國立中山大學物理研究所 碩士論文. 運用第一原理研究矽原子團 (Sin, n=1-16) 參雜金原子 (Au) 之原子與電子結構. Atomic and electronic structures of AuSin(n=1-16) clusters. from first-principles

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1 國立中山大學物理研究所 碩士論文 運用第一原理研究矽原子團 (Sin, n=1-16) 參雜金原子 (Au) 之原子與電子結構 Atomic and electronic structures of AuSin(n=1-16) clusters from first-principles 研究生 : 許志強撰 指導教授 : 莊豐權 中華民國九十八年一月

2 Acknowledgment 首先, 當然要先感謝我的指導教授莊豐權, 他的指導與支持是能完成這篇論文的主要原因 同時也要感謝口試委員郭萬銓教授 林德鴻教授和郭建成教授所提出的珍貴想法與建議 而對於電腦室的同學, 謝昀頤 許嘉修 黃志權 林文煥 羅迪 林煌鈞, 感激他們在實驗上的幫忙 理論上的討論 技術上的探討和生活上的相處, 這些對於不管是這篇論文或是我的研究生活都有很大的助益 另外, 其他實驗室的同學雖然沒有對於論文有直接的影響, 但在我碩士生涯中, 也在生活 課業上給予我幫助, 所以也同樣感謝他們, 何金潤 劉丞勛 黃治融 吳俊逸 林員梃 蔡銘宮 張晉健 林瑋歆 杜毅洲 等中山物理的同學們 也感謝中山物理的各個份子, 是你們造就了這個環境, 讓我可以在其中完成學業 最後, 要感謝我的父母, 畢竟沒有他們就不會有我, 而在這段期間也謝謝他們各方面的支持與理解 This work was supported by the Taiwan National Center for Theoretical Science and the Taiwan National Science Council under Grant No. NSC M MY3. We are grateful to the Taiwan National Center for High performance Computing for computer time and facilities. i

3 摘要 利用第一原理整體性探討 AuSi n (n=1-16) 原子團之結構 而在 n=1-16 中, 同樣大小的原子團裡, 其最低能量結構表現出一種趨勢 -- 傾向把金原子擺在結構外圍勝於用矽原子來包覆 我們的研究表示在 AuSi n 原子團 n=5 與 10 時, 是相對其他尺度更為穩定的結構 在 n= 尺度下, 我們發現即使參雜了金原子, 矽原子團仍然維持了穩定時的結構 此外, 也詳細地分析了裂解能 (Fragmentation energy) 研究更進一步指出對於 n>7 的尺度, 金原子的參雜會縮小最高填滿與最低未填滿分子軌道間的能量差距 再者, 我們在 CuSi n (n=1-16) 與 AgSi n (n= ) 呈現了類似的分析, 也將其與 AuSi n 原子團之結果做一個比較 接著, 在特定尺度 (n=10-16) 的 CuSi n 原子團中, 選定一些較低能量的同分異構物, 讓其在 Gaussian03 package 下做進一步的最佳化 我們發現對於 CuSi n (n=12-16), 包覆金屬粒子的結構比金屬粒子依附在外的結構擁有更低的能量, 而此種趨勢也跟 Janssens et al. Phys. Rev. Lett. 99, (2007) 最近的研究相符 ii

4 Abstract The structures of AuSi n (n = 1-16) clusters are investigated systematically using first-principles calculations. The lowest energy isomers exhibit preference toward exohedral rather than endohedral structure. Our studies suggest that AuSi n clusters with n = 5 and 10 are relatively stable isomers. We found no significant alteration in the cluster s inner core structure for sizes n= 6, 7, 10, 11, 12, 14, and 15 even in the presence of doping. Moreover, analysis of fragmentation energies is presented in detail. Our studies further indicate that doping of Au atom significantly decreases the gaps between the highest occupied molecular orbital and the lowest unoccupied molecular orbital for n > 7. Additionally, we report on similar results obtained for CuSi n (n = 1-16) and AgSi n (n = 14, 15, and 16) and compared them with those on AuSi n clusters. Next, the low energy isomers for certain sizes of CuSi n (n = ) clusters are selected for further optimizations using Gaussian 03 package. We found that for CuSi n (n = ), the endohedral isomers have lower energies than their exohedral counterparts, consistent with a recent study by Janssens et al. [15] in which a similar trend was observed. iii

5 Contents ACKNOWLEDGMENT ABSTRACT LIST OF FIGURES LIST OF TABLES i iii vi vii 1 Introduction 1 2 Theory Density functional theory (DFT) Thomas-Fermi model The Hohenberg-Kohn theorem The Kohn-Sham equation with local spin density approximation (LSDA) and generalized gradient approximation (GGA) The pseudopotential method Norm-conserving pseudopotential Efficient form for the model pseudopotentials Projector augmented waves (PAW) Geometry optimization Hellmann-Feynman theorem Steepest descent method and conjugate gradient method Simulated annealing Generating initial structure by cluster growth method Details of Computational Packages Vienna Ab initio Simulation Package Gaussian 03 package iv

6 CONTENTS v 3 Results and discussions Structures of AuSi n ( n = 1 to 16) Structures of AuSi n ( n = 1 to 5) Structures of AuSi n ( n = 6 to 8) Structures of AuSi n ( n = 9, 10) Structures of AuSi n ( n = 11) Structures of AuSi n ( n = 12, 13) Structures of AuSi n ( n = 14) Structures of AuSi n ( n = 15) Structures of AuSi n ( n = 16) Exohedral versus endohedral structures Relative Stability of AuSi n Embedding energy Electronic properties of MSi n (M = Au, Ag, and Cu) HOMO and LUMO Charge transfer Conclusion 41

7 List of Figures 2.1 The self-consistent flow chart for solving the Kohn-Sham equation AuSi 1 AuSi AuSi 6 AuSi AuSi 9 AuSi AuSi AuSi 12 AuSi AuSi AuSi AuSi Relative energy of the lowest endohedral isomer with respect to the lowest exohedral isomer versus the size of cluster Binding energy per atom and second difference in binding energy versus the size of cluster Fragmentation energy of Au, Ag, and Cu versus the size of cluster The embedding energy of the listed endohedral isomers in Table 3.1 for MSi n (M= Cu, Ag, and Au; n= 10-16) are plotted as a function of the number of Si atoms in the clusters HOMO-LUMO gap versus the number of silicon atoms in a clusters The HOMOs and LUMOs of AuSi 10, AuSi 14, AuSi 15, and CuSi Plot of charge transfer for AuSi 10, AuSi 14, AuSi 15, and CuSi vi

8 List of Tables 3.1 Table of the lowest energy isomer of various metals, sizes and structure types (endohedral or exohedral) vii

9 Chapter 1 Introduction Metal-doped silicon clusters or cages have been the focus of extensive theoretical and experimental investigations [1, 2, 3]. However, unlike bulk materials, complete understanding of the properties of these clusters has yet to be reached, despite significant headway gained in this field over the years. Among theoretical material scientists, the clusters novel properties in relation to the bulk present an opportunity to develop and refine existing theories on materials. For materials engineers, the challenge has been to explore potential applications in the microelectronics industry that take advantage of the clusters unique behavior. Thus, numerous theoretical studies [3, 4, 5] have been devoted to the determination of equilibrium geometries, electronic and bonding structures, as well as structural transitions of different sizes of both pure silicon and metal-doped silicon clusters. Compared with other metal-doped silicon clusters [2, 3, 4, 5], studies on Group I-B (Cu, Ag and Au) metals are mostly limited to CuSi n clusters [1, 6, 7, 8, 9, 10, 11]. Following Beck s [1] observation of CuSi n clusters of various sizes particularly n=10, interests in copper-silicon clusters as evidenced by a number of theoretical studies [7, 8, 9, 10] has remained high and sustained through the years. Recently, in a new attempt to fully explore group IB metal doped silicon clusters, Duncan and his coworkers performed [12, 13] studies on the MSi n (M= Cu, Ag, and Au) clusters. Prior to this work, only copper-silicon clusters had been produced, and production of silver and gold had so far been missing. In their report, Duncan s group explored and highlighted some of the most desirable properties of group IB metal-silicon clusters [12, 13]. Their studies confirmed the similarities in the mass spectra of these systems [12]. Furthermore, they indicated that for copper-silicon and silver-cluster [13], MSi n (n = 7 and 10) are the most abundant products. Photodissociation data, on the 1

10 CHAPTER 1. INTRODUCTION 2 other hand, revealed that the most abundant fragment ions are Si + 7 and Si+ 10 for AgSi+ 7 and AgSi + 10, respectively. For CuSi n the primary fragments detected are the Si + 7 from CuSi + 7, and Si+ 6, CuSi+ 6, and Si+ 10 from CuSi+ 10. Together, these findings suggest that the clusters structure should have an external metal and further reinforce the idea that both Ag-Si and Cu-Si bondings are weaker on average than Si-Si bonding. However, despite the obvious similarities in the mass spectral data among the three metals, differences exist in the photodissociation data in the form of available fragmentation channels for n = 10. Partly in response to the new findings by Jaeger et. al. [13], we have performed a first-principles study on AgSi n ( n = 1-13) and theoretically confirmed that the Ag atom is indeed exohedral [14]. In a related experimental paper, Janssen and co-workers reported that the physisorption of Ar is very sensitive to the position of the transition metal (TM) atom(s) on TM doped silicon clusters, and thus could serve as an indication of whether the TM atom(s) is exohedral or endohedral [15]. They found that for CuSi n clusters with n 12 the argon-complex formation is unlikely, thus suggesting a caged structure. Next, in analogy with AgSi n and CuSi n clusters, the most abundant gold-silicon product turns out to be AuSi 7, with AuSi 10, AuSi 11, and AuSi 12 somewhat less so [12], a phenomenon not replicated in the AgSi n and CuSi n cluster spectral data. However, aside from the published result in Ref. [12], no additional photodissociation data on AuSi n clusters has been reported to date. This seeming lack of data notwithstanding, it should be pointed out that the caged structure for AuSi n clusters have been included in numerous comparative studies in relation to other transition metals [16, 19, 17, 18]. Nevertheless, whether Au atom is exohedral or endohedral remains an open question. Thus, we believe an extensive and definitive study on AuSi n clusters which aims to address unresolved issues, as well as one that highlights the similarities and differences among the group IB metals, is still relevant and useful. In this thesis, we first focus on AuSi n clusters and later extend the work by performing comparative studies between AuSi n and the other Group IB metal-silicon clusters with the overall goal of making general observations regarding this family of metals. This thesis is organized as follows: In Chapter 2, we describe the basic theory and our computational methods. In Chapter 3, we present the bulk of our work which consists mostly of our efforts to determine the optimized structures for each metal cluster size, along with an analysis of the stability of these clusters. We end this article with Chapter 4 highlighting the key findings and insights derived from this study.

11 Chapter 2 Theory 2.1 Density functional theory (DFT) Density functional theory is a quantum mechanical theory used to investigate the ground-state electronic structure of many-body systems. Nowadays, DFT is popularly adopted in condensed matter physics and computational physics. The aim of density functional theory is to replace the complicated N-electron wavefunction by the much simpler electron density as the basic quantity Thomas-Fermi model The first step of this theory was the Thomas-Fermi model, developed by Thomas and Fermi in 1920s. They expressed a total electronic kinetic energy as a functional of the electron density in a single-atom system. T T F [n] = C F n 5/3 (r)d r (2.1) Hence, they were able to calculate the energy of an atom with this functional. Although this is the start of a remarkable idea, the Thomas-Fermi theory is lack of the accuracy for most applications. The error mainly comes from the representation of the kinetic energy, the exchange energy term and the neglect of electron correlation. Therefore, the Thomas-Fermi model isn t a theoretical footing of density functional theory, and DFT has the fundamental theorems until the paper published by Hohenberg and Kohn (1964). 3

12 CHAPTER 2. THEORY The Hohenberg-Kohn theorem In quantum mechanics, all information we can possibly observe are stored in a wavefunction. The outcome of a physical measurement is contained in the probability amplitude that belongs to a complete set of observables for the system. Due to nature has the tendency to decrease its own energy, the question people most want to figure out is the formulation of the ground-state wavefunction. Hohenberg and Kohn [21] first show a method to calculate the ground-state energy through the electronic density. There are two important points in this theorem. One is the groundstate electron density and the ground-state wavefunction of a many-particle system exist a relation that is a one-to-one mapping. The other is that choosing the groundstate density n(r) as the variable, the minimization of total system energy is the ground state energy. The time-independent Hamiltonian in an N-electron atomic or molecular system within the Born-Oppenheimer approximation is H = T + V ne + V e e, (2.2) where T = V ne = N ( 1 2 2) (2.3) i i =1 V e e = N ( Z α ) (2.4) r i α i =1 α N ( 1 ) (2.5) r i j i <j T is the total kinetic energy operator, V ne is the external potential operator including the electron-nucleus attraction energy, and V e e is the total Coulomb repulsion energy operator. We can clearly know that n(r) is a functional of an external potential v (r), and Hohenberg and Kohn demonstrate that the mapping from v (r) to n(r) is reversible. We first assume that there are two different external potentials, v (r) and v (r). They correspond to different non-degenerate ground eigenstates, Ψ and Ψ, but have the same ground-state density n(r). E 0 and E are the ground state energies 0 with Hamiltonian H and H, respectively. Hence, we have E 0 = Ψ H Ψ < Ψ H Ψ = Ψ H Ψ + Ψ H H Ψ = E 0 + n(r)[v (r) v (r)]d r. (2.6)

13 CHAPTER 2. THEORY 5 On the other hand, taking Ψ as a trial function that goes with the Hamiltonian H E = 0 Ψ H Ψ < Ψ H Ψ = Ψ H Ψ + Ψ H H Ψ = E 0 + n(r)[v (r) v (r)]d r. (2.7) Adding Eq. (2.6) and Eq. (2.7), we obtain E 0 + E < E + E (2.8) This inconsistent result proves that the assumption at first is wrong, so one n(r) corresponds to only one v (r). Therefore, we can say that v (r) is a unique functional of the ground state electron probability density. Hence, the charge density can substitute for the wavefunction to determine all the properties of a system s ground state. Note that the argument above applies only if the ground state is non-degenerate The Kohn-Sham equation with local spin density approximation (LSDA) and generalized gradient approximation (GGA) Hohenberg and Kohn showed that the total energy E can be expressed in terms of the electron charge density n(r). That is E [n(r)] = n(r)v (r)d r + F [n(r)] (2.9) where F [n(r)] is a universal functional. Hence, one can calculate the energy by the density, not through the wavefunction. Kohn and Sham [21] further simplified the many-body problem of interacting electrons, and suggested a successful way in numerical calculations through replacement of the many-body Schrödinger equation by a single-particle one within an effective local potential. In Eq. (2.9), F [n(r)] can be decomposed into F [n(r)] = T s [n(r)] + 1 n(r)n(r ) d rd r + E 2 r r xc [n(r)], (2.10) The first term T s [n(r)] is the non-interacting kinetic energy at a given electron density n(r). The second term is the electron-electron Coulomb repulsion energy, and the last term E xc [n(r)] is the exchange-correlation energy functional. Here, the manybody interactions are included in the exchange-correlation energy term. Kohn and

14 CHAPTER 2. THEORY 6 Sham assumed that E xc [n(r)] can be approximated as a function of local density (LDA). That is E xc [n] = n(r)ε xc (n(r))d r, (2.11) which follows from the stationary property, and is subject to the constraint n(r)d r = N. By the variational principle, one can write δ E [n(r)] µ[ n(r)d r N ] = 0, (2.12) where µ, acting as a lagrange multiplier, is the chemical potential. Also, using the following µ = v (r) + δf [n] δn(r) = v e f f (r) + δt s [n] δn(r) (2.13) and considering Eq. (2.10), we know v e f f (r) = v (r) + n(r ) r r d r + v xc (2.14) in which v xc = δe xc [n] δn(r) is the exchange-correlation potential. Therefore, for a given effective potential v e f f, we are able to obtain n(r) which minimizes the total energy simply by solving the single-electron Schrodinger equation: ħh 2m 2 + v e f f (r) Ψ i (r) = ε i Ψ i (r). (2.15) Eq. (2.15) is a Schrodinger-like equation called Kohn-Sham equation. And the charge density can be expressed as occ n(r) = 2 Ψ i (r) 2. (2.16) i =1 The process, from Eq.(2.14) to Eq. (2.16), has to be solved in a self-consistent way. The schematic diagram is shown in Fig Beginning with an initial trial density n(r), we can construct v xc and v e f f (r) from Eq. (2.14), and obtain a new Ψ(r) by solving Eq. (2.15). With Eq. (2.16), one can calculate both a new density and a new total energy. This procedure is repeated to obtain energies of two consecutive iterations. The iterative method will terminate when the energy difference of two consecutive iterations is less than a threshold energy. When concerning a spin-polarized system, the density and wavefunction in DFT must be modified into a spin-dependent form. Kohn-Sham equation and other key

15 CHAPTER 2. THEORY 7 equations become to be occ n σ (r) = Ψ i σ (r) 2, (2.17) i =1 ħh 2m 2 + v e f f (r,σ) Ψ i σ (r) = ε i σ Ψ i σ (r), (2.18) and form v e f f (r,σ) = v (r) + n(r,σ) r r d r + v xc,σ, (2.19) v xc,σ = δe xc,σ[n(r,σ)]. (2.20) δn(r,σ) In general, one can write down the LSDA exchange-correlation energy in this E LSDA xc [n σ ] = n(r,σ)ε xc (n(r,σ))d r. (2.21) However, the local density approximation (LDA) will fail in systems where charge densities rapidly oscillate, like molecular systems. An improvement to this problem can be made by considering the gradient of the electron density in a scheme called generalized gradient approximation (GGA). It can be expressed as E GG A xc = E GG A xc [n σ, n σ ], (2.22) and work in both LSDA and LDA.

16 CHAPTER 2. THEORY 8 Figure 2.1: The self-consistent flow chart for solving the Kohn-Sham equation.

17 CHAPTER 2. THEORY The pseudopotential method Norm-conserving pseudopotential For numerical calculations, the worst situation is that we can t accomplish our job and obtain the result within a finite time. Hence, how to save computing time is a crucial question, but it s not easy to balance the importance of both time and accuracy. For this, one should drop the part that can be neglected, and use approximate models for those systems to reduce calculating time. One way to this problem is done by means of a pseudopotential method in which the real potential is modified to a fictitious one. Pseudopotential methods are based on following two characteristics : 1. Valence electrons play the dominant role and are unable to neglect in a chemical reaction. By contrast, core electrons are less important than valence electrons for chemical bonding. 2. The core states have larger kinetic energy. In other words, the wave function near the nucleus fluctuates greatly, and this causes that the expansion using the complete set will converge slowly. Therefore, Pseudopotentials eliminate the need for considering atomic core states. In the early days, one constructs a pseudopotential and chooses the value of parameters by fitting with the experimental data. This methodology is known as the empirical pseudopotential method (EPM). Nevertheless, eigenfunctions that are generated by this method are crude. Subsequently, researchers determined pseudopotentials in a new way without using empirical parameters, called the "first principles" pseudopotentials. This method no more needs experiments to complete a pseudopotential. D. R. Hamann [37] derived a norm-conserving pseudopotential. This potential is built to have the following properties: 1. Pseudo valence eigenvalues are identical with the real ones. 2. Pseudo and real wavefunctions agree outside a chosen core cutoff radius. 3. The sum of the pseudo charges inside the cutoff radius is the same as that in the real system. (norm conservation) 4. The logarithmic and first energy derivatives of the pseudo systems are identical with the real systems outside the cutoff radius.

18 CHAPTER 2. THEORY 10 Property (3) guarantees that the pseudo and real valence charges experience the same electrostatic potential. Property (4) ensures that the scattering properties of the real nuclei are reproduced with the minimum energy error in pseudo systems. These two properties are important for better transferability of pseudopotentials. Outside the core radius, the norm conserving pseudopotential(ncpp) has not only the same eigenvalue but also the same amplitude of wavefunctions. On the other hand, the waves inside the core radius are modified smoothly, and thus fewer waves are required for basis set expansion. In order to construct a norm-conserving pseudopotential, we first determine an intermediate pseudopotential V ps 1l (r ), which can be written as V ps 1l (r ) = [1 f ( r )]V (r ) + c l f ( r ), (2.23) r cl r cl where r cl is the core cutoff radius. Pseudopotentials with larger r cl result in a smoother wave, that is convergent more rapidly, but lose more accuracy instead. A small r cl means that psudopotentials are more transferable and accurate to reproduce realistic features. f ( r r cl ) is a parameter function that can be obtained by fitting the pseudo eigenvalue with the real one, and D. R. Hamann set it as exp( ( r r cl ) 4 ). The intermediate pseudo wave w 1l, which is the solution of the Schrödinger equation with Eq. (2.23), is also identical with the real wave u l (r ) outside the core radius within a multiplicative constant γ l. It means that for r > r cl γ l w 1l (r ) u l (r ). (2.24) Considering the situation inside the core radius, one should modify the intermediate pseudo wave function w 1l (r ) to smear the wave and transform it into w 2l (r ). That is w 2l (r ) = γ l [w 1l (r ) + δ l g l ( r r cl )], (2.25) where g l ( r r cl ) behaves as ( r r cl ) l +1 inside the core radius and drop to 0 when r > r c. D. R. Hamann set g l (x)=x l +1 exp( x 4 ), and δ l can be obtained from the normalized constraint, γ 2 l 0 [w 1l (r ) + δ l g l ( r r cl )] 2 d r = 1. (2.26) We are able to derive the atomic pseudopotential V 2l (r ) and the valence pseudo charge density from the wave functions w 2l (r ). The atomic pseudopotential is composed of

19 CHAPTER 2. THEORY 11 two parts, the ionic part V ps 2l (r ) and the electronic part. That is V 2l = V ps 2l + V e e, (2.27) where V e e is the Coulomb and exchange-correlation potentials due to this charge density, and V ps 2l is the pseudopotential that we want to construct Efficient form for the model pseudopotentials The norm-conserving pseudopotential, which is semilocalized (nonlocal in angular coordinates and local in radial coordinate), provides us a more accurate method to transform a real system into a pseudo one. However, compared with previous nonlocal pseudopotentials, the norm-conserving pseudopotential is extremely expensive in terms of numerical calculations, and that means it needs more computer resources to complete a work. Fortunately, Kleinman [38] accelerated the computing process by reducing the number of integrals within the pseudopotential. Hamann s pseudopotential can be written as V HSC M = V v a l (r) + l m Y l m V HSC l (r) Y l m (2.28) where V HSC M (r) -Z/r for r> r c, and V v a l (r) is the Coulomb and exchange potential due to the valence electrons. Next, Kleinman modified Eq. (2.28) to the following form: where and V M = V v a l (r) + Y l m V SO (r)[l S + V ion r] Y l l l m, (2.29) l m V SO (r) = 2 HSC [V l 2l + 1 l V ion r = 1 HSC [(l + 1)V l 2l + 1 l V HSC ], (2.30) l l V HSC ]. (2.31) l 1 2 The integral of Y l m V (r ) Y l m between two plane waves basis is proportional to j l (k r )V (r )j l (k r )r 2 d r P l (cos θ k k ), (2.32) in which j l is the spherical Bessel function, P l (cos θ k k ) is the Legendre polynomial and θ k k is the angle between wave vectors k and k. For n different wave vectors and

20 CHAPTER 2. THEORY 12 m fractions in the Brillouin zone, there are mn(n + 1)/2 points we should evaluate for each l. For an arbitrary function V L (r ), one can obtain an expression equivalent to Eq. (2.29) and its form is V (j =l ± 1 2 ) M = V v a l (r) + V L (r) + V (j =l ± 1 2 ) SL, (2.33) where and V (j =l ± 1 2 ) SL = l m Y l m δv l ± 1 (r) Y l m (2.34) 2 δv l ± 1 (r) = V HSC (r) V 2 l ± 1 L (r). (2.35) 2 For reducing the computing time, Kleinman replaced the semilocal potential V (j =l ± 1 2 ) SL by a nonlocalized one V (j =l ± 1 2 ) N L, which can be written as V (j =l ± 1 2 ) N L (r) = l m δv l ± 1 (Φ 0 ) 2 l ± 1 2,m (Φ0 ) l ± 1 2,m δv l ± 1 2 (Φ 0 ) l ± 1 2,m δv l ± 1 (Φ 0 ) 2 l ± 1 2,m, (2.36) where (Φ 0 1 (j =l ± 2 ) l ± 1,m is an eigenfunction of V ) 2 SL (r) and can be expressed as (Φ 0 ) l ± 1 2,m (r) = (ϕ0 ) l ± 1 2,m (r )Y l,m (θ,ϕ). (2.37) We note that V (j =l ± 1 2 ) N L (Φ 0 ) l ± 1 2,m = δv l ± 1 (Φ 0 ) 2 l ± 1,m (2.38) 2 and V (j =l ± 1 2 ) SL (Φ 0 ) l ± 1 2,m = Y l m δv l ± 1 (ϕ 0 ) 2 l ± 1 2,m = δv l ± 1 (Φ 0 ) 2 l ± 1,m. (2.39) 2 Comparing these two equations, Eqs. (2.38) and (2.39), we have V (j =l ± 1 2 ) N L (Φ 0 ) l ± 1 2,m = V (j =l ± 1 2 ) SL (Φ 0 ) l ± 1,m. (2.40) 2 Eq. (2.40) shows that V (j =l ± 1 2 ) N L and V (j =l ± 1 2 ) SL has the same eigenfunction. Moreover, the integral of ψ c ψ c in Eq. (2.36) is proportional to j l (k r )ψ c (r )r 2 d r j l (k r )ψ c (r )r 2 d r P l (cos θ k k ), (2.41) where ψ c (r ) = δv l ± 1 (Φ 0 ) 2 l ± 1,m. With the substitution, there are only mn points needed 2

21 CHAPTER 2. THEORY 13 to be integrated for n different wave vectors and m points in the Brillouin zone. Kleinman in effect decreases the number of grid points from mn(n + 1)/2 to mn, and through this way one can save the calculating time Projector augmented waves (PAW) Just like other pseudopotentials methods, projector augmented wave method (PAW) [36] also transforms the real system into a pseudo one. The difference between PAW and NCPP methods is that PAW does not change the form of the real potential into the pseudopotential in the first step. Instead, it begins from transforming the wave into a pseudowave. That is, Ψ = T Ψ, (2.42) where Ψ is the all-electron wavefunction and Ψ is the pseudowave function. T, on the other hand, is the linear transformation from the pseudowave function to the all-electron wave functions. It can be divided into T = 1 + T, (2.43) R where T R acts only within the core region in order to rebuild the core wave. One can expand the pseudowave function with pseudo partial waves, φ i c i. That is R Ψ = φ c i i, (2.44) i where c i s are the coefficients. Next, one can insert Eq. (2.44) into Eq. (2.42), and since φ i = T φ i, the result become this form Ψ = T Ψ = φ i c i. (2.45) The previous two transformations hold within the core region. Then, the all-electron wave function can be obtained from the pseudo wave function by i Ψ = Ψ φ c i i + φ i c i. (2.46) i This equation means that all-electron wave = pseudowave "pseudowave within core" + "all-electron wave within core", and it holds only if there is no interaction between cores. Any operator A in the all-electron wave can be transformed into one i

22 CHAPTER 2. THEORY 14 operator A in the pseudowave via A = T + AT. (2.47) 2.3 Geometry optimization Hellmann-Feynman theorem To figure out the force acting on ions is necessary for geometry optimizations. An effective approximation for the eigenstates is done via the Hellmann-Feynman Method [40]. Since the ground state is one of the eigenstates, we can use this method in relaxing. Supposing that an ion located at {R I } is affected by a force F, we have F I = I E 0 (R) = R I Ψ 0 (R) H(R) Ψ 0 (R) = I Ψ 0 (R) H Ψ 0 + Ψ 0 (R) I H Ψ 0 + Ψ 0 (R) H I Ψ 0 = I Ψ 0 (R) E 0 Ψ 0 + Ψ 0 (R) I H Ψ 0 + E 0 Ψ 0 I Ψ 0 (R) = I E 0 Ψ 0 Ψ 0 + Ψ 0 I H Ψ 0 (2.48) = Ψ 0 I H Ψ 0. (2.49) This method holds if the wave function is an eigenstate of the Hamiltonian. The process for relaxing one structure into a local energy minimum is as follows: 1. Calculate the ground-state wave using the Kohn-Sham self-consistent method. 2. The force acting on an ion can be obtained via the Hellmann-Feynman method. 3. Move the atom by the steepest descent method or conjugate-gradient method. 4. Repeat steps 1 3 until the force is less than a threshold value Steepest descent method and conjugate gradient method Steepest descent [40] is a method wherein ions move a short distance in the direction of a force, on the other hand, the direction of the potential decreasing most rapidly. In many cases, the result of the steepest descent method is limited to the assignment of the initial structure. Hence, the result will tend to obtain the energy local minimum near the initial structure.

23 CHAPTER 2. THEORY 15 Another way to move an ion to a local minimum or the global minimum is the conjugate gradient (CG) [40] method. This method involves determination of a complete basis set of space that we want to describe, and finding out the minimum (or local minimum) along every basis in turn until the minimum (or local minimum) is reached. The CG method usually could reach the local minimum faster than the steepest descent method and skip the metastable structures. The directions of motion are different between the conjugate gradient and steepest descent methods. In steepest descent method, the direction of motion is along to the calculated forces acting on the ion. In CG scheme, one should locate the minimum along the given direction-called line minimization. Then, locate the next minimum along the direction conjugated to the proceeding direction Simulated annealing A physical system tends to decrease its total energy by nature. Hence, the ion configurations of a system with a lower energy are more interesting for us than a higher one. The initial prediction of structures is the first thing in obtaining physical information, but is difficult to be done since the fact that there are extremely large possible configurations. Simulating the annealing process in the real system the so-called "simulated annealing" [39] is one of the ways of solving this problem. In first principles calculations, simulated annealing method isn t usually effective in the structure optimization and is further disadvantaged by the fact that global optimization problem is limited by a narrow energy surface landscape Generating initial structure by cluster growth method It s easy to generate initial candidate structures through randomly placing the desired number of atoms in a cubic box for further optimizations. However, this socalled random search method is unskillful due to the unreasonable bond length and many loose structures that are initially mass-produced. In order to solve these problems, the cluster growth method whereby an atom is adsorbed on any outer surface of the cluster is introduced: 1. Generating the first "surface": To construct the first surface, we place 3 ions with a reasonable distance between each of them( bond length). 2. Recognizing every possible "surface" on this cluster.

24 CHAPTER 2. THEORY Adsorbing an ion on any "outer" surface. 4. Repeating steps 2 3 until the total number of ions is reached. Here, "surface" is defined as a triangle with 3 ions on its corner and with a reasonable distance on each side( bond length) just like what constructed in step 1. On the other hand, "outer surface" is defined as the surface with enough room for absorbing an ion. The cluster growth method guaranties that no additional dangling bond was produced during the adsorption and is more efficient than the common random search. 2.4 Details of Computational Packages Vienna Ab initio Simulation Package The calculations for both metal-silicon clusters and pure silicon clusters were done within the generalized gradient approximation (GGA) as parameterized by Perdew, Burke, and Ernzerhof (PBE) [20] to spin polarized density functional theory (DFT) [21] using projector-augmented-wave potentials (PAW) [22], as implemented in Vienna Ab initio Simulation Package (VASP) [23]. The kinetic energy cutoff is set to e V (18.03 Ry ), e V (18.36 Ry ), and e V (20.08 Ry ), for MSi n (M = Au, Ag,and Cu), respectively. The structural optimization was done with the conjugate gradient algorithm and with symmetry until the forces on the atoms were less than ev/å. The length of the supercell was set to 15 Å. Every cluster is rotated such that the vector defined by the longest pair distance between any two atoms within the cluster lies along the diagonal direction of the simulation box. In order to optimize certain proposed models, a quasi-newtonian algorithm was used to relax systems to their local minima. Next, initial test runs to ensure the validity of our methodology [14] were done by verifying that the results on pure silicon clusters Si n ( n = 2-16) are in agreement with previous studies by Ho et al. [24], Liu et al. [25], and Lu et al. [26]. To classify the Au atom s structure (in relation to the Si cluster) as either endohedral or exohedral, we generated numerous initial structures for each Au-doped silicon cluster size, then allow each of these structures to reach their optimized geometry via a series of structural optimizations using first-principles total energy calculations. For this purpose, proposed candidates from the literature are taken as the

25 CHAPTER 2. THEORY 17 natural starting point [16, 30, 31, 28, 32, 17, 29, 27, 2, 14, 8] for the optimization procedures that follow. Two other techniques are used to supplement these candidate structures. In one such technique, we replace one Si atom in a pure Si n+1 with an Au atom, resulting in a new structure for AuSi n. In the other, we cap a pure Si n cluster with an Au atom at selected positions to create a AuSi n cluster. Further supplementing these aforementioned techniques is an approach using cluster growth model to generate hundreds of initial candidates for further optimizations. Together, these techniques yielded a total of over 150 (for sizes n= 9 and 10), 220 (for sizes n = 11 and 12) and 600 (for sizes n 13 ) initial structures. However, we caution the reader that our current approach may not be appropriate in cases where a silicon cluster is doped with two or more additional metal atoms. More complicated cases such as these may require more sophisticated approaches such as genetic algorithm [9, 34]. Here we should mention that in order to make the comparison among the lowest energy structures for the three metals valid, we deem it necessary to redo CuSi n (n =1-16) clusters calculations instead of relying on previously known results for Cu [8]. This would ensure that the same level of accuracy is employed. Also, we extend our calculations to include results for AgSi n clusters ( n= 14, 15, and 16), which was not the focus of our prior work [14] Gaussian 03 package Furthermore, the low energy isomers for certain sizes of CuSi n (n = 10-16) clusters are selected for further optimizations using Gaussian 03 program [33]. We chose the G(d) basis set for both Cu and Si and applied the B3LYP method as used in previous studies [8]. Gaussian 03 is the edition in the Gaussian series of electronic structure programs announced in Based on the laws of quantum mechanics, Gaussian 03 can calculate the energies of a given system, optimize molecular structures, and compute vibrational frequencies of molecular systems, along with numerous molecular properties derived from these basic computation types. A variety of basis sets can be chosen in the program and can be specified easily by name. The use of Gaussian 03 program will help scientists to concentrate their efforts on applying the methods to research problems and to the development of new methods, rather than on the mechanics of performing the calculations [33, 41].

26 Chapter 3 Results and discussions In this chapter, key results of our comprehensive search for the lowest energy structures of AuSi n (n = 1-16) clusters are first presented. Then we will compare and contrast it with AgSi n and CuSi n clusters in order to make general observations regarding the properties of the entire class of group IB metals. Figs illustrate selected low energy isomers for each size. We would like to stress that the discussions in this subsection are centered on the results of our VASP (PBE/PAW) calculations. However, meaningful comparison with previous related studies on Cu [8] means that for CuSi n ( n = 10-16) we need to use the Gaussian 03 package (B3LYP/6-311+G(d)) as this was the one employed in those studies. 3.1 Structures of AuSi n ( n = 1 to 16) Structures of AuSi n ( n = 1 to 5) Starting with the smallest cluster AuSi, our calculations reveal that the bond lengths of Si dimer, Au dimer and AuSi dimer are 2.28, 2.52, and 2.25 Å, respectively. The bond length of AuSi turns out to be shorter than that of AgSi by 0.11 Å( 4.7 %). This calculated AuSi bond length (2.25 Å ) agrees with values found in both experiment (2.26 Å ) [6] and another theoretical study (2.25 Å ) [7]. For AuSi 2, the optimized geometry shown in Fig. 3.1(a) is an isosceles triangle wherein the bond lengths are 2.43 Å for Au-Si and 2.33 Å for Si-Si. Here, we remark that AuSi 2, AgSi 2, and CuSi 2 [14] all share the same lowest structure. Previously, we found that AgSi 3 shows preference for a planar rhombus structure for its lowest energy isomer [14]. Our data on AuSi 3 and CuSi 3 indicate that this structure as shown in Fig. 3.1(e) is common for all Group IB metals, though naturally Si-Si angles and 18

27 CHAPTER 3. RESULTS AND DISCUSSIONS 19 metal-si bond lengths vary in each metal cluster [8, 14, 19]. For the next bigger cluster, AuSi 4, our analysis indicates that a structure in Fig. 3.1(j) in which the Au atom is bonded to 3 Si atoms and which together form a distorted rectangular pyramidal structure is the most stable. For comparison, our analysis shows that CuSi 4 also share this lowest structure, in agreement with a previous study [8]. As the Au-doped clusters gets bigger, a trend toward increasing propensity for adsorption of an additional Au atom on stable pure silicon clusters slowly emerges. A case in point is AuSi 5, where the lowest structure appears to be that of an Au atom adsorbed on a bipyramid (hexahedron) of Si 5, as illustrated in Fig. 3.1(o). We note that other metals in this family, AgSi 5 and CuSi 5, similarly adopt this optimized geometry. Figure 3.1: Si dimer, Au dimers and Au silicides, along with their bond lengths. The lowest energy isomers of neutral AuSi n (n = 2-5) are shown. The numbers under the structures are relative energies per atom (in mev/atom) with respect to that of the lowest energy isomer of the same size. The Si atoms and Au atoms are shown in yellow (light gray) and grey (dark gray), respectively.

28 CHAPTER 3. RESULTS AND DISCUSSIONS Structures of AuSi n ( n = 6 to 8) For AuSi 6, one can create stable isomers such as the ones presented in Fig. 3.2(a), (b), and (c) by having one Au atom and one Si atom cap a bipyramid (hexahedron) of Si 5. Of these, the one shown in Fig. 3.2(a) is the lowest energy isomer for AuSi 6. For comparison, we included its lowest energy isomer counterparts for AgSi 6 and CuSi 6 in Fig. 3.2(b) and (c), respectively. The latter in particular shows an isomer which can be generated by substituting one Si in a pentagonal bipyramid of Si 7 with a metal atom. Other isomers of interest are also shown for completeness. The low energy isomers for AuSi 7 are similarly presented in Fig. 3.2(f), (g), (h), (i), and (j). Of these, the one with the lowest energy (Fig. 3.2(f)) appears to be a pentagonal bipyramid of Si 7 in which an Au atom is attached to the side and slightly off the plane defined by the pentagon. By contrast, the corresponding metal atom in the lowest energy isomer counterparts for AgSi 7 and CuSi 7 is coplanar with the pentagon as shown in Fig. 3.2(i). Reminiscent of AgSi n clusters behavior in our previous work, certain relatively large clusters increasingly show a tendency to form lowest structures by substituting a Si atom with a metal atom which is then subject to further relaxations. This trend slowly emerges starting with AuSi 8, and continues for bigger clusters. This new trend is evident in Fig. 3.2(k) in which the lowest energy isomer of Si 9 is such that one of the Si atoms has been replaced by an Au atom. This same structure (Fig. 3.2(k)) is shared by CuSi 8, whereas AgSi 8 prefers a different structure which is displayed in Fig 3.2(l). To arrive at the latter, it is helpful to imagine a top Si atom of the lowest energy isomer for Si 9 (bicapped pentagonal bipyramid) being replaced by a metal atom which then gradually moves away from this initial position. Shown in Fig. 3.2(m) is one Au atom capping the lowest energy isomer of Si 8 [25].

29 CHAPTER 3. RESULTS AND DISCUSSIONS 21 Figure 3.2: The lowest energy isomers of neutral AuSi 6, AuSi 7 and AuSi 8 clusters. The numbers under the structures are relative energies per atom (in mev/atom) with respect to that of the lowest energy isomer of the same size. The Si atoms and Au atoms are shown in yellow (light gray) and grey (dark gray), respectively Structures of AuSi n ( n = 9, 10) However, for AuSi 9, the lowest energy isomer illustrated in Fig. 3.3(a) is not structurally related to Si 9, and have not in fact been identified previously. On the other hand, the next lowest energy isomer shown in Fig. 3.3(b) is something which we have already seen in AgSi 9. Fig. 3.3(c) is the lowest energy structure for CuSi 9. Moving up to the next bigger cluster, we found that in fact, all three Group IB metals adopt the same lowest energy isomer for n = 10. In addition, doping is generally favored outside for MSi 10 as illustrated in Fig. 3.3(e). These structures tend to retain the tricapped trigonal prism (TTP) of Si 9 with one silicon and one group IB metal atom capping the TTP. Having covered them previously in Ref. [14], however, it suffices to show two representative open structures from that study [14] in Fig. 3.3(g) and (h). We also examined the other known structural models of caged metal-doped Si 10 [8, 10, 16, 30, 31, 14] as shown in Figs. 3.3 (i), (j), (k), and (l). All these are indicative that the center-site structures are unfavorable for MSi 10 (M= Cu, Ag, and Au) as they are relatively high in energy compared to those in the adsorption and substitu-

30 CHAPTER 3. RESULTS AND DISCUSSIONS 22 tional structures. Here, we reiterate that all three Group IB metals adopt the same lowest energy isomer for n = 10 in Fig. 3.3(e). However, the earlier study [8] done for CuSi 10 indicates that Fig. 3.3(f) is the lowest energy isomer. Two CuSi 10 isomers as seen in Fig. 3.3(e) and (f) have the same TTP unit but with different adsorption site for the Cu atom and are nearly degenerate in energy by mev per atom in our VASP calculations. Furthermore, a separate Gaussian 03 package calculation is in agreement with a previous study [8] and thus confirming that the isomer in Fig. 3.3(f) is indeed the most stable. In addition, the same calculations show that the isomers in Fig. 3.3(e) and (f) are degenerate and differ only by 0.48 mev per atom. Lastly, we verified that the caged and open structures as shown in Fig. 3.3(g)-(l) are unfavorable for CuSi 10. We note that for CuSi 10 the isomer in Fig. 3.3(h) relaxes to the structure in Fig. 3.3(i), whereas the isomers in Fig. 3.3(k) and (l) are reduced to the same configuration.

31 CHAPTER 3. RESULTS AND DISCUSSIONS 23 n = 9 (a) 0.00 (b) (c) (d) (18.59) (6.04) (0.00) (150.47) n = 10 (e) 0.00 (f) (g) (h) (0.00, 0.48) (12.66, 0.00) (86.57, 87.40) (101.25, N/A) (i) (70.42, 36.85) (j) (k) (l) (64.64, 35.56) (N/A, N/A) (28.92, 38.79) Figure 3.3: The lowest energy isomers of neutral AuSi 9 and AuSi 10. The numbers under the structures are relative energies per atom (in mev/atom) with respect to that of the lowest energy isomer of the same size. The Si atoms and Au atoms are shown in yellow (light gray) and grey (dark gray), respectively. The numbers in the parentheses are for CuSi 9 and obtained using the VASP package only, whereas those are for CuSi 10 and obtained using the VASP and Gaussian 03 packages, respectively.

32 CHAPTER 3. RESULTS AND DISCUSSIONS Structures of AuSi n ( n = 11) The six low energy isomers for AuSi 11 are likewise plotted in Fig Of these, the lowest structure, shown in Fig. 3.4(a), is one wherein two silicon atoms and one gold atom cap the Si 9 cluster. A similar structure can be generated using the same procedure in which a pure Si cluster is capped with a metal atom and two Si atoms, but with the three atoms assuming different positions, resulting in slightly higher energies. This is illustrated in Fig. 3.4(b) which shows the lowest energy isomer for AgSi 11. Also, for CuSi 11, the most stable structure is shown in Fig. 3.4(c). n = 11 (a) 0.00 (35.88, 15.27) (b) (c) (37.61, 13.15) (0.00, 0.00) (d) (e) (f) (81.71, 86.93) (57.95, 43.91) (133.16, N/A) Figure 3.4: The lowest energy isomers of neutral AuSi 11. The numbers under the structures are relative energies per atom (in mev/atom) with respect to that of the lowest energy isomer of the same size. The Si atoms and Au atoms are shown in yellow (light gray) and grey (dark gray), respectively. The numbers in the parentheses are obtained using the VASP and Gaussian 03 packages, respectively Structures of AuSi n ( n = 12, 13) For AuSi 12 and AgSi 12, the lowest structures favored vary, with a metal latching on to different Si atoms from the lowest energy isomer for pure Si 12 obtained from Ref. [25]. The isomers shown in Fig. 3.5(a), (b) and (d) are the lowest energy isomer for AuSi 12, AgSi 12, and CuSi 12 respectively. Fig. 3.5(e), (f), and (g) are the known Si 12 caged clusters formed by encapsulation of a metal atom [17, 2]. Just like in CuSi 10 and CuSi 11, we compare our result with the previous study [8]. For CuSi 12, we found that the energies

33 CHAPTER 3. RESULTS AND DISCUSSIONS 25 of isomers as shown in Fig. 3.5(b), (c), and (e) are very close to the one in Fig. 3.5(d). As we can see, the result for CuSi 12 based on VASP calculations shows that the energy is almost degenerate, making it impossible to make definitive conclusions. Therefore, we further optimized CuSi 12 clusters using Gaussian 03 package and confirmed that the caged isomer as shown in Fig. 3.5(g) is indeed the lowest energy isomer. The isomers in Fig. 3.5(e) and (g) are energetically degenerate. Four low energy isomers for AuSi 13 are presented in Fig. 3.5, with the lowest energy isomer shown in Fig. 3.5(h). The next lowest isomer, shown in Fig. 3.5(i), is also the lowest energy isomer for AgSi 13 and is generated simply by capping the lowest energy isomer of Si 13 with a metal atom. The isomer shown in Fig. 3.5(j), on the other hand, is formed via a substitution of a metal atom with a Si atom in the lowest energy isomer of Si 14. This isomer is the lowest energy isomer for CuSi 13.

34 CHAPTER 3. RESULTS AND DISCUSSIONS 26 n = 12 (a) 0.00 (b) (c) (d) (34.71, 41.28) (13.73, 33.98) (7.85, 26.44) (0.00, 40.86) n = 13 (e) (f) (1.84, 0.99) (54.32, 52.06) n = 13 (g) (42.85, 0.00) (h) 0.00 (16.14, 21.65) (i) 1.86 (j) (k) (l) (1.14, 7.07) (0.00, 14.62) (15.73, 7.36) (12.55, 0.00) Figure 3.5: The lowest energy isomers of neutral AuSi 12 and AuSi 13. The numbers under the structures are relative energies per atom (in mev/atom) with respect to that of the lowest energy isomer of the same size. The Si atoms and Au atoms are shown in yellow (light gray) and grey (dark gray), respectively. The numbers in the parentheses are obtained using the VASP and Gaussian 03 packages, respectively Structures of AuSi n ( n = 14) The trend in which the lowest energy isomer is a cluster of Si atoms capped with an Au atom continues for AuSi 14, as illustrated in Fig. 3.6(a). Its structure can be characterized as consisting of a TTP core plus 5 Si atoms and one Au atom on one side. The next two lowest energy isomers shown in Fig. 3.6(b) and (c) can be described as a pure Si 13 capped with a Au atom as well as Si atoms pointed to by the red arrows. For n = 14, 15, and 16, we also illustrate other low energy caged structures based on

35 CHAPTER 3. RESULTS AND DISCUSSIONS 27 several ZrSi n structures from Ref. [29]. Here, we remark that AuSi 14 and AgSi 14 share the same lowest structure using VASP package. For CuSi 14, we note that the caged isomer in Fig. 3.6(e) is nearly energetically degenerate with Fig. 3.6(a) by 2.55 mev per atom. We further optimized CuSi 14 using Gaussian 03 package. our result reveals that the caged isomer in Fig. 3.6(e) is indeed the lowest energy isomer. n=14 (a) 0.00 (b) (c) (0.00, 56.83) (55.49, 78.13) (55.33, 92.23) (d) Top (e) (f) (g) (h) Side (2.55, 0.00) (112.08, ) (149.82, ) (4.40, 7.12) Figure 3.6: The lowest energy isomers of neutral AuSi 14. The numbers under the structures are relative energies per atom (in mev/atom) with respect to that of the lowest energy isomer of the same size. The Si atoms and Au atoms are shown in yellow (light gray) and grey (dark gray), respectively. The numbers in the parentheses are obtained using the VASP and Gaussian 03 packages, respectively Structures of AuSi n ( n = 15) For AuSi 15 the two lowest energy isomers as illustrated in Figs. 3.7(a) and (b) can be regarded as that of the lowest energy pure silicon cluster isomer (Si 15 from Ref. [24]) and a Au atom positioned at specific locations at its exterior. Here, we note that AuSi 15, AgSi 15, and CuSi 15 all share the same lowest structure. Another low energy isomer is formed by joining two TTP units and capping the resulting structure with a Au atom as illustrated in Fig. 3.7(c). Lastly, the calculations using Gaussian 03 package shows that this endohedral isomer in Fig. 3.7(d) is the lowest energy isomer for CuSi 15.

36 CHAPTER 3. RESULTS AND DISCUSSIONS 28 n=15 (a) 0.00 (b) 6.58 (c) 7.24 (0.00, 14.96) (49.00, 48.14) (66.01, 63.13) (d) (e) (f) (17.06, 0.00) (15.68, 41.92) (35.78, 28.44) Figure 3.7: The lowest energy isomers of neutral AuSi 15. The numbers under the structures are relative energies per atom (in mev/atom) with respect to that of the lowest energy isomer of the same size. The Si atoms and Au atoms are shown in yellow (light gray) and grey (dark gray), respectively. The numbers in the parentheses are obtained using the VASP and Gaussian 03 packages, respectively Structures of AuSi n ( n = 16) Finally, the lowest energy isomer of AuSi 16, displayed in Fig. 3.8(a) shows lowest energy Si 13 isomer which abuts a cluster consisting of an Au and three Si atoms. The next lowest energy isomer shown in Fig. 3.8(b), on the other hand, is formed by capping the lowest energy AuSi 15 isomer (Fig. 3.7(a)) with a Si atom. Alternatively, one can regard it as a Si (pointed to by the red arrow) and a Au atom embedded on Si 15 in which these two atoms are slightly distorted. Moreover, this isomer shown in Fig. 3.8(b) is the lowest energy isomer for AgSi 16. The next isomer shown in Fig. 3.8(c) is similar to that in Fig. 3.8(b) except the Si atom (pointed to by the red arrow) is found at the bottom of the structure. It also can be generated from the lowest energy isomer for AuSi 15 in Fig. 3.7(a) by placing the Si atom at the bottom. Additionally, Fig. 3.8(d) is essentially a TTP unit bonded to a Si 7 -like cluster wherein a corner Si atom in the latter has been shifted to the surface. A metal atom at the tip completes the structure.

37 CHAPTER 3. RESULTS AND DISCUSSIONS 29 n=16 (a) 0.00 (b) 3.25 (c) 7.65 (d) 8.71 (47.59, 64.31) (16.12, 45.16) (7.60, ) (59.91, 56.16) (e) (2.42, 0.00) (f) (g) (h) (0.00, 1.47) (66.12, 34.63) (118.90, ) Figure 3.8: The lowest energy isomers of neutral AuSi 16. The numbers under the structures are relative energies per atom (in mev/atom) with respect to that of the lowest energy isomer of the same size. The Si atoms and Au atoms are shown in yellow (light gray) and grey (dark gray), respectively. The numbers in the parentheses are obtained using the VASP and Gaussian 03 packages, respectively. We have also calculated the zero point vibrational energies, E v ib, of the MSi n clusters (M= Au and Cu; n 10) using the VASP package. We did not observe significant change in the lowest energy isomers when the stability of isomers are evaluated according to E t ot = E 0 + E v ib. In general, the zero point vibrational energies of the clusters increase with increasing cluster size, n. Furthermore, we report that the uncertainty due to the zero point vibrational energies for the different motifs considered here can be up to 13 mev per atom. However, for isomers of the same size and sharing a similar motif, this uncertainty is only less than 4 mev. The only change resulting from this re-recalculation is that the isomer for CuSi 13 shown in Fig. 3.5(k) now has a lower energy than that in Fig. 3.5(l). Here we should mention that these values compare well with the corresponding uncertainties due to the zero point vibrational energies obtained for CuSi n (n 10) using Gaussian 03 package, which are all less than 5.2 mev per atom. Our recalculations reveal that the energy orderings for CuSi 10, CuSi 12, and CuSi 13 is affected, specifically those which are degenerate in energy. The new ordering in ascending total energy are reflected in the isomers shown in Fig. 3.3(l), Fig. 3.3(i), and Fig. 3.3(j). For CuSi 12, the isomer in Fig. 3.5(e) is lower than that in Fig. 3.5(g) by 0.1 mev per

38 CHAPTER 3. RESULTS AND DISCUSSIONS 30 atom, whereas for CuSi 13 the isomer in Fig. 3.5(k) is lower than that in Fig. 3.5(i) by 0.4 mev per atom. Overall, we found that for sizes n = 1-16, the MSi n clusters, in which M is a Group IB metal, tends to be exohedral, except for CuSi n (n 12) in which both the lowest energy exohedral and endorderal isomers are energetically degenerate. Our studies further suggest that for exohedral isomers the Group IB metal generally favors capping a pure silicon cluster rather than substituting a silicon atom in the cluster. This latter tendency is in sharp contrast with group VI-B transition metals (W and Cr) behavior in which endohedral atoms is the preferred structure. Furthermore, for n= 6, 7, 10, 11, 12, 14, and 15, doping does not alter the inner core structure in any significant way. However, for the relatively unstable Si clusters( n = 5, 8, 9, and 13) substitution may compete with capping as the dominant mode in forming the lowest energy structure in certain cases, and which unlike the latter, may cause significant structural changes in the clusters. Moreover, our studies suggest that the TTP unit is the dominant motif for the exohedral MSi n (n > 9) clusters. For the largest clusters considered here (n = 14 16), a pattern in which the Au atom prefers to cap the clusters emerges. Also, we found evidence that MSi n clusters behavior (as regards the lowest energy structures preferred for a particular size) shows some uniformity for certain sizes (2, 3, 5, 7, 10, 14, and 15). However, for n =4, 6, 8, 9, 11, 12, 13, and 16, we found that three group IB metals have different lowest energy structures. This might be attributed to the difference in mass spectral data in the range n > Exohedral versus endohedral structures To illustrate the structural preferences of MSi n (n= 10-16) clusters, we identified both the lowest energy endohedral and exohedral isomers for each cluster size and summarized the results in Table 3.1. The relative energy per atom of the lowest endohedral isomer (in mev/atom) with respect to the lowest exohedral isomer as a function of the number of Si atoms in the clusters are plotted in Fig For AgSi n and AuSi n, the endohedral isomers have much higher relative energies than their exohedral counterparts. Here we notice a pattern in which the relative energy gets smaller as the cluster size increases, signifying that the critical size beyond which endohedral structures are favored over exohedral ones may not be smaller than 17. By contrast, our studies indicate that for CuSi n there is a critical size n= 12. In fact, for cluster size larger than 12, the energy difference between the endohedral and exohedral isomers

39 CHAPTER 3. RESULTS AND DISCUSSIONS 31 for each size is small and less than 16 mev per atom as shown in Fig Notice that the lowest energy endohedral and exohedral isomers are energetically degenerate. As these are general trends, certain nuances are revealed by our VASP calculations. Specifically, we found that for CuSi n, where n= 12 and 14 exohedral isomers are slightly lower in energy than endohedral ones by 1.84, and 2.55 mev per atom. For n = 16, the caged isomer is preferred over exohedral structures by 7.60 mev per atom. As mentioned previously, we also recalculated the CuSi n (n= 10-16) with the Gaussian 03 package. We found that for n = 12, 13, 14, 15, and 16, the endohedral isomers have lower energies than their exohedral counterparts by 26.44, 7.07, 56.83, 14.96, and mev per atom, respectively. This is reminiscent of a recent experimental study by Janssens et al. [15] which demonstrated that there is a critical size beyond which the argon-complex formation is unlikely, suggesting that Cu metals are caged inside the Si clusters. Table 3.1: Summary of the predicted lowest energy isomers of various metals, sizes and structure types (endohedral or exohedral). The corresponding figure for the lowest isomer is indicated in the table. exohedral endohedral VASP Gaussian VASP Gaussian size, n Au Ag Cu Cu Au Ag Cu Cu Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig. 10 3(e) 3(e) 3(e) 3(f) 3(i) 3(j) 3(l) 3(j) 11 4(a) 4(b) 4(c) 4(c) 4(d) 4(d) 4(e) 4(e) 12 5(a) 5(b) 5(d) 5(c) 5(e) 5(e) 5(e) 5(g) 13 5(h) 5(i) 5(j) 5(i) 5(k) 5(k) 5(l) 5(l) 14 6(a) 6(a) 6(a) 6(a) 6(d) 6(d) 6(e) 6(e) 15 7(a) 7(a) 7(a) 7(a) 7(d) 7(e) 7(e) 7(d) 16 8(a) 8(b) 8(c) 8(c) 8(e) 8(e) 8(f) 8(e)

40 CHAPTER 3. RESULTS AND DISCUSSIONS relative energy (mev/atom) CuSi n AuSi n AgSi n number of Si atoms, n Figure 3.9: The relative energy per atom of the lowest endohedral isomer (in mev/atom) with respect to the lowest exohedral isomer as a function of the number of Si atoms in the clusters. 3.3 Relative Stability of AuSi n A commonly employed method to evaluate relative stability of clusters is to focus on the binding energy per atom, defined here as E b (MSi n ) = [E (MSi n ) E (M ) n E (Si )]/(n + 1) (3.1) and E b (Si n ) = [E (Si n ) n E (Si )]/n, (3.2) where E (M ) and E (Si ) are the single atom energies and E (MSi n ) and E (Si n ) are the total energies of clusters. The binding energies (per atom) for metal-doped Si clusters and pure Si clusters as a function of cluster size are plotted in Fig. 3.10(a). It shows that for pure silicon clusters, those with size n = 7, 10, 12, and 15 are stable, whereas for group IB metal doped Si clusters only n= 10 appears to be stable. Our current analysis points toward slightly higher binding energies for pure Si n clusters compared to their doped (MSi n ) counterparts and that doping does not necessarily lead to more stable clusters.

41 CHAPTER 3. RESULTS AND DISCUSSIONS 33 Another indicator of relative stability of clusters is the second difference in binding energy per atom defined by and 2 E (MSi n ) = 2 E b (MSi n ) + E b (MSi n+1 ) + E b (MSi n 1 ) (3.3) 2 E (Si n ) = 2 E b (Si n ) + E b (Si n+1 ) + E b (Si n 1 ). (3.4) In Fig. 3.10(b), we plotted 2 E (n) as a function of cluster size. Stability analysis based on this plot suggests that among the various sizes for pure Si n, those with n = 7, 10, 12, and 15 are relatively stable. For AuSi n, the same analysis points toward stability of size n = 5, 10, and 13, in addition to n = 10 which we already know based on the earlier binding energy curve analysis. We note that the spike in the curve for AgSi n, which is indicative of the stability of the cluster with n = 7, is missing in both AuSi 7 and CuSi 7 plots. From the same plot, we can see that metal clusters with n = 5, 10, and n=13 are stable for both AuSi n, whereas for CuSi n only n= 5 and 10 is stable. On the other hand, for AgSi n, n =5, 7, 10, and 13 are stable. The observation regarding the stability of AuSi n (n =5, 10, and 13) and CuSi n (n=5 and 10) but excluding n = 7 shows partial agreement with the mass spectral data obtained by Jaeger et. al. [13], whose group studied group IB metal-silicon clusters.

42 CHAPTER 3. RESULTS AND DISCUSSIONS 34 Figure 3.10: (a) Binding energy per atom and (b) second difference in binding energy (per atom) versus the number of silicon atoms in a cluster.

43 CHAPTER 3. RESULTS AND DISCUSSIONS 35 In order to determine specific fragmentation pathways of clusters, as well as evaluate the stabilities of the resulting products, it is often useful to turn to dissociation energy, or the energy needed to dissociate or break up a neutral cluster. The dissociation energy of MSi n cluster needed to break it up into MSi m and Si n m clusters is provided below, E n,m = E (MSi m ) + E (Si n m ) E (MSi n ), (3.5) where E (MSi m ) is the total energy of the cluster with m silicon atoms (0 m n 1). We have evaluated the fragmentation energies of the neutral clusters for all possible pathways and provided plots for key pathways in Fig Analysis of fragmentation energies reveals AuSi n s propensity toward dissociation into an Au atom and Si n for n < 11 and n= 14, excepting n = 8, as AuSi 8 prefers to split into Si 7 and a AuSi dimer instead. For n = 11, 15, and 16, AuSi n clusters tend to dissociate into a Si 10 and either a AuSi dimer, AuSi 5 or AuSi 6. For n = 12 and 13, clusters both break up into the stable Si 7 cluster and either AuSi 5 or AuSi 6. By contrast, for AgSi n clusters the preferred fragmentation channel is via evaporation of an Ag atom up to n= 15. For n= 16, clusters tend to break up into Si 10 and AgSi 6. Next, we have also calculated all the possible pathways for CuSi n. We found that the primary pathway for the smaller CuSi n ( n 11) clusters is via evaporation of one Cu atom which is similar to AgSi n. For the next largest clusters examined, n 12, CuSi n fragmentation behavior is likewise similar to that in AuSi n as discussed in the preceding paragraph.

44 CHAPTER 3. RESULTS AND DISCUSSIONS 36 Figure 3.11: (a), (b), and (c) Fragmentation energy ( E n,m = E (MSi m ) + E (Si n m ) E (MSi n ), M = Au, Ag, and Cu) vs. the number of silicon atoms in a cluster. The legend explains the labeling of the smaller fragments.

45 CHAPTER 3. RESULTS AND DISCUSSIONS Embedding energy We also point out an alternative way of measuring the endohedral cluster s stability via the so-called embedding energy [35]. This quantity can be obtained from the fragmentation energy introduced in Eq. 3.5 above by setting m=0 and letting E (MSi n )=E n ), defined as the total energy of the encapsulated isomers. The equation is now rewritten as follows: E B E 1 = E (M ) + E (Si n ) l ow e s t E n ), (3.6) where E (Si n ) l ow e s t is the lowest energy of the Si n cluster. The same definition can also be found in Ref. [10]. The embedding energy of the aforementioned endohedral isomers in Table 3.1 for MSi n (M= Cu, Ag, and Au; n= 10-16) are plotted as a function of the number of Si atoms in the clusters in Fig As we can see, embedding energies of CuSi n are higher than that of the other two metals, signifying that the MSi n (n= 10-16) cluster is most stable when the particular metal M is Cu. Embedding Energy (ev) CuSi n AuSi n AgSi n Number of Si atoms (n) Figure 3.12: The embedding energy of the listed endohedral isomers in Table 3.1 for MSi n (M= Cu, Ag, and Au; n= 10-16) are plotted as a function of the number of Si atoms in the clusters.

46 CHAPTER 3. RESULTS AND DISCUSSIONS Electronic properties of MSi n (M = Au, Ag, and Cu) HOMO and LUMO Certain cluster stability analyses invariably rely on the HOMO-LUMO gap (the difference between the highest occupied molecular orbital and the lowest unoccupied molecular orbital) being the prototypical electronic property and thus a staple in first-principles calculations. In this work, we set out to evaluate the HOMO-LUMO gaps of the lowest energy isomers for each cluster size using spin-polarized DFT calculations. Analysis of the data in our previous study indicates that the AgSi n clusters with n = 1, 5, and 12 have relatively wider HOMO-LUMO gaps, contrasting our current results which indicate that the same is true only for AuSi n and CuSi n with n = 1 and 5. The new result for n=12 might simply be explained as due to the fact that all three metals have different stable configurations for this size, as discussion in Section III. Fig also shows that doping group IB atom (Au and Cu) significantly decreases the HOMO-LUMO gaps for n > 6. Furthermore, for pure Si n clusters, the result strongly indicate a correlation between the HOMO-LUMO gaps and the energetic stability, whereas for the MSi clusters (M= Cu, Ag and Au) our data is inconclusive. Figure 3.13: HOMO-LUMO gap versus the number of silicon atoms in a clusters. We show the HOMOs and LOMOs of two relatively stable isomers (AuSi 10 and AuSi 15 ) and two picked-up isomers (AuSi 14 and CuSi 16 ) in Fig The HOMOs are

47 CHAPTER 3. RESULTS AND DISCUSSIONS 39 marked in blue and the LOMOs are marked in red. For each size, the HOMOs and LOMOs are the spin-up and spin-down of the same orbital due to the use of the spinpolarized calculation and the odd number of total electrons. Figure 3.14: The HOMOs and LOMOs of (a) AuSi 10, (b) AuSi 14, (c) AuSi 15, and (d) CuSi Charge transfer Ideally, it s desirable to see the changes of the electronic structure in the silicon cluster with the addition of a metal dopant. To do this, we first calculate the total charge density of the isomers of certain sizes, usually the ones having the lowest energy. Next, we divide the structure into two parts, the doping metal and the silicon cluster, and fix their position to calculate charge densities again and separately. Then, one can calculate the charge transfer from the following definition: δn(r) = n MSi n (r) n M (r) n Si n (r) (3.7)

48 CHAPTER 3. RESULTS AND DISCUSSIONS 40 where n MSi n (r), n M (r), and n Si n (r) are the charge densities of MSi n, M, and Si n, respectively, and M = Au and Cu. δn(r) are plotted in Fig Red and blue areas indicate charge accumulation and depletion zones, respectively. Figure 3.15: Plot of (a) Charge transfer of AuSi 10. (b) Charge transfer of AuSi 14. (c) Charge transfer of AuSi 15. (d) Charge transfer of CuSi 16.