III. Au adatom on pure Pt(111), IV. Au adatom on Pt(111) doped with Au, V. Pt adatom on Au(111) and VI. Au adatom on Au(111). part. When reactants approach the surface, the surface may structurally react to form active sites. The induction time, i.e., the time that goes by before the reaction starts, seen in NP catalysis can be an indication for such processes [80, 132, 145]. A canonical step in surface restructuring is the movement (diffusion) of atoms on the surface. Therefore, we chose the height of the barriers for the diffusion of Au and Pt adatoms on pure and doped Ptand Au-surfaces as a measure for the surfaces’ ability to restructure. As an aside we note that this line of thinking is also supported by the observation that bimetallic effects as well play a role in Au-Pt surfaces [115] as mentioned before. system Eads fcc Eads hcp Ebarrier Pt/Pt 4.60 4.82 312 Pt/Pt+Au 4.61 4.76 287 Au/Pt 2.77 2.87 177 Au/Pt+Au 2.79 2.86 165 Pt/Au 4.05 4.15 143 Au/Au 2.50 2.56 115 Table 3.3 : Adsorption energies (in eV) of Ptand Au-adatoms in pure and doped Ptand Au surfaces for the start and the end position of each NEB calculation. For a definition of the diffusion barrier (in meV) see text. For calculating these diffusion barriers we rely on the NEB method (see Sec. 2.5). It allows to determine the minimum energy path between a given initial and final state. If this minimum energy path goes through one or several maxima as a function of the reaction coordinate the largest of these barriers can be regarded the rate-determining step of the reaction. Calculations were carried out with VASP using the PBE GGA [24] and employing the PAW method to describe the electron-ion interaction [64, 67] (see Sec. 2.4). Technical details regarding these calculations can be found in Appendix C. The cutoff-energy for the plane wave basis set was set to 450 eV. Pure and doped Pt and Au fcc(111) surfaces were modeled by 4-layer slabs containing 4x4 atoms in each layer and being separated by 20 ˚ A of vacuum. The bottom two layers were held fixed at the Pt (Au) bulk equilibrium distance of 3.977 (4.174) ˚ A, while the other layers were allowed to fully relax. The Brillouin-zone integrations were performed using Monkhorst-Pack k-point meshes of 5x5x1 size. We considered a diffusion step from a fcc to a hcp hollow site for different combinations of surfaces and adatoms, as depicted in Fig. 3.20. In all cases the diffusion barrier was defined as the energetic difference between the transition state and the hcp adsorption site, this being the largest energy difference and thus determining the rate of this particular diffusion 3.4. The interplay between fluxionality and electronic structure 43 step. In addition to the diffusion barriers we determined the adsorption energies (see Tab. 3.3) of the adatoms at the fcc and hcp hollow site as Eads =Eslab +Eadatom −Eslab+adatom (3.7) 100 120 140 160 180 200 220 240 260 280 300 320 Pt/Pt Pt/Pt+Au Au/Pt Au/Pt+Au Pt/Au Au/Au System Diffusion barrier [meV] Figure 3.21: Barrier for Ptand Auadatom diffusion on Pt-surfaces with and without Au doping and on a pure Ausurface. An explanation of the notation can be found in Fig. 3.20. Fig. 3.21 shows that diffusion of a Pt adatom on a pure Pt surface has the highest barrier of all considered systems. This is in line with the Pt adatom being most strongly adsorbed to the pure Pt surface. Yet, the insertion of one Au atom into the Pt surface lowers the diffusion barrier by about 25 meV. A similar effect can be seen by comparing the diffusion barrier of an Au adatom on a pure Pt surface and on a Pt surface with one Au dopant. It is particularly interesting that the diffusion barrier decreases even more for a Pt adatom on a Au surface. In this case the adsorption energy is almost as high as that of Pt on Pt(111) leading one to expect an equally high diffusion barrier. The facilitated surface diffusion on surfaces with a higher Au content can therefore not only be explained by Au adatoms being more weakly bound to the surface than Pt adatoms. We interpret this finding as an increased ability of Au-alloyed Pt-surfaces to undergo structural changes and thus as a higher surface fluxionality with increasing Au-content. In the second type of study that we did in order to show that increasing amounts of Au increase the atomic mobility in Au-Pt NP we moved away from the surface perspective and regarded the NP as finite clusters. The fluxionality of small Au clusters has recently been shown by extensive scans of their potential energy surfaces [146]. This method, however, is not feasible for Au-Pt nanoalloys in the experimentally relevant size range. Therefore, we performed one nanosecond long MD simulations for randomly mixed truncated octahedral clusters (38, 201 and 586 atoms) and for 60-atom and 100-atom Gupta potential based core-shell clusters to obtain a measure of the mobility of the cluster atoms in terms of their mean square displacement (MSD) hˆr2i=*1 M M X i=1 (Ri(t)−Ri(t= 0))2+,(3.8) where M is the number of atoms in the cluster and Ri ( t ) is the position of atom i at time t . The angle brackets indicate a time average. The MD simulations were done using the DL POLY program [126] and the NVT Evans ensemble with the same many-body Sutton-Chen potential as in Sec. 3.3 to describe the potential energy of the system. Randomly mixed Au-Pt NP segregate to Au shell Pt core structures even at comparably modest temperatures and short simulation times. As our aim was to retain the homogenous 44 Chapter 3 — The bimetallic effect in Au-Pt and Au-Pd nanoalloys mixing pattern of the truncated octahedra all simulations were carried out at 300 K. This lies well below the melting point of most of these systems 6 , meaning that the MSD averaged over the whole simulation time is approximately zero. However, at shorter timescales of 1000 fs a diffusive motion of Au and Pt atoms takes place. The time average in Eq. 3.8 was therefore computed for subsequent time windows of 1000 fs. For each Au-Pt mixing ratio three simulations were carried out to minimize statistical uncertainties. Note that even though this method yields only an approximation to the diffusion of cluster atoms, it can be used to compare the short-time atomic mobility of nanoalloys with varying mixing ratios. 0.02 0.04 0.06 0.08 0.10 0.12 38 atoms 201 atoms 586 atoms 0.02 0.04 0.06 0.08 0.10 0 20 40 60 80 100 Au content [%] 60 atoms 100 atoms MSD [ ] Figure 3.22: Top: MSD from MD simulations of truncated octahedral clusters that are solid solutions of Au and Pt with 38, 201 and 586 atoms. Bottom: MSDs of 60-atom and 100atom core-shell particles. For NP in the size regime that we are concerned with the high surface-to-volume ratio can have significant impact on the average MSD of the NP. Atoms at lowly coordinated positions, e.g., at kinks or edges can show MSDs that are considerably larger than those of highly coordinated volume atoms or atoms in a surface plane. For irregular NP like the 60atom and the 100-atom NP this effect is more pronounced than it is for the regular truncated octahedra with their magic atom numbers of 38, 210 and 586. To obtain comparable MSD values it is therefore necessary to keep the geometries of NP with the same number of atoms but different Au-Pt ratios as close as possible. For the irregular 60-atom and 100-atom clusters this makes a full geometry optimization prior to MD simulations necessary. Nevertheless, the MSDs of these NP show large statistical variations as compared to the MSDs of the truncated octahedra. For this reason we want to stress that we are not concerned with evaluating the exact value of the MSD in our NP. In contrast, we want to study the trend that the MSD shows with increasing Au content. The resulting MSDs are depicted in Fig. 3.22: the truncated octahedra at the top and the 60-atom and 100-atom core-shell particles at the bottom. It can clearly be seen that the composition-dependent trend is the same for all clusters: The MSD increases with increasing Au-content. This is again independent of particular geometric configurations or mixing patterns. Thus, although being of very different nature, the MD study confirms the conclusions drawn from the surface diffusion barrier calculations: structural flexibility increases with an increasing amount of Au. 6 The melting point of the 60-atom clusters lies below 300 K. Therefore in this case MD simulations were carried out at a temperature of 200 K. 3.4. The interplay between fluxionality and electronic structure 45 3.4.4 Combining electronic structure and fluxionality 0.0 0.2 0.4 0.6 0.8 1.0 0 20 40 60 80 100 MSDxDOS at FL [arb. units] Au content [%] 38 atoms 60 atoms Figure 3.23: Product of a polynomial least squares fit to the DOS at the Fermi level (FL) and MSD for 38-atom and 60-atom Au-Pt clusters as a function of Au content. Quantities are scaled so that for 0 and 100% Au content the curves intersect the y-axis at 0 and have a maximum value of 1. We can now put together all the findings. The DOS of Au-Pt NP develops properties that one naturally associates with being favorable for catalysis with increasing Pt content, whereas the structural flexibility increases with increasing Au content. Therefore, for each reaction there can be an ideal ratio of Au and Pt that yields overall optimal properties. This is visualized in Fig. 3.23. It shows the volcano-like plot that one obtains by multiplying a polynomial least squares fit to the DOS at the Fermi level and the MSD. This has here been done for the 38-atom and 60-atom clusters, as for these systems we could access both quantities (see Fig. 3.24). An ideal Au-Pt mixing ratio emerges as a consequence of the interplay between structural fluxionality provided by Au and a favorable electronic structure contributed by Pt. Of course what this ideal mixing ratio is will sensitively depend on the reaction, experimental conditions, as well as on the exact structure and mixing pattern of the catalysts. Yet, the point that Fig. 3.23 emphasizes is that synergistic effects Figure 3.24 : Left: MSD and linear fit to MSD of 38atom (black triangles) and 60-atom (red triangles) clusters. Right: DOS at the Fermi level for 38-atom and 60-atom clusters. For constructing the ”volcano” plot the data was fitted with 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 content [%] 38 atoms 60 atoms DOS at FL/electron 0 1 2 3 4 5 6 7 8 0 20 40 60 80 100 Au content [%] 38 atoms 60 atoms in Au-Pt NP are expected on rather general grounds and regardless of the specific chemical reaction and environment. Additionally, Fig. 3.23 illustrates another point. We do not expect ”the whole to be greater than the sum of its part”. The synergistic or rather bimetallic effect in Au-Pt nanoalloys emerges as a consequence of adding up catalysis-favorable fundamental physical properties of Au and Pt. The effect of the combined properties might be amplified by the high surface to volume ratio in NP, the presence of steps and kinks that act as active sites in catalysis [144] and the elongated average bond lengths resulting from thermal expansion (see Sec. 3.3) or NP-matrix interactions (see Appendix A). Different to what the buzzword ”synergy” suggests, Au-Pt nanoalloys might not possess special catalytic properties that go beyond a combination of the properties of pure Au and Pt NP. 4 Magnetic-to-nonmagnetic transition in Mn-doped Si-clusters 4.1 Dilute magnetic semiconductors Semiconductors doped with few impurities that carry a magnetic moment are called dilute magnetic semiconductors. During the last decades they have been subject of intense study, after it had been observed that the introduction of a small fraction of magnetic impurities would leave the favorable transport and optical properties of the semiconductors unchanged, at the same time being accompanied by large magnetic moments and many interesting magnetic order phenomena. To name only one example, it has been shown by Ohno et al., that (Ga,Mn)As exhibits ferromagnetic order up to temperatures as high as 110 K [147]. Later theoretical work provided a model for the origin of this ferromagnetic ordering and predicted materials with Curie temperatures even exceeding room temperature. Even though it is believed today that dilute magnetic semiconductors with such high Curie temperatures may not be realizable, the field nevertheless bears plenty of fundamental questions, many of which are still unanswered. The seemingly simplest of these questions is how a single magnetic impurity interacts with the semiconducting host, i.e., whether the impurity’s magnetic moment is quenched or survives. Pioneering studies have been conducted by Ludwig and Woodbury in the 1960s using electron-paramagnetic resonance spectroscopy to investigate magnetic properties of Si doped with a range of 3 d transition metal impurities at substitutional and interstitial positions [148]. Furthermore, Ludwig and Woodbury provided a simple and descriptive model to predict quenching or survival of the magnetic moment of the transition metal impurity. 54 Chapter 4 — Magnetic-to-nonmagnetic transition in Mn-doped Si-clusters MnSi7 +MnSi8 +MnSi9 +MnSi10 + MnSi11 +MnSi12 +MnSi13 +MnSi14 + Figure 4.5: Energetically most favorable structures for MnSi + n ( n = 7 − 14) as predicted by PBE0. Contrary to experiment MnSi+ 11 is exohedrally doped. For the simulated annealing simulations different starting structures were heated to temperatures between 3500 and 4500 K and propagated at this temperature for 3 ps. Subsequently, the temperature of the system was reduced by a factor of 0.9 every 0.2 ps until it fell below 300 K. At this point the structure usually was trapped in a stiff local minimum. It is therefore of great importance to start simulations from a large number of different geometries to scan as much of the potential energy surface as possible. The lowest energy isomers from these simulations were then locally relaxed using a quasi-Newton method and larger TZVPP basis sets until the total energy changed by less than 10 −8 Hartree and the forces acting upon the ionic cores were smaller than 0.001 a.u.. The modified big bang approach consisted of randomly generating different sets of starting coordinates with bond lengths compressed by about 10% of the equilibrium bond length and propagating these coordinates at a fixed temperature of 300 K for a short time span of 1.5 ps. Again, lowest energy geometries were locally relaxed as described before. Figure 4.6: Structure of the energetically lowest lying endohedrally doped MnSi + 11 as predicted by the PBE0 xc functional. The energetically lowest lying structures that PBE0 predicts are shown in Fig. 4.5. Consistent with experiment clusters with n = 7 − 10 are exohedrally and clusters with n = 12 − 14 are endohedrally doped. Similarly, in perfect agreement with the XMCD spectra of Fig. 4.4 clusters with n = 7 − 10 exhibit a large magnetic moment of 4 µB , while the magnetic moment of those with n = 12 − 14 is completely quenched. MnSi + 11 is, contrary to the experimentally predicted exohedral-toendohedral transition, exohedrally doped. Furthermore, the calculated structure carries a magnetic moment of 2 µB in contradiction to the vanishing XMCD signal of MnSi + 11 in Fig. 4.4. The energetically lowest lying endohedral structure, although 1.1 eV higher in energy than the PBE0 ground state, (shown in Fig.4.6) would however agree with experiment. As the experimentally observed structures are expected to be the ground state structures, a closer look at the theoretical results reveals that the failure of PBE0 for MnSi+ 11 can indeed be explained in terms of the effect of self-interaction on the eigenvalue spectrum of both isomers. To clarify the situation we calculated the orbitalSIE, Eq. (2.14) , as introduced in Sec. 2.2. To determine eiσ a local developers version of the program package PARSEC [59] was used to calculate the electronic structure of MnSi + n ( n = 7 − 14). We employed Troullier-Martins pseudopotentials in Kleinman-Bylander form constructed within the PBE GGA [24] with a non-linear core correction as discussed in Sec. 2.4. The pseudopotentials were based on the electronic configuration 4 s1.75 4 p0.25 3 d5 with s/p/d cut-off radii of 1.884/2.573/1.980 a.u. for Mn and 3 s2 3 p2 with s/p cut-off radii of 4.3. A DFT-study of the magnetic-to-nonmagnetic transition in Mn-doped Si-clusters 55 2.493/2.390 a.u. for Si. It was tested that the influence of the used pseudopotentials on the eigenvalue spectrum was negligible by comparing them with our all-electron TURBOMOLE calculations. A grid spacing of 0.3 a.u. was used. eiσ was evaluated using the PBE GGA xc functional. Following Perdew and Zunger and neglecting the difference between selfinteraction corrected and uncorrected orbitals, the self-interaction corrected eigenvalues can be estimated as εiσ ≈εapprox iσ −eiσ,(4.9) where εapprox iσ results from a self-consistent calculation using vapprox xc , which is PBE in our case [21]. We can now compare these approximately self-interaction corrected eigenvalues with those that PBE0 yields. The smaller the difference ∆ εiσ = εiσ −εPBE0 iσ , the more reliable the PBE0 eigenvalues εPBE0 iσ are. Figure 4.7 : Eigenvalue spectrum (bars), total DOS (dotted line), and Mn-projected local DOS (solid line) of endohedral MnSi+ 11 (top) and the exohedral PBE0 ground state (bottom). ∆ εi is a measure for how much the PBE0 eigenvalues are affected by the 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 the occupied eigenvalue spectrum interpreted as a DOS and the respective ∆ εi for both MnSi + 11 isomers, the endohedrally doped at the top and the exohedrally doped PBE0 ground state at the bottom. While for the endohedrally doped isomer ∆ εi≈ 0 for all upper orbitals, the situation is considerably different for the exohedrally doped isomer. Here, the PBE0 eigenvalues are strongly affected by the SIE. This is not only true for the orbitals 4-5 eV below the Fermi level, that can be associated with the Mn dopant 5 , but also for higher lying states that participate in bonding. All in all, Fig. 4.7 illustrates that the energetic ordering of MnSi+ 11 isomers as predicted by PBE0 is unreliable. In a second step we used a completely different approach to further elucidate the situation. Using the RSH xc functional ω PBE [43] (see Sec. 2.2) as implemented in the QChem program package [174], we checked the energetic ordering of the exohedral and the endohedral MnSi + 11 isomers as a function of the range-separation parameter ω . For values of ω = 0 . 2 − 0 . 4 a.u. the ω PBE functional predicts the energetic ordering that is in agreement with experiment. This finding highlights the delicate balance of exchange and correlation effects that determines the energetic ordering in MnSi+ 11. I will explain in the following, why the SIE, although also plaguing the other exohedral systems, only becomes decisive for the energetic ordering in case of MnSi + 11 . To this end 5 This can be done by projecting the KS eigenvalues onto atomic orbitals via a Mulliken population analysis as explained in Sec.3.4.1. 56 Chapter 4 — Magnetic-to-nonmagnetic transition in Mn-doped Si-clusters Fig. 4.8 shows the total DOS, the DOS projected onto the Mnd -orbitals and ∆ εi for MnSi + 7 (top) and MnSi + 14 (bottom). These two systems are exemplary for the exohedral and the endohedral size regime, respectively. Additionally, isosurface plots of the orbitals with the largest Mn-3dcharacter and of the HOMO are shown. −200 0 200 −2 0 −400 0 400 −2 0 −7−6−5−4−3−2−1 0 1 MnSi7 + MnSi14 + DOS DOS Figure 4.8 : Eigenvalue spectrum (bars), total (dotted line), and local Mn 3 d -projected (solid line) DOS with isosurface plots (at ± 0.04 a.u.) of the Mn 3 d orbitals and the highest occupied orbital for MnSi + 7 and MnSi + 14 . Positive (negative) values represent the spin-up (spin-down) channel. The d -like orbitals are localized on the Mn atom in MnSi + 7 , whereas they are delocalized in MnSi + 14 . Also shown is ∆ εiσ , the effect of the SIE on the PBE0 eigenvalues. Fig. 4.8 illustrates several points. First, for the endohedral size regime, as represented by MnSi + 14 , PBE0 faithfully predicts the electronic structure, while in the exohedral size regime this is not the case for all relevant states. However, the eigenvalues most severely affected by the SIE (4-5 eV below the Fermi level) are energetically well separated from the Si-states. It is to be expected that the shift of these orbitals in a fully self-interaction corrected calculation would hardly increase the hybridization between the Mnand the Si-states and therefore the electronic and magnetic properties of these clusters are expected to be only slightly affected. Furthermore, the relative shift between delocalized and localized orbitals that results from the SIE is less critical when structures with localized and delocalized orbitals are energetically well separated [52]. Here, this is the case for all structures except MnSi + 11 . The latter is the only cluster for which the global geometry optimization resulted in both exohedral and endohedral isomers close in energy. For MnSi + n ( n = 7 − 10) exohedral isomers are consistently lower in energy than endohedral ones. We checked, however, that the energetic ordering of exohedrally and endohedrally doped isomers remains the same for cluster sizes at the structural transition, i.e., for MnSi + 10 and for MnSi + 12 . To this end we calculated the total energy of the ground state geometry as predicted by PBE0 and the energetically lowest lying endohedral (exohedral) isomer for MnSi + 10 (MnSi + 12 ) using ω PBE as before with a range separation parameter of ω =0.2 a.u. as this ω leads to the correct energetic ordering for MnSi + 11 . Tab. 4.1 shows the energetic difference between exohedrally and endohedrally doped isomers calculated with PBE0 and ω PBE energies for MnSi + n (n=10-12) and independently supports the view that the SIE only affects MnSi + 11 in such a way as to change the energetic ordering of exohedrally and endohedrally doped isomers. For MnSi + 10 and MnSi + 12 the energetic ordering as well as the 4.3. A DFT-study of the magnetic-to-nonmagnetic transition in Mn-doped Si-clusters 57 Table 4.1 : Total energies differences of MnSi + n (n=10-12) exohedral and endohedral isomers ∆ E = Eexo −Eendo calculated using PBE0 and ω PBE with a range separation parameter of ω = 0 . 2 a.u. Positive numbers mean that the endohedral isomer is energetically more favorable. system ∆EωPBE in eV ∆EPBE0 in eV MnSi+ 10 -0.67 -1.09 MnSi+ 11 1.67 -1.13 MnSi+ 12 2.05 1.05 relative energy difference between the exohedral and the endohedral isomer roughly stay the same for PBE0 and ω PBE. Consequently, only MnSi + 11 needs special consideration and the endohedral structure shown in Fig. 4.6 can in agreement with experiment considered to be the ground state structure of MnSi+ 11. The second point that Fig. 4.8 emphasizes is that the change in local electronic structure of the Mn dopant that was observed in the x-ray absorption spectra (Fig. 4.4) is reflected in the occupied eigenvalue spectrum and also by the shape of the KS orbitals. −200 0 200 400 −7−6−5−4−3−2−1 0 1 DOS [arb. units] Figure 4.9: Eigenvalue spectrum (bars) and DOS of atomic Mn + with 4 unpaired electrons. Note that the ground state of Mn+has a spin moment of 6 µB. For MnSi+ 7 , the occupied states of Mn 3 d character are mostly isolated at ≈ 4 − 5 eV below the Fermi level and are tightly localized at the Mn dopant, i.e., they largely preserve a (perturbed) atomic character and only weakly interact with Si states, as can be seen from the DOS and the isosurface plots. Note also the similarity between the eigenvalue spectrum of atomic Mn + with 4 µB as shown in Fig. 4.9 with the Mn-3 d -PDOS of MnSi + 7 . Because of this weak interaction, the localized orbitals of Mn 3 d character are qualitatively very similar for all exohedral MnSi+ n clusters and lead to the nearly identical x-ray absorption spectra in Fig. 4.4. This notion of, at least partly, atomic Mn 3 d states is lost in the endohedral size regime, represented by MnSi+ 14 in Fig. 4.8. Here, orbitals with partial Mn 3 d character are strongly hybridized with Si states and are shifted to ≈ 0 . 5 − 2 eV below the Fermi level, i.e., they participate strongly in bonding and are delocalized over the Si frame. Consequently, the Mn 3 d derived PDOS sensitively depends on the structure of the Si cage, which is reflected in the variation of the x-ray absorption spectra of endohedral MnSi+ n for different n ( n = 11 − 14) in Fig. 4.4. The strong spd hybridization with the participation of all Mn valence orbitals explains the complete quenching of the magnetic moment from 4 µB in exohedrally to 0 µB in endohedrally doped clusters, which is again in accordance with the experimental XMCD spectra. Thirdly, Fig. 4.8 is exemplary for the well-known connection between the SIE and the orbital localization [51]. The tightly localized, nearly atomic 3 d -orbitals of MnSi + 7 are more strongly affected by the SIE than the rather delocalized orbitals of MnSi + 14 , despite the partial Mn-3 d -character of the latter. For a thorough discussion of orbital localization and self-interaction, the reader is referred to Ref. [51]. 58 Chapter 4 — Magnetic-to-nonmagnetic transition in Mn-doped Si-clusters The structural change at the exohedral-to-endohedral transition of MnSi+ n can be quantified by the coordination number Nc of Mn, i.e., the number of Si atoms in the first coordination sphere as exemplified for MnSi+ 8 in the inset of Fig. 4.11. In exohedral clusters, Mn adopts a minimal coordination number of Nc = 2 − 4, while in endohedral clusters Nc is maximized to 11−14, i.e., all Si atoms are within the first coordination sphere of Mn. 2.35 Figure 4.10: Lowest endohedral isomer of MnSi + 10 . The Mn-Si bond length is compressed to only 2.35 ˚ A in this cluster. The average Mn-Si bond length a elucidates why encapsulation of the Mn dopant becomes energetically favorable only for n≥ 11: In the ground state structures, the Mn-Si nearest neighbor distance expands from a = 2 . 43 − 2 . 58 ˚ A in exohedral to a = 2 . 53 − 2 . 67 ˚ A in endohedral clusters. In contrast, it would be compressed to a = 2 . 35 ˚ A in the energetically most favorable endohedral isomer of MnSi+ 10 (as illustrated in Fig 4.10). Even though Mn favors high coordination in Si [175], this strain, which becomes even more pronounced in smaller clusters, precludes endohedral ground states for n≤10. The abrupt change in coordination at the structural transition is interrelated with the quenching of the magnetic moment as illustrated in Fig. 4.11: Here, the calculated magnetic moments of MnSi+ n are plotted versus the weighted coordination number d0Nc/a , which takes into account both the number of Si nearest neighbors Nc as well as their average distance a to the Mn atom, normalized to the nearest neighbor distance d0 in bulk Si. Low-coordinated exohedral clusters with d0Nc/a = 1 . 9 − 3 . 7 ( Nc = 2 − 4) carry a magnetic moment of 4 µB , which is quenched to 0 µB in high-coordinated species with d0Nc/a = 10 . 2 − 12 . 3 ( Nc = 11 − 14). This relation of magnetic moment and weighted coordination also holds for higher-energy isomers that are included in Fig. 4.11 and mark the transition from magnetic to nonmagnetic impurities around d0Nc/a ≈4. # of bonds MnSi8 + 2 3 4 5 121086420 magnet ic moment 0 1 2 3 4 5 6 14 Mn-Si distance 6 0 1 2 3 weighted coordination number d0Nc/a Figure 4.11 : Magnetic moment versus weighted coordination d0Nc/a for ground state (open circles) and higher energy (solid circles) isomers of MnSi+ n ( n = 7 − 14), isolated neutral Mn impurities in bulk ([151], solid squares) and amorphous Si ([154], open triangles), and in a Si nanocrystal ([156], solid diamond). Inset: bond length distribution of MnSi+ 8 with first coordination sphere (dotted line). Fig. 4.11 shows that this observation can be generalized to extended systems, i.e., to a neutral Mn impurity at a substitutional site in crystalline Si [151] or in hydrogen-passivated 4.3. A DFT-study of the magnetic-to-nonmagnetic transition in Mn-doped Si-clusters 59 Si nanocrystals [156]. It also applies to very low concentrations of Mn in amorphous Si, for which a possible relation between magnetic moment and coordination has been pointed out [154]. Yet, although Nc is the leading term, it does not account for the dependence of the local magnetic moment on the nearest-neighbor distance [151] that becomes important around Nc = 4 and is included in d0Nc/a . As can be seen in Fig. 4.11, Mn-doped bulk Si is just at the transition from high-spin to low-spin states and therefore reacts very sensitively to changes in d0Nc/a . This might explain the large scatter in experimental results on Mn-doped Si and indicates that high-spin states could be stabilized by an appropriate expansion of the lattice parameter, e.g., in ultra-thin films or passivated nanocrystals. In summary, the magnetic moment of Mn-doped Si has been investigated over a wide range of structural parameters, including extreme coordination numbers from 2 - 14. The study of singly doped, size-selected MnSi+ n clusters avoids impurity-band formation or interaction between impurities that might be present in experiments on bulk samples, but also in calculations with periodic boundary conditions. Thus the observed quenching of the magnetic moment is not a result of impurity band formation but of the electronic interaction with the Si host. While the SIE strongly affects tightly localized orbitals with Mn-3 d character and is thus persistent in exohedrally doped clusters in which the dopant only weakly interacts with the Si host, it only perturbs the energetic ordering of exohedral and endohedral isomers for MnSi + 11 . Finally, we found an universal correlation of the magnetic moment and the weighted coordination number. Bulk Si doped with Mn lies at the transition from high to low magnetic moments and therefore is particularly sensitive to the nearest neighbor bond lengths of the Mn dopant. 5 Summary and outlook 5.1 Summary In the present thesis structural and electronic properties of Au-Pt, Au-Pd nanoalloys and Mn-doped Si clusters were studied using non-empirical DFT and MD simulations employing empirical Sutton-Chen potentials. Au-Pt and Au-Pd nanoalloys show a bimetallic effect, i.e., increased activity and selectivity in many different catalytic reactions. This work contributes to the long-standing debate between theory and experiment on the mixing pattern of Au-Pt nanoalloys. We were able to show that a prominent characterization technique based on a combination of XRD and Vegard’s law relies on questionable assumptions. Au shell Pt core and homogeneously mixed NP in the experimentally relevant size range of ≈ 4 nm are indistinguishable in a typical XRD experiment. Furthermore, it was shown that Vegard’s law is valid for both types of mixing patterns. Theory consistently predicts core-shell structures as energetically most favorable, but homogeneously mixed Au-Pt nanoalloys are observed in experiment using element-specific characterization techniques. Our findings show that the formation of mixing patterns other than the core-shell one must be linked to effects such as interactions with the support material or the synthesis procedure. The study of the electronic structure of Au-Pt and Au-Pt nanoalloys showed that the DOS at the Fermi level depends on the Pt and Pd content of the NP, respectively. The DOS at the Fermi level is related to the binding energy between the catalyst and possible adsorbates and it increases with increasing Pt (Pd) content. Additionally, the spatial distribution of the HOMO is connected to the position of the Pt (Pd) atoms. Empirical MD simulations were used to extract the MSD of the atoms in an Au-Pt NP. The MSD is related to the atomic mobility or structural flexibility of the NP. The latter can be beneficial in that it allows the catalyst to structurally adjust to reactants, form element-specific binding sites 62 Chapter 5 — Summary and outlook and regenerate after a reaction has taken place. It was shown that the MSD increases with increasing Au content. This result could be supported by modeling the NP as surface slabs and studying the diffusion of a single Au or Pt adatom on these surfaces. The activation barrier for adatom diffusion, the simplest type of surface reconstruction, was shown to decrease with increasing Au content. A combination of the fundamental properties DOS at the Fermi level and atomic mobility might explain why a bimetallic effect is observed for Au-Pt nanoalloys under very varying experimental conditions. A specific Au-Pt ratio could amount to an optimal interaction between catalyst and reactants mediated by an optimal DOS at the Fermi level and an optimal amount of structural flexibility. In the second part of this thesis the magnetic properties of Mn-doped Si clusters were studied. These systems can be seen as well-defined models for a semiconducting host that interacts with a single magnetic impurity. A global geometry optimization was carried out to identify the ground state structures of MnSi + n (n=7-14). In agreement with experiment, it could be shown that a structural transition from exohedral to endohedral doping from MnSi + 10 to MnSi + 11 is accompanied by complete quenching of the magnetic moment. The size of the magnetic moment is correlated with the coordination number of the Mn dopant weighted with its average nearest neighbor distance. This relation holds also for Mn in amorphous and crystalline Si and suggests ways to stabilize the magnetic moment of a Mn impurity by appropriate lattice expansion. Systems for which the interaction between strongly localized and delocalized orbitals plays a major role pose serious difficulties to existing approximations of DFT because of the well-known self-interaction problem. By evaluating its orbital-SIE we found that MnSi + 11 suffers particularly severely from self-interaction. The one-parameter hybrid xc functional PBE0 which correctly predicted structures and magnetic moments of all other clusters therefore yields a wrong, exohedrally doped ground state structure in this case. By using a RSH functional we saw that the correct energetic ordering of the endohedral and the exohedral isomer is obtained for certain values of the range separation parameter. Together with the orbital-SIE this finding highlights how delicately exchange and correlation effects determine the energetic ordering in MnSi+ 11. 5.2 An outlook to Ni-Pd nanoalloys The search for bimetallic catalysts that perform efficiently in large-scale catalysis is frequently guided by the simple principle to find the combination of materials that yields the highest turnover rates and is as cheap as possible. Ni-Pd nanoalloys are in this sense optimal. Additionally, they constitute a system covering a variety of enticing fundamental questions. Firstly, small Ni and Pd clusters are known to exhibit high magnetic moments [11, 140]. The sizeand composition-dependent magnetic moment of Ni-Pd nanoalloys is therefore expected to exhibit interesting features. However, Chap. 4.3 showed that the correct description of 5.2. An outlook to Ni-Pd nanoalloys 63 Table 5.1 : Adsorption energies (in eV) of Pdand Ni-adatoms on pure and doped Pdand Ni surfaces for the start (ini) and the end (fin) position of each NEB calculation. The diffusion barrier (in meV) is defined as the energetic difference between the inital position and the saddle point of the minimum energy path. system Eads ini Eads fin Ebarrier 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 the geometrical and electronic structure even of allegedly simple magnetic transition metal compounds is a huge challenge for theory. DFT studies of small Ni-Pd clusters have up to now only been carried out using standard semilocal xc functionals [176, 177] and global geometry optimizations are rare [178] for these systems. Attempts to construct a reliable pseudopotential for Ni that could be used in conjunction with orbital-dependent functionals have so far proved challenging as well [179]. ini fin ini fin (a) (b) Figure 5.1: Setup for Pd/Ni adatom diffusion on Ni/Pd surface (a) without and (b) with a single Pd/Ni atom in the upper surface layer. The diffusion step takes place from one hcp hollow to another hcp hollow site. Secondly, a bimetallic effect in catalysis has recently been observed for Ni-Pd nanoalloys immobilized in so-called metal-organic frameworks [180] as well as in other reactions [78]. A close lying question is therefore whether the general concepts that apply for Au-Pt nanoalloys, are also suited to explain the special properties of Ni-Pd particles. This would again require a faithful description of their electronic structure, at least close to the Fermi level. Furthermore, the atomic mobility or fluxionality of Ni-Pd systems has to be tested. Initial steps into this direction have been taken following a similar line of argument as described in Sec. 3.4.3. Pure and doped Niand Pd-surfaces have been modeled by 4x4x4-layer slabs with equilibrium lattice parameters of 3.520 ˚ A for Ni and 3.939 ˚ A for Pd. A surface diffusion step between two equivalent hcp hollow sites as depicted in Fig. 5.1 has been considered. CI-NEB calculations were carried out in order to determine the activation energy of this diffusion step (see Sec. 2.5). Furthermore, the adsorption energies of the adatom at the inital and the final position have been determined using Eq. (3.7) . The resulting barriers and adsorption energies are listed in Tab. 5.1. Note that both the adsorption energies and the barriers are considerably smaller than for Au-Pt surfaces (cf. Tab. 3.3). This finding is in line with the recent observation that large Ni-Pd clusters (with diameters of ≈ 4 nm) preferably form solid solutions and that their surface is corrugated due to the large lattice mismatch of Ni and Pd [180]. The strong tendency for segregation of the two components found in Au-Pt NP thus seems not to be present in Ni-Pd NP. Furthermore, no specific trend of the diffusion barriers can be seen in Tab. 5.1. Exceptions to the overall small barriers of between 65 and 80 meV are found for the diffusion of a Ni adatom on Pd with a barrier of almost 300 meV and diffusion of a Ni adatom on a Ni surface doped with a 70 Appendix A — Temperature and support effects Figure A.4: DOS at the Fermi level with increasing lattice parameter for bulk Au (left) and Pt (right). Different integration thresholds are shown, that demonstrate that the Au bulk DOS increases more rapidly, as soon as the 5d-derived DOS is taken into account. Au Pt 3.90 3.95 4.00 4.05 4.10 4.15 4.20 lattice parameter [Å] up to -1 eV up to -2 eV 0 1 2 3 4 5 4.00 4.05 4.10 4.15 4.20 DOS at Fermi level [arb. units] up to -1 eV up to -2 eV up to -3 eV substitution of one Au atom in, e.g., Au 38 by a Pt atom (upon which the mean interatomic distance even decreases, as shown in Sec. 3.3), leads to a relative increase of the DOS at the Fermi level of nearly 100%, while the position of the Fermi level remains approximately constant. The effect of thermal expansion on the electronic structure of Au-Pt NP is therefore negligibly small as compared to the increase that is observed for alloying Au clusters with Pt atoms. A.2 Influence of the nanoparticle support The practical usefulness of catalytic NP is often dictated by how successfully they can be stabilized against coagulation, by their handling and their re-usability. Stabilization by end groups that strongly interact with the NP’s surface can severely alter and even limit the catalytic activity. Recently, it has been shown that highly catalytically active Au-Pt nanoalloys with a diameter of ≈ 3-4 nm can be synthesized and immobilized in so-called spherical polyelectrolyte brushes (SPBs) [80]. SPBs consist of a solid polystyrene core onto which long cationic polyelectrolyte chains are densely grafted as sketched in Fig. A.5. The polyelectrolyte is 2-amino-ethyl methacrylate (AEMH) which is positively charged in aqueous solution. Cl − anions represent the counterions which are mostly confined to the surface layer of the SPB, that is indicated in light gray in Fig. A.5. The Cl − counterions can be replaced by metal ions like [AuCl 4 ] − and [PtCl 6 ] −2 . These are then reduced to elemental Au and Pt using NaBH 2 . During the process of Au-Pt nanoalloy formation the surface layer considerably decreases from about 70 nm to only 20 nm. This can be attributed to the nanoalloys carrying a negative surface charge and therefore strongly interacting with the cationic polyelectrolyte chains. However, this interaction seems to be weak enough not to obstruct the catalytic activity of the Au-Pt nanoalloys [80]. A.2. Influence of the nanoparticle support 71 Figure A.5 : Au-Pt nanoalloys can be synthesized within the dense polyelectrolyte layer of a spherical polyelectrolyte brush. C H CH3 O CH2 NH3 C C H O CH2 n + polystyrene 70nm ClClClClClpolystyrene 20nm [AuCl4]- [PtCl6]2NaBH4 + 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 + polyelectrolyte layer with NP In this section, I want to present first results on how the interaction between the NP and the SPB support affects the fundamental structural and electronic properties of the Au-Pt nanoalloys that have been established in the main part of this thesis. As the simplest possible model setup we assume that the metal NP interact with monomeric AEMH units and that the combined system consisting of NP and AEMH units is neutral. The polystyrene core is assumed to be inert and will not be considered here. Furthermore, we disregard the aqueous solution in which the reaction takes place, as we want to concentrate on the effect of the NP-AEMH interaction. pure Au random core-shell Figure A.6: Two AEMH monomer units are placed close to the surface of an Au 20 , a randomly mixed Au10Pt10 and a core-shell Au10Pt10 cluster. Our first model setup is shown in Fig. A.6. Two AEMH monomer units are placed close to the surface of an Au 20 , a randomly mixed Au 10 Pt 10 and a core-shell like Au 10 Pt 10 cluster 1 . The dynamical evolution of these systems is simulated using constant temperature MD simulations via a Nos´e-Hoover thermostat of 300 K temperature. In the original formulation of Nos´e, a canonical ensemble is achieved by augmenting the classical Hamiltonian by a time scale variable s , its conjugate momentum ps and a parameter Q HNos´e = M X i=1 p2 i 2mis2+U(q1, ..., qM) + ps 2Q+ (Nf+ 1)kBTln s. (A.4) The time scale variable s fluctuates which amounts to a coupling to a heat bath of temperature T. Nf is the number of degrees of freedom. It can be shown that the microcanonical distribution in the augmented set of variables is equivalent to a canonical distribution of the variables {q1, ..., qM} , {p0 1, ..., p0 M} , where p0 i = pi/s [182]. As it is inconvenient to work with fluctuating time intervals in practical simulations, one often uses the formulation of Hoover, in which ps/Q is replaced by a thermodynamic friction 1 Similar to Sec. 3.2, the label ”core-shell” indicates that all Pt atoms are as highly coordinated and all Au atoms are as lowly coordinated as possible. 72 Appendix A — Temperature and support effects coefficient η, which can be expressed via a phenomenological relaxation time τ[183] η=ps NfkBTτ2,(A.5) which is here chosen four times as large as the simulation time step ∆t= 80 a.u. b= 1.9 fs. All systems were propagated for about 16 ps. The electronic structure in each timestep was calculated using the PBE GGA and a SVP basis set augmented as implemented in the TURBOMOLE program package [184]. As van-der-Waals forces are expected to play a role in the interaction between the Au-Pt nanoalloys and the AEMH molecules, the empirical dispersion correction of Grimme was employed [185], which adds a term Edisp =−s6 M−1 X i=1 M X j=i+1 Cij 6 R6 ij fdamp(Rij) (A.6) to the total energy of the system. Here, s6 is a global scaling parameter, which depends on the xc functional employed, Cij 6 are empirical dispersion coefficients of atom pair i and j , Rij = |Ri−Rj| , fdamp is a damping function that avoids singularities at small interatomic distances Rij and M denotes the number of atoms. To differentiate between the effects of temperature and those of the AEMH monomer units, a second set of MD simulations of just the bare systems without AEMH molecules was carried out. random pure Au core-shell Figure A.7: Structure snapshot taken at the end of each simulation, showing that considerable structural changes have severely distorted the tetrahedral geometry of all three systems. Fig. A.7 provides a visual impression of the first important result of these simulations. It shows an exemplary structure snapshot taken at the end of the simulation. The geometry of all three systems is considerably different from the tetrahedral structure we started with. That this is not only an effect of temperature can be seen by comparison with the structures of the bare Au 20 and Au 10 Pt 10 systems that overall retain their tetrahedral shape. The structure of the bare system is even stable up to temperatures of 600 K, as mentioned in Sec. 3.2 and Ref. [94]. In fact, during the whole simulation strong structural changes take place in the combined NP-AEMH system, while in the bare systems mainly vibrations of the atoms about their equilibrium positions can be seen. To quantify these effects we calculated the average nearest neighbor bond length for the NP with and without AEMH monomers for every timestep. Fig. A.8 shows the nearest neighbor bond length of Au 20 (left), randomly mixed Au 10 Pt 10 (middle) and core-shell Au 10 Pt 10 (right). The nearest neighbor bond length was defined as the average distance of every atom to its four nearest neighbors. Note that the relaxed core-shell Au 10 Pt 10 was taken as the initial geometry for all NP. For Au 20 all Pt atoms in this NP were then replaced by Au atoms. For the randomly mixed cluster, the Pt atoms were redistributed over the cluster as depicted in Fig. A.6. This means, that one has to compare the nearest A.2. Influence of the nanoparticle support 73 Figure A.8: Average nearest neighbor bond length for every time step of the simulation for the Au 20 (left), the Au 10 Pt 10 randomly mixed (middle) and the Au 10 Pt 10 core-shell system with (top) and without (bottom) AEMH monomers. The yellow line, a moving window average, is a guide to the eye. 2.75 2.80 2.85 2.90 2.95 0 2000 4000 6000 8000 10000 12000 14000 16000 nearest neighbor distance [Å] Au20 2.75 2.80 2.85 2.90 2.95 0 2000 4000 6000 8000 10000 12000 14000 16000 Au20 with AEMH without AEMH simulation time [fs] 2.70 2.75 2.80 2.85 2.90 0 2000 4000 6000 8000 10000 12000 14000 16000 random 2.70 2.75 2.80 2.85 2.90 0 2000 4000 6000 8000 10000 12000 14000 16000 random 2.70 2.75 2.80 2.85 2.90 0 2000 4000 6000 8000 10000 12000 14000 16000 core-shell 2.70 2.75 2.80 2.85 2.90 0 2000 4000 6000 8000 10000 12000 14000 16000 core-shell neighbor bond length of the NP-AEMH system with the equilibrium bond length of the bare NP, after all transient processes have decayed. On average, the nearest neighbor bond length increases from 2.84 ˚ A to 2.85 ˚ A in Au 20 , from 2.78 ˚ A to 2.79 ˚ A in the randomly mixed Au 10 Pt 10 cluster and from 2.78 ˚ A to 2.80 ˚ A in the core-shell Au 10 Pt 10 cluster. More significant than this slight increase of the average bond length by less than 1% is that the nearest neighbor bond length fluctuates much more strongly in the NP-AEMH systems than in the bare clusters. This is consistent with the larger structural changes in the former. Additionally, we counted the number of Au-Au, Au-Pt and Pt-Pt bonds of the two Au 10 Pt 10 systems in each simulation step. A bond is here defined as every interatomic distance ≤ 2.9 ˚ A. For the bare NP these numbers are constant during the whole simulation. For the combined NP-AEMH systems one has to distinguish two time domains, similarly as for the nearest neighbor bond length. The transient time domain from 0 to ≈ 6000 fs and the steady state until the end of the simulation. Of course, transient effects are also present for the bare clusters, but they do not affect the number of Au-Au, Au-Pt and Pt-Pt bonds here. Tab. A.1 shows the average steady state number of bonds and the respective standard deviations for Au 10 Pt 10 with and without AEMH monomers. Again, we compare the average number of bonds of the combined NP-AEMH system with the number of bonds of the bare NP. For both Au 10 Pt 10 +AEMH systems the number of Au-Au bonds decreases as compared to their bare counterparts. For the randomly mixed cluster also the number of Au-Pt bonds decreases, while the number of Pt-Pt bonds increases. For the core-shell cluster the number of Au-Pt bonds slightly increases and the number of Pt-Pt bonds decreases. I want to stress that the overall number of Au-Au, Au-Pt and Pt-Pt bonds is too small to allow conclusive 74 Appendix A — Temperature and support effects without AEMH random core-shell Au-Au 4 ±1 8 ±1 Au-Pt 22 ±2 13 ±2 Pt-Pt 12 ±1 19 ±1 with AEMH random core-shell Au-Au 2 ±1 5 ±2 Au-Pt 20 ±2 14 ±2 Pt-Pt 14 ±1 16 ±1 Table A.1 : Number of Au-Au, Au-Pt and PtPt bonds ≤ 2.9 ˚ A of randomly mixed and coreshell clusters without and with two AEMH monomers. The averages for the bare system is taken over the complete simulation time. For the combined system the average was computed from 6000 to 16 000 fs. statements here. A first cautious interpretation of these results, however, indicates that the core-shell cluster partly looses its clear Pt-core/Au-shell characteristics, while the randomly mixed cluster undergoes structural changes making it more core-shell like. In other words, in the combined NP-AEMH system the differentiation between randomly mixed and core-shell clusters is not meaningful anymore. Firstly, because on average clusters will exhibit an intermediate mixing pattern in between core-shell and a random distribution of Au and Pt. Secondly, at realistic experimental conditions, i.e., at finite temperature and embedded in a supporting matrix, Au-Pt nanoalloys cannot be described in a static, monostructure fashion anymore [146]. Indeed, in our case at 300 K interaction with the AEMH monomers seems to be the dominant factor that induces structural fluxionality. Figure A.9: Total energy of randomly mixed and core-shell Au 10 Pt 10 in the MD simulation of the combined NP-AEMH system and the bare NP system. −93394 −93392 −93390 −93388 −93386 −93384 −93382 −93380 0 2000 4000 6000 8000 10000 12000 14000 16000 total energy [eV] simulation time [fs] random core-shell −69464 −69462 −69460 −69458 −69456 −69454 −69452 −69450 0 2000 4000 6000 8000 10000 12000 14000 16000 simulation time [fs] random core-shell with AEMH without AEMH These conclusions are further supported by considering the total energy of the two Au 10 Pt 10 systems with and without AEMH support as depicted in Fig. A.9. For the bare system the total energy of the core-shell cluster is consistently lower than that of the randomly mixed particles. This result is not surprising considering that structure and mixing pattern in these simulations remain largely undisturbed. The situation is completely different for the combined NP-AEMH system. Here the core-shell cluster-AEMH system is energetically more favorable at the beginning of the simulation. From ≈ 6000 fs on the total energies of the two systems are nearly identical. Finally, we will take a closer look at the electronic structure of our model system. To A.2. Influence of the nanoparticle support 75 this end, we took 11 structure snapshots of each system, being 194 fs apart from each other ( b= 100 MD steps) starting at 13 545 fs. The electronic structure of these systems was calculated using a TZVPP basis set. No structural relaxation was carried out, as we are interested in the electronic structure of each particular snapshot here. For each structure snapshot we calculated the total DOS and the metal DOS of the combined NP-AEMH system (in the following called metal-PDOS), which is obtained by projecting the total DOS onto the Au and Pt atoms. Fig. A.10 shows the total DOS of the bare Au 20 and Au 10 Pt 10 systems and the respective metal-PDOS of the combined NP-AEMH systems. The depicted DOS’s represent the arithmetic mean of the DOS’s of each of the 11 structure snapshots of each system. The metal-PDOS of the combined system is interesting, because we assume that possible reactants in a catalytic process are adsorbed to the NP and react at its surface. Thus, following the considerations of Sec. 3.4.1, the interaction strength between the catalysts and the adsorbates is governed by the metal-PDOS at the Fermi level. Figure A.10: Total DOS of bare NP (black) and DOS projected onto the metal atoms of the combined NP-AEMH system (yellow) averaged over 11 structure snapshots each, as explained in the text. 0 500 1000 1500 2000 2500 3000 −10 −8−6−4−2 0 density of states NP +AEMH PDOS bare NP DOS −10 −8−6−4−2 0−10 −8−6−4−2 0 pure Au random core-shell The left panel of Fig. A.10 again demonstrates the special electronic structure of the bare Au 20 tetrahedron [68]. It’s DOS exhibits discrete features and is high close to the Fermi level as compared to other Au clusters (see Sec. 3.4.1). In contrast, the DOS of the bare Au 10 Pt 10 clusters, is more continuous and band-like. Note that contrary to the visual impression the DOS at the Fermi level of Au 10 Pt 10 is significantly higher than that of Au 20 in accordance with the results of Sec. 3.4.1. Noticeably, Fig. A.10 shows that the metal-PDOS at the Fermi level is considerably reduced as compared to the bare NP case. This is most pronounced for Au 20 . Here, the metal-PDOS is more continuous than that of the bare NP. More importantly, the metal-PDOS at the Fermi level is reduced by 60% as compared to the total DOS of the bare NP. The same effect, though less distinct, is seen for Au 10 Pt 10 . The metal-PDOS at the Fermi level is reduced by 16% for the randomly mixed and by 30% for the core-shell cluster. The large deviations between the DOS reduction of the two Au 10 Pt 10 systems can be ascribed to the overall small number of structural snapshots for which the DOS was evaluated. A reduction of the DOS at the Fermi level is expected to lead to weaker interactions with possible adsorbates. Whether it results in an increased or decreased catalytic activity, however, depends on the specific reaction one is interested in. Quite generally the results of this section indicate that the interaction of 76 Appendix A — Temperature and support effects our 20-atom Au and Au-Pt NP with AEMH monomer units leads to a marked increase of the fluxionality of the clusters on the one hand. On the other hand the DOS at the Fermi level decreases in the combined NP-AEMH system. In terms of the volcano picture introduced in Sec. 3.4.4, this might correspond to a shift of the volcano peak as compared to the bare NP case. Our findings support the notion that liquid-phase catalysis cannot be modeled using the same concepts that have successfully been applied to gas-phase catalysis on surfaces [131]. Under realistic experimental conditions the description of supported Au-Pt nanoalloys as static entities being governed by their (vacuum and zero-temperature) ground state properties is likely to be flawed. Yet, even though the results presented in this section allow valuable first insights into the interaction between support molecules and Au-Pt NP further simulations have to be carried out to achieve conclusive results and to judge the significance of the findings presented above. First, simulations have to be done for larger NP, e.g., 40-atom Au-Pt nanoalloys, possibly with a less ”special” geometry than the tetrahedron, which has only surface and no volume atoms and is therefore expected to be particularly reactive. Second, it has to be clarified how significant the type of NP support in terms of the reported effects is. Typical supports that could be modeled with reasonable computational effort are, e.g., carbon [87] or silica [79] supports. B Optical properties of Au-Pt nanoalloys The use of finely dispersed Au particles in ruby glass and the colorful stained-glass windows of European’s medieval cathedrals shows that their special optical properties have been known for centuries. The first scientific examination of the subject is attributed to Michael Faraday, who studied the interaction of light with colloidal Au particles, as well as thin Au films and leaves as early as 1857 [186]. Today it is well understood that the beautiful color of suspensions of colloidal Au is a result of collective excitations of the conduction electrons that dominate the absorption spectrum, accumulating in a so called Mie resonance or surface plasma resonance 1 . The position of the Mie resonance depends on the size of the Au colloids, but the optical material functions that describe the linear response of the clusters to electromagnetic waves, are largely size-independent and similar to the bulk values. With these optical material functions one can determine the optical response of (rather large) NP using classical electrodynamics and the so called Mie theory [188]. For NP smaller than ≈ 10 nm, that are still too large to assess them with quantum chemical methods such as time-dependent DFT, the situation is more difficult. The optical response functions become size dependent and can deviate substantially from the bulk. A comprehensive account on optical properties of metal clusters is given in Ref. [187] and pertinent references therein. Here, the focus will be on spherical clusters of 10 nm diameter ( R = 5 nm) interacting with visible light of wavelength λ , i.e., Rλ . The optical response of clusters in this size range can be determined by virtue of Mie theory. For Rλ (quasi-static case) phase retardation effects and higher multipole orders can be neglected and the absorption cross section in the dipolar limit is σ(ω) = 4πω c√εm=[α(ω)],(B.1) 1 The term surface plasma resonance refers to the surface polarization being the main restorative force in clusters. This surface polarization is due to charges within the electronic screening length [187]. 78 Appendix B — Optical properties of Au-Pt nanoalloys where = [ α ( ω )] is the imaginary part of the polarizability of the cluster, which for a homogeneous sphere 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 dielectric function energy [eV] ε1 ε2 dielectric function Gold Platinum Figure B.1: Real and imaginary part of the dielectric function of bulk Au and Pt, taken from Ref. [189] and [190]. In Eq. (B.1) and (B.2) , ε is the (complex and ω -dependent) dielectric function of the cluster. We will here assume that ε is similar to its bulk value. The validity of this assumption will be discussed below. The real and imaginary part of the dielectric function ε = ε1 + iε2 describe polarization and energy dissipation of matter, respectively, and are related by the famous Kramers-Kronig relations. Fig. B.1 shows ε1 and ε2 for bulk Au and Pt as tabulated in Ref. [189] and [190]. Furthermore, Eq.(B.1) and B.2 contain the relative permittivity εm of the matrix in which the particle is embedded. For all following calculations εm = 2 . 7 was used, which corresponds to the relative permittivity of polystyrene, one of the major ingredients of the polyelectrolyte matrices used for Au-Pt nanoalloys stabilization in Ref. [80]. Finally, c is the vacuum speed of light. Fig. B.2 depicts the optical absorption spectra of a pure Au and a pure Pt NP of 10 nm diameter. The Mie resonance of the Au NP can clearly be seen at around 2.25 eV. The Pt NP exhibits no Mie resonance in the visible energy range. 0 200 400 600 800 1000 1200 1400 1600 1800 0 1 2 3 4 5 absorption cross section [nm2] energy [eV] pure Au pure Pt Figure B.2 : Optical absorption spectrum calculated using Mie theory in the dipolar limit for spherical metal spheres of 10 nm diameter. Contrary to Pt, Au NP show a pronounced Mie resonance. 79 At this point the question arises how the optical response of an Au-Pt nanoalloy will evolve as a function of the Au/Pt ratio and how the mixing pattern of Au and Pt influences the optical absorption spectrum of the NP. The optical properties of Au-Pt NP have been studied experimentally [191, 192] and for core-shell clusters also theoretically using Mie-theory as will be explained in the following [192]. Our approach follows a study of Ni-Ag clusters by Gaudry et al. [193]. Mie theory can be extended to the scattering of an electromagnetic wave by two concentric spheres [194]. In the dipolar limit one then obtains for the polarizability αcore-shell(ω) = (εshell −εm)(εcore + 2εshell) + fv(εcore −εshell)(εm+ 2εshell) (εshell + 2εm)(εcore + 2εshell)+2fv(εshell −εm)(εcore −εshell)εmR3,(B.3) where εcore and εshell are the dielectric functions of the core and the shell, respectively. Here, fv = Rc/R and Rc is the radius of the core, that one can estimate using the Au-Pt ratio (1 −x)/x and the atomic radii of Au and Pt for a AushellPtcore cluster Vcore =4 3πR3 c(B.4) Vshell =4 3π(R3−R3 c).(B.5) Assuming the cluster to consist of MAu Au and MPt Pt atoms MAu =Vshell VAu (B.6) MPt =Vcore VPt ,(B.7) where VAu,Pt =4 3πr3 Au,Pt is the volume occupied by a single Au or Pt atom, one obtains MAu MPt =1−x x=r3 Pt r3 Au R3−R3 c R3 c .(B.8) With this, the core radius is determined to be approximately Rc=R3 sxr3 Pt (1 −x)r3 Au +xr3 Pt .(B.9) For εcore and εshell we take the bulk dielectric functions for Pt and Au. This might not be justified, notably for the shell region and only small percentages of Au. However, for the case of Ni-Ag NP (R=1–2.5 nm) this approximation reproduced the experimentally observed Mie resonance and its blueshift with increasing Ni-content qualitatively well [193]. For a Au-Pt solid solution, the optical absorption can be determined using Eq. (B.1) and (B.2) with an averaged dielectric function [187] ε=εPtx0Au1−x0(ω) = x0εPt(ω) + (1 −x0)εAu(ω),(B.10) 86 Appendix C — Surface slab calculations with VASP The selective dynamics flag allows to determine which atomic positions are supposed to be kept fixed during a geometry optimization or a transition state search using the NEB method (see Sec. 2.5). Fixing, e.g., the bottom few layers of the surface slab at the bulk equilibrium lattice constant can reduce computational cost considerably and simultaneously improve the convergence performance. This, however, has to be tested carefully, as will be explained in the following section. Similarly, the number of atoms per surface layer depends on the specifics of the problem one is interested in. In the present work 4x4 atoms per layer were necessary to avoid interactions between adatoms in adjacent unit cells. C.3 Slab calculations Prior to any slab calculation one has to determine the number of k-points that yield accurate results at reasonable computational cost. In the direction normal to the surface one k-point is usually sufficient 1 . In the two other directions one can start with the bulk values and then test convergence of the free energy when decreasing the value along these two directions. For the 4x4x4 atoms surface slabs a 5x5x1 k-point mesh was used throughout all calculations. 620 625 630 635 640 645 650 655 660 2 3 4 5 6 7 8 9 surface energy/atom [meV] number of layers unrelaxed relaxed Figure C.3: Surface energy per atom of a 2x2 atoms per layer Pt(111) slab as a function of the number of layers. One then has to determine how many layers of bulk are necessary to get converged results for the surface energy σ=1 2(Eslab −MEbulk),(C.3) where Eslab is the free energy of the slab, Ebulk is the free energy per atom of the bulk system and M is the number of atoms used for construction of the surface slab. The factor 1 / 2 arises as a consequence of the surface slab having two surfaces. Fig. C.3 depicts σ per atom as a function of the number of layers (using 4 atoms per layer) of a Pt(111) surface slab. As 16 atoms per layer were used for the final calculations, a minimum number of 4 layers is chosen. Note, that even this seemingly small setup yields unit cells containing 64 atoms which in the case of a pure Pt slab, using the PAW method to describe the electron-ion interaction (see Sec. 2.4), corresponds to a total of 640 electrons. The computational cost can be reduced by fixing the bottom two layers at the equilibrium lattice parameter of the bulk using VASP’s selective dynamics flag. It was tested that reaction barriers change only insignificantly as compared to a full relaxation of the slab. 1 Using more than one k-point in this direction only improves the description of the interaction between adjacent unit cells that has to be minimized anyway by choosing an adequate amount of vacuum space in between neighboring slabs. C.3. Slab calculations 87 -164.5175 -164.5170 -164.5165 -164.5160 -164.5155 -164.5150 5 10 15 20 25 30 35 energy [eV] vacuum height [Å] free Figure C.4: Free energy of a 2x2x7 Pt(111) surface slab as a function of the vacuum between adjacent unit cells. Finally, the convergence with respect to the vacuum height in between neighboring slabs has to be tested. Fig. C.4 shows the free energy of a 2x2x7 layer Pt(111) slab as a function of vacuum height. For the final calculations a vacuum height of 20 ˚ A was chosen. The initial and final configuration between which the NEB algorithm is supposed to find a transition state, have to be relaxed prior to setting up the NEB calculation. The initial guess for the elastic band can be a linear interpolation between start and end configuration. If one is interested in the energy of the saddle point it is recommended to use the CI-NEB together with a minimum amount of elastic band images. For the simple diffusion step considered in case of the Au-Pt surfaces one single image was sufficient to determine the reaction barrier. Using three instead of one image did not change the height of the barrier significantly. However, if a truthful map of the MEP is required, more images might be necessary depending on the particulars of the system one is interested in. A typical input file used for the NEB calculations presented in Sec. 3.4 is SYSTEM = Pt/Pt(111) #insert system name ENCUT = 450 #plane wave cutoff ISTART = 0 #start calculation from scratch ISPIN = 2 #allow spin polarized solution PREC = accurate #controls several precision parameters #relax NSW = 100 #maximum of 100 ionic steps IBRION = 2 #conjugate gradient algorithm EDIFFG = -0.02 #relax tolerance SPRING = -5 #turns on NEB IMAGES = 1 #number of images LCLIMB = .TRUE. #turn on CI-NEB IALGO=48 #algorithm for electronic relaxation LREAL=auto #projection operators evaluated in real space LCHARG=.TRUE. #write electronic density LWAVE=.TRUE. #write wavefunctions #these can be turned off if memory is an issue List of abbreviations AEMH 2-amino-ethyl-methacrylate Au Gold (CI)-NEB (Climbing-image)-Nudged Elastic Band DFT Density Functional Theory GGA Generalized gradient approximation (G)KS (Generalized) Kohn-Sham HOMO Highest occupied (generalized) Kohn-Sham orbital LDA Local density approximation LUMO Lowest unoccupied (generalized) Kohn-Sham orbital MEP Minimum energy path Mn Manganese MD Molecular dynamics Ni Nickel NP Nanoparticle(s) OEP Optimized effective potential PAW Projector augmented waves PBE xc functional of Perdew, Burke and Ernzerhof Pd Palladium Pt Platinum RSH Range-separated hybrid SIC Self-interaction correction SIE Self-interaction error Si Silicon SVP Split valence basis set TZVPP Triple-ζbasis set VASP Vienna-Ab-initio-Simulation Package XAS X-ray absorption spectroscopy xc exchange-correlation XMCD X-ray magnetic circular dichroism XRD X-ray diffraction List of Publications 1. V. Zamudio-Bayer, L. Leppert , K. Hirsch, A. Langenberg, J. Rittmann, M. Kossick, R. Richter, A. Terasaki, T. M¨oller, B. v. Issendorff, S. K¨ummel, and J. T. Lau. Coordination-driven magnetic-to-nonmagnetic transition in manganese-doped silicon clusters. Phys. Rev. B, 88, 115425 (2013). 2. L. Leppert , R. Q. Albuquerque, A. S. Foster, and S. K¨ummel. The interplay of electronic structure and atomic mobility in nanoalloys of Au and Pt. J. Phys. Chem. C117, 17268 (2013). 3. L. Leppert , R. Q. Albuquerque, and S. K¨ummel. Gold-platinum alloys and Vegard’s law on the nanoscale. Phys. Rev. B, 86, 241403(R) (2012). 4. J. Kaiser, L. Leppert , H. Welz, F. Polzer, S. Wunder, N. Wanderka, M. Albrecht, T. Lunkenbein, J. Breu, S. K¨ummel, Y. Lu, and M. Ballauff. Catalytic activity of nanoalloys from gold and palladium. Phys. Chem. Chem. Phys. 14, 6487 (2012). 5. L. Leppert , S. K¨ummel. The Electronic Structure of Gold-Platinum Nanoparticles: Collecting Clues for Why They Are Special. J. Phys. 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Acknowledgment I thank the following people for their help and support with this thesis: Stephan K¨ ummel . I have it on good authority, that ≈ 8.23 out of 10 of my sentences during an average week in the last couple of years started with ”Stephan says that...”. I thank him for the most inspiring quantum mechanics lecture of all time (in my second year of undergraduate study), for convincing me to stay just a little bit longer in Bayreuth (in my third and fifth year) and for all the advice, discussions and patience during the last years that were of inestimable value to me. Rodrigo Albuquerque , who has been incredibly patient in performing yet another molecular dynamics run. I thank him for inspiring discussions about the bimetallic effect in Au-Pt nanoalloys, valuable insight into molecular dynamics simulations and his indestructible enthusiasm for our project. Adam Foster , for always lending a hand when it came to Nudged Elastic Band calculations. I enjoyed his and his group’s hospitality in Tampere, Finland, for six winterly weeks and learned a lot about VASP, transition state search and (not so) Finnish beer. Tobias Lau and Vicente Zamudio-Bayer , for countless physics discussions, the most amusing science gossip, a desk in Tobias’ office whenever I traveled to Berlin, for more support than I ever expected to receive and for letting me convince them that DFT can actually be a lot of fun. Andreas Karolewski has been my best friend for the last 8 years and the best office vis-`a-vis one could wish for. From physics to politics and especially ”beyond”; there is little that Andreas does not know about. He always impressed me with his great curiosity and stamina. Besides, he’s just pretty cool. Matthias Dauth , Tobias Schmidt and all the other members and alumni of the K¨ ummel group . I thank you for making these last years most exciting and instructive but first and foremost for making them an awful lot of fun. Furthermore, I thank Andreas, Matthias and Tobias for their thorough proof-reading of parts of this thesis. Markus Hilt , Bernhard Winkler and Monika Birkelbach for technical and administrative support whenever it was needed. My family, but especially Karena and J¨ org , the second physicist in the family, who aroused an early interest for physics in me. Konstantin Hirsch , for patient proof-reading of this thesis and for many helpful comments, advice and discussion. Most of all, I want to thank him for proving an easy and a difficult thing to me, that changed my life in countless ways: 1. Only a physicist can be a physicist. 2. All can be well. For financial and intellectual support I am grateful to the SFB 840 of the DFG, the Wilhelm und Else Heraeus Stiftung and the Elite Study Programme Macromolecular Science. Erkl¨arung Hiermit erkl¨are ich, dass ich die vorliegende Arbeit selbstst¨andig verfasst und keine anderen als die angegebenen Quellen und Hilfsmittel verwendet habe. Die Arbeit wurde weder in gleicher noch in ¨ahnlicher Form bei anderen Prüfungsbehörden zur Erlangung eines akademischen Grades vorgelegt. Ich erkl¨are, dass ich keine Hilfe von gewerblichen Promotionsberatern bzw. -vermittlern oder ¨ahnlichen Dienstleistern in Anspruch genommen habe und auch nicht beabsichtige diese zuk¨unftig in Anspruch zu nehmen. Weiterhin erkl¨are ich, dass ich bisher keinen anderweitigen Promotionsversuch unternommen habe. Bayreuth, den 29. Mai 2013 Linn Leppert