II! 1:11.! It. I: 1,311.1!!!) . 4:.) a). .Pd.0)l)whl.§¢. fl... J ,nzal‘frn ,.(.1 I35)! . .4174llfvlhh..lv‘|.. .‘ln 1".t .- w :flpqumtmfl‘l'” . - ,un . l. V. [Zr . . .I. . . V Ewfiwwmuif. 25f. . . I .. ._ . . V :5}... 1.6 "munmqhwflhfivflk..xu . . , . I u..b...vl:r....:. .105“. , . .A 1 1:32.39... . . . IVERSITY LIBRARIES l lllll llllll llllllllllllllllllllllll 3 129300 This is to certify that the dissertation entitled "Electronic and Structural Properties of Atomic Clusters" presented by Yang Wang has been accepted towards fulfillment ofthe requirements for Ph.D. degree in PhYSiCS WW1 35mm )3 Major professor Date W MSUIJ an Atrirmunw' Action ’Equal' Opportunity Institution 0712771 r LIBRARY University \ Michigan State fix 1" PLACE IN RETURN BOX to remove this checkout from your record. TO AVOID FINES return on or before date due. DATE DUE DATE DUE DATE DUE 7___l : MSU Is An Affirmative Action/Equal Opportunity Institution . emunS-DJ ELECTRONIC AND STRUCTURAL PROPERTIES OF ATOMIC CLUSTERS By YANG WANG A DISSERTATION Submitted to Michigan State University in partial fulfillment of the requirements for the Degree of DOCTOR OF PHILOSOPHY Department of Physics and Astronomy 1993 ELEC In atomic my res found PristjI ABSTRACT ELECTRONIC AND STRUCTURAL PROPERTIES OF ATOMIC CLUSTERS BY YANG WANG In this Thesis I have studied the electronic and structural properties of small atomic clusters, namely alkali metal clusters and the C60 carbon fullerene cluster. In my research, I addressed the possibility of collective excitations in small clusters, and found similarities between the excitation spectra of these two type of systems. In alkali metal clusters, specifically N an and Lin, I calculated the equilibrium struc- ture and electronic excitations, as well as their damping, using ab initio methods. I used the Local Density Approximation (LDA) to describe the ground state proper- ' ties of these systems, and the Random Phase Approximation (RPA) for the electronic excitations. Results for the excitation spectra of clusters with the “magic number” of n = 2,8 atoms are in good agreement with experimental data. I found that even in small clusters, one single excitation exhausts most of the total oscillator strength, which has also been observed experimentally. My calculations indicate that the cou- pling of electronic levels to vibrational degrees of freedom accounts quantitatively for the observed width of the collective electronic excitations in alkali dimers. In the C60 system, my main interest has been devoted to the dielectric response of isolated clusters, and elastic properties and stability of the C60 fullerite solid in its pristine and intercalated phases. ln tl at high transfo C60 bat stable hly C50 cl calcul. tions, This 1 tire e deloce In the C60 solid, 1 found a very low value of the bulk modulus at zero pressure while at high pressures the bulk modulus of fullerite exceeds that of diamond and a phase transformation to diamond is likely. My calculations for the stability of potential C60 based superconductors indicate that alkali and some alkaline earth elements form stable fullerite intercalation compounds. My calculations on the static polarizability and hyperpolarizability of the isolated Ceo cluster have shown that the valence electrons are quite delocalized. Our RPA calculations of the dynamical response indicate a strong screening of low—lying excita- tions, which transfer their oscillator strength to a giant collective mode at he; s: 20 eV. This mode has been subsequently observed experimentally. This high energy collec- tive excitation which is very similar to a Mie plasmon results from a large electron delocalization across the fullerene and a large charge density in the C60 cluster. . “C Ya ant Publications “Stiffness of a Solid Composed of C60 Clusters”, Yang Wang, David Tomanek, and George F. Bertsch, Phys. Rev. BR 44, 6562 (1991). “Collective Plasmon Excitations in C60 clusters”, George F. Bertsch, Aurel Bulgac, David Tomanek, and Yang Wang, Phys. Rev. Lett. 67, 2690 (1991). “Hyperpolarizability of the C60 Fullerene Cluster”, Yang Wang, George F. Bertsch, and David Tomanek, Z. Phys. D (1993). “Stability of C60 Fullerite Intercalation Compounds”, Yang Wang, David Tomanek, George F. Bertsch, and Rodney S. Ruoff, Phys. Rev. B (1993). “Lanthanide and Actinide Based Fullerite Compounds: Potential AzCso Superconductors?” Rodney S. Ruoff, Yang Wang, and David Tomanek, Chem. Phys. Lett. (1993). “Fullereneynes: A New Form of Porous Fullerenes”, Ray H. Baughman, Douglas S. Galvao, Changxing Cui, Yang Wang, and David Tomanek, Chem. Phys. Lett. (1993). “Collective Electronic Excitations and their Damping in Small Alkali Clusters”, Yang Wang, Caio Lewenkopf, David Tomanek, George F. Bertsch, and Susumu Saito, (submitted for publication). to UN for pu: ins; adx fina Scit ACKNOWLEDGMENTS With great respect and deep gratitude, I would like to express deepest thanks to both of my advisers, Professors David Tomanek and George F. Bertsch. Without their professional guidance and endless encouragement, it would have been impossible for me to complete my graduate study. Prof. Tomanek’s scientific research ethics and pursuit for physics have had a great influence on me. Prof. Bertsch’s enormous physical insight and elegant way of doing physics have always been and will be an inspiration to me. I thank both of them for their understanding, support and patient advising during the pass three and half years. I would also like to acknowledge the financial support by the Center for Fundamental Material Research and the National Science Foundation. I am very grateful to Prof. N. Birge, Prof. D. Stump, and especially Prof. S. D. Mahanti for their service on my guidance committee and the encouragement I have received. I would like to thank other members in our weekly discussions: Prof. A. Bulgac, Dr. G. Overney, Dr. C. Lewenkopf, Mr. W. Zhong, Mr. N. Ju, and Mr. J. Foxwell. I would like to express my special thanks to Prof. Kovacs for his help I received in my pursuit of graduate study at Michigan State University, and Mrs. S. Conroy, Mrs. S. Holland, and Mrs. J. King for their help. Thanks are also due to my friends, B. Lian, Q. Yang, Z. Sun, J. Chen, N. Mousseau, H. Seong, C. Hsu, F. Liu, W. Yang, A. Azhari, E. Ramakrishnan, Dr. L. Zhao, Dr. Y. S. Li, Dr. Y. Cai, Dr. S. Saito, and many others for their help and friendship. Finally, I heartily thank my parents for their unconditional love, constant moral support, and for their stressing the importance of education. Cc LIST LIST 1 Int 1.1 1.2 Contents LIST OF TABLES LIST OF FIGURES 1 Introduction 1.1 Alkali Metal Clusters ........................... 1.1.1 Cluster synthesis and experimental techniques ......... 1.1.2 Structural properties ....................... 1.1.3 Electronic shell structure ..................... 1.1.4 Optical response ......................... 1.2 Carbon Clusters .............................. 1.2.1 Synthesis of C60 and the C60 crystal ............... 1.2.2 Structural properties of C60 and the C60 crystal ........ 1.2.3 Electronic properties of C60 and the C50 crystal ........ 1.2.4 Optical response of C60 and the C60 crystal .......... 2 Computational Techniques 2. 1 Density Functional Theory (DFT) .................... ii vi 12 20 21 22 28 3O 38 39 .6). 4St 4.2 4.3 5 Elle: 5.1 2.1.1 Local Density Approximation (LDA) .............. 39 2.1.2 Norm conserving pseudopotentials ................ 41 2.2 Jellium model ............................... 44 2.3 Tight—binding Hamiltonian ........................ 47 2.4 Random Phase Approximation (RPA) .................. 49 3 Alkali Clusters J 56 3.1 Equilibrium geometry of small alkali clusters .............. 58 3.2 Collective electronic excitations (Mie plasmon) in small alkali clusters 62 3.3 Damping of the Mie plasmon in small alkali clusters .......... 69 3.4 Conclusions ................................ 71 4 Structural Properties of Cso and Solid C60 76 4.1 Structural and elastic properties of the C60 based solid ........ 76 4.2 Stability of donor and. acceptor intercalated C60 solid ......... 87 4.2.1 Born—Haber cycle ......................... 88 4.2.2 Structural and cohesive properties of fullerite intercalation com- pounds ............................... 96 4.2.3 Discussion ............................. 104 4.3 Conclusions ................................ 113 5 Electronic Properties of the C60 Clusters 124 5.1 Linear and nonlinear static polarizability of C60 ............ 124 iii 5.2 Collective electronic excitations of C60 ................. 5.3 Conclusions ................................ 6 Summary and Conclusions iv List of Tables 3.1 Ground state properties of sodium and lithium dimers: Equilibrium bond length dc, dissociation energy D,, and vibration frequency w... . 61 3.2 Collective electronic excitations in small sodium and lithium clusters. Our results for the plasmon frequency hwpza,mon and its width F are listed together with results based on spherical jellium [25], thELLYRpA, and results of the classical Mie theory, thge. ............. 64 4.1 Total ionization energy [tot of C60 .................... 93 4.2 Total electron affinity Am of C60 .................... 93 4.3 Madelung constants a for the structures considered in this work. . . 94 5.] Calculated and observed optical susceptibilities of C60 and CeHe molecules. 130 List of Figures 1.1 1.2 1.3 1.4 Typical experimental setup for cluster spectroscopy: Cluster source and time-of-flight spectrometer [From de Heer et al, Solid State Physics 40, 128 (1987)]. ............................. Equilibrium geometries of small sodium clusters [From Bonacic et al, Phys. Rev. B 37, 4369 (1988) and Moullet et al, Phys. Rev. Lett. 65, 476 (1990)]. ............................. Mass abundance spectrum of N aN clusters. (a)Mass abundance spec- trum of NaN clusters, N =4—75. The inset corresponds to N =75—100. (b) The calculated second derivative A2(N) of the total energy E(N) of jellium clusters, defined in Eq. (1.1), as a function of cluster size [From Knight et al, Phys. Rev. Lett. 52, 2141 (1984)]. ........ Optical response of the Na; cluster using the depletion technique. [From Wang et al, Chem. Phys. Lett. 166, 26 (1990)]. ........ vi 1.5 A classical picture of the Mie plasmon. (a) If no electrical field is applied, the positive and negative charge background coincide with each other. (b) Under an external electrical field with a particular frequency, the negative charge background will move back and forth with respect to the positive background. Since the collective motion results in a nonvanishing total charge density only near the surface, it is often called the surface plasmon of the cluster. ........... 1.6 Dependence of the collective electronic excitation energy on the cluster size in N aN clusters. Energies derived from reflectivity change spectra (solid circles) and energies calculated via sum rule from experimental static polarizabilities (open circles) are compared with jellium calcula- tions (crosses) [From Parks et al, Phys. Rev. Lett. 62, 2301 (1989)]. 1.7 Damping mechanisms for collective electronic excitations. (3.) Static fragmentation due to aspherical shape. (b) Landau damping. (c) Electron—vibration coupling. ...................... 1.8 Schematic diagram of the pulsed supersonic nozzle used to generate carbon cluster beams [From Kroto et al, Nature 318, 162 (1985)]. 1.9 Experimental mass spectrum of carbon clusters [From Kroto et al, Nature 318, 162 (1985)]. ......................... 1.10 Structure of the C60 “buckyball” cluster. ................ 1.11 Mass production technique for fullerite [From Huffman, Physics Today, 44, 22 (1991)]. .............................. vii 15 17 19 23 24 25 1.12 1.13 2.1 2.2 (a) Single—particle energy level spectrum of a C30 cluster, as obtained using the tight—binding Hamiltonian described in Section 2.3. The levels have been sorted by symmetry. (b) Expanded region of the energy level spectrum near the Fermi level. Allowed dipole transitions between states with gerade (g) and ungerade (u) parity are shown by arrows [From G. F .Bertsch, A. Bulgac, D. Tomanek, and Y. Wang, Phys. Rev. Lett. 67, 2690 (1991)]. ................... Optical response of the C60 cluster [From H. Ajie et al, J. Phys. Chem. 94, 8630 (1990)]. ............................. The 33, 317 and 3d pseudo wave functions and the corresponding all— electron wave functions of the sodium atom. Outside the core radius re, the pseudo wave functions and the all-electron wave functions are the same. Inside the core, the pseudo wave functions are nodeless and smooth. At re, the spatial derivative of the all—electron and the pseudo wave functions, as well as their first energy derivatives, agree with each other. The eigenvalues associated with the pseudo wave function agree with those of the all—electron wave function. The bottom panel shows the pseudopotentials for the s, p and d states. Inside the core radius, the pseudopotentials are finite, and at large radii, the pseudopotentials approach — e2 / r. ............................. Energy spectrum and self—consistent potential of the Nag cluster ob- tained from the spherical jellium model. ................ viii 31 3.1 3.2 3.3 4.1 Franck—Condon broadening of the collective electronic excitations in (a) N a2 and (b) Liz. The lowest levels are the LDA dissociation energies D(d) of the dimers as a function of the bond length d. The higher levels give the excitations energies, which are presented as D(d) + ERpA(d) [From Y. Wang et al, (submitted for publication)]. .......... Calculated spectral function of N a; (in arbitrary units) and its broad- ening due to nuclear zero—point motion (dashed line), as compared to the observed photoionization spectrum of Ref. [6] (solid line). The width of the Gaussian envelop is 0.10 eV and the displayed theoretical data are red—shifted by 0.5 eV with respect to the calculated results [From Y. Wang et al, (submitted for publication)]. .......... Calculated oscillator strength distribution in the excitation spectrum of Nag [From Y. Wang et al, (submitted for publication)]. ...... Schematic drawing of the the elastic parameters describing the inter- action between neighboring C60 clusters in fullerite. The weak Van der Waals bond between these clusters can be mapped onto an anhar- monic soft spring (spring constant CI). The compressibility of the stiff C60 fullerene cluster itself can be described by a stiff anharmonic spring (spring constant Cg) [From Y. Wang, D. Tomanek, and G. F. Bertsch, Phys. Rev. BR 44, 6562 (1991)]. ................... ix 60 66 68 4.2 4.3 4.4 4.5 (3.) Binding energy of hexagonal graphite (with respect to isolated lay- ers, per carbon atom) as a function of the interlayer spacing d. The solid line represents a modified Morse fit [Eqs. (4.1) and (4.2)] to ab initio LDA results of Overney et al, J. Phys. C 4, 4233 (1992). (b) Negative gradient of the energy given in (a), corresponding to the in- terlayer force [From Y. Wang, D. Tomanek, and G. F. Bertsch, Phys. Rev. BR 44, 6562 (1991)]. ....................... (a) Interaction energy between two C60 fullerene clusters as a func- tion of the closest approach distance d. (b) Negative gradient of the interaction energy in (a), corresponding to the pairwise force between neighboring C60 clusters. (c) Binding energy of an isolated C60 fullerene cluster as a function of the cluster radius R. ((1) Negative gradient of the binding energy given in (c). Note the difference in scales between (b) and (d) [From Y. Wang, D. Tomanek, and G. F. Bertsch, Phys. Rev. BR 44, 6562 (1991)]. ....................... (a) Binding energy of fee-fullerite (per C60 cluster, with respect to isolated carbon atoms) as a function of cell volume V. (b) Pressure dependence of the equilibrium cell volume V of fullerite. (c) Pressure dependence of the bulk modulus B of fullerite (solid line), as compared to diamond (dashed line, from Yin et al, Phys. Rev. Lett. 50, 2006 (1983)) [From Y. Wang, D. Tomanek, and G. F. Bertsch, Phys. Rev. BR 44, 6562 (1991)]. .......................... Phonon dispersion relation u(k) of bulk fullerite with fee structure [From Y. Wang, D. Tomanek, and G. F. Bertsch, Phys. Rev. BR 44, 6562 (1991)]. ............................... 81 83 84 4.6 4.7 4.8 4.9 4.10 4.11 Born—Haber cycle used to predict the formation enthalpy AH? of (a) donor and (b) acceptor C60 fullerite intercalation compounds [From Y. Wang, D. Tomanek, G. F. Bertsch, and R. S. Ruoff, Phys. Rev. B (1993)]. .................................. Predicted equilibrium lattice constant a, bulk modulus B, and forma- tion enthalpy AH? for C60 fullerite intercalation compounds ACso(fcc structure). Results are presented for elements A from the 1A, 2A, 6A and 7A groups of the periodic table [From Y. Wang, D. Tomanek, G. F. Bertsch, and R. S. Ruoff, Phys. Rev. B (1993)]. ........ Predicted equilibrium lattice constant a, bulk modulus B, and forma- tion enthalpy AH? for C60 fullerite intercalation compounds A3C60(fcc structure). Results are presented for elements A from theilA, 2A, 6A and 7A groups of the periodic table [From Y. Wang, D. Tomanek, G. F. Bertsch, and R. S. Ruofl', Phys. Rev. B (1993)]. ........ Predicted equilibrium lattice constant a, bulk modulus B, and forma- tion enthalpy AH}J for C60 fullerite intercalation compounds A6C60(bcc structure). Results are presented for elements A from the 1A, 2A, 6A and 7A groups of the periodic table [From Y. Wang, D. Tomanek, G. F. Bertsch, and R. S. Ruoff, Phys. Rev. B (1993)]. ........ Phonon band structure of (a) KCGO, (b) K3C60, and (c) Rbngo [From Y. Wang, D. Tomanek, G. F. Bertsch, and R. S. Ruoff, Phys. Rev. B (1993)]. .................................. Phonon density of states (solid line) and integrated density of states (dashed line) of K3C50 [From Y. Wang, D. Tomanek, G. F. Bertsch, and R. S. Ruoff, Phys. Rev. B (1993)]. ................ xi 5.1 5.2 5.3 5.4 The Graphs for (a) the second order, and (b) the fourth order pertur- bation theory expressions in Eqs. (5.4) and (5.5). —t- represents a particle state and -t— denotes a hole state. .............. Free response (a) and RPA response (b) of C60 clusters to an external electromagnetic field (solid line). The sharp levels have been broadened by adding an imaginary part hr] = 0.2 eV to the energy. The dashed line indicates the integrated oscillator strength. (c) Observed pho- toabsorption spectrum of C60 in solution [From H. Ajie et al, J. Phys. Chem. 94, 8630 (1990) and G. F. Bertsch, A. Bulgac, D. Tomanek, and Y. Wang, Phys. Rev. Lett. 67, 2690 (1991)]. ........... Dipole response of C60 clusters to an external electromagnetic field, shown in an expanded energy region. (a) Free response, (b) RPA re- sponse based on the charge term D9), and (c) RPA response based on both the charge and the dipole terms D9) and D12) in Eq. (5.6). (d) Interacting response of a thin jellium shell, describing the electron— electron interactions in LDA. The response function is given by the solid line, and the integrated oscillator strength is shown by the dashed line [From G. F. Bertsch, A. Bulgac, D. Tomanek, and Y. Wang, Phys. Rev. Lett. 67, 2690 (1991)]. ................... Observed C30 photoion yield as a function of photon energy displaying excitation of the giant plasmon resonance [From I. V. Hertel et al, Phys. Rev. Lett. 68, 784 (1992)]. .................... xii Chapter 1 Introduction A cluster is a stable aggregate of atoms or molecules, the size of which can range from few atoms to a vaguely defined limit of thousands of atoms or molecules. Clusters seem to be the bridge connecting the discrete atomic and molecular limit and the continuum crystal limit. These systems have many unique and interesting properties, such as unconventional atomic packing [1], electronic shell structure [2], collective electronic excitations [3], magnetic properties vastly different from the bulk [4]. Clusters have attracted many researchers from different disciplines, and a great effort has been devoted to understand their physical properties [5, 6, 7, 8, 9, 10]. Cluster science is a fascinating new field and one of the fastest growing fields in the past ten years. For a long time, cluster research concentrated on alkali metal clusters exhibiting a rich variety of phenomena, many of which have been explained by an electronic shell model taken from nuclear physics [2, 6, 11]. A more timely and exciting example of cluster research is the C60 molecule, which aroused a tremendous research activity and many speculations about potential applications in the past two years. Five years after it was first proposed in 1985 [1], it was crowned “molecule of the year” in 1991 by the Science magazine. C60 is the first existing example of a homonuclear molecule that has the icosahedral symmetry [12]. These moleculars, packed into a crystal, have established a third form of crystalline carbon, based on the C60 cluster, which is called “fullerite” [13]. Atomic clusters can be composed of a single atomic species, as homonuclear clus- ters, or a mixture of elements, composing a heteronuclear aggregate. Many different types of clusters have been investigated. Among them, the structural and thermal properties of noble gas clusters, such as argon clusters [14], have been studied. Clus- ters composed of metallic elements, such as alkali and alkaline earth elements, have attracted great attention, and many theoretical [10] and experimental [6] studies have been devoted to understand their behavior. Clusters with nonmetallic elements, such as silicon and carbon, have many interesting properties and potential applica- tions [1, 15], in the semiconductor technology and in novel materials based on carbon clusters. As mentioned before, clusters exhibit properties of both molecules with a discrete spectrum and the bulk with a continuum spectrum. These properties include the electronic structure [10], response to an external electromagnetic field [3], magnetic [4] and thermal properties [16]. In this work I will discuss the atomic packing and electronic properties of clusters, as well as their response to external perturbations (pressure, external field, temperature). The stacking of atoms in a cluster is crucial for the stability of the cluster. Since clusters have a large fraction of surface atoms, and the environments of the bulk and surface atoms are quite different, one cannot simply consider them as a piece of the crystalline structure. The structure of a cluster will determine the formation process, the thermal properties of the cluster [16], the evaporation and dissociation patterns [17]. It will also have an effect on the electronic structure [18]. The electronic structure plays a major role in the bonding between the atoms. Hence it affects the cluster stability, the interaction of the cluster with other atoms or the embedding matrix, its electromagnetic response, and other properties. Since one is dealing with finite Fermi systems, quantum size effects, such as shell structure of the electrons, can be seen. The character of electronic excitations in a cluster will be reminiscent of both the discrete single—particle excitation spectrum in the atomic limit and the collective plasmon behavior of the electron gas in the bulk limit. In this Thesis, I will focus on describing the equilibrium atomic structure and opti- cal response of small alkali metal clusters, namely Liz, N a2, and Nag, and a prominent carbon cluster, namely the C60 molecule. The Thesis is organized as follows. In the remaining part of this Chapter, I will present the basic concepts used in this field, which will provide the background for further discussions. In Chapter 2, I present the theoretical tools I used in this study. The equilibrium structure of metal clusters will be studied by state-of-the-art ab initio LDA calculations. The optical response, mainly due to the collective motion of the delocalized electrons, will be described using the Random Phase Approximation (RPA). The corresponding results are pre- sented in Chapter 3. In Chapter 4, I will present results for the structural properties of fullerite, a novel bulk material based on the C60 cluster, and will also address the stability of fullerite intercalation compounds. Finally, in Chapter 5, I will present results on the static polarizability, hyperpolarizability, and the optical response of C60, obtained using a tight—binding Hamiltonian. 1.1 Alkali Metal Clusters In this Section, I will discuss some basic concepts related to alkali metal clusters which will be useful for subsequent discussions in Chapter 3. Alkali metal clusters are one of the best studied types of clusters so far, both theoretically and experimentally [6]. Among the alkali metal clusters, the sodium systems have been most thoroughly investigated [2]. Sodium is heavier than lithium which may exhibit nuclear quantum effects [19]. The heavier potassium has many stable isotopes, which makes the interpretation of mass abundance spectra difficult. Knight and coworkers [2, 20] found that the shell structure in sodium clusters is more pronounced than in potassium clusters, due to the larger Fermi energy (Fermi energy in the free electron model is inversely proportional to the squared Wigner—Seitz free electron radius r,). This fact implies unique features for the stability of Na clusters. In Li clusters, the electronic shell structure is also not well pronounced for the following reason. The large optical electronic mass in bulk lithium m“ = 1.56mc [21] indicates a strong electron—phonon coupling, which would make a static model of the ion lattice questionable for the lithium clusters. Therefore, sodium clusters are the simplest object to study if one wants to understand the collectivity of electronic excitations and, at a later stage, the coupling of electronic excitations to nuclear degrees of freedom. 1.1.1 Cluster synthesis and experimental techniques A major experimental achievement in cluster physics is that now one can generate and investigate free clusters. This comes as a significant advantage when compared to the previous experimental setup, where clusters were investigated in a supporting ma- trix [22]. The latter experimental setup does not allow for a clear distinction of cluster features from those of the matrix [23]. More important, the cluster—matrix interac- tion can modify cluster properties significantly. With the development of molecular beam techniques, one is able to produce and detect free clusters over a large range of size and to analyze their basic properties [2, 24]. Alkali clusters can be produced in oerecroe uv some: oer-Leonora om moms "“55 , A H-J C2: 1 _fi Cl A CZ C3 Ifigure 1.1: Typical experimental setup for cluster spectroscopy: Cluster source and t’Lne~Oi'-flight spectrometer [From de Heer et al, Solid State Physics 40, 128 (1987)]. sufficiently large numbers to obtain statistically relevant information. In Figure 1.1, I show a typical experimental setup for generating and analyzing cluster beams. At the far left is a supersonic nozzle source, in which the bulk metal is evaporated and the metal vapor mixes and condenses in a cold inert carrier gas. At the exit of the source, the mixture is ejected into the vacuum via a pinhole nozzle. The metal vapor undergoes an adiabatically rapid cooling, and condenses into clusters. Following the source, the cluster beam goes through a series of collimating slits and mechanic choppers. A laser light is used to ionize the cluster. Cluster ions are subsequently accelerated to generate a time-of-flight mass spectrum. 1 . 1 .2 Structural properties The equilibrium geometry is an important property of a cluster. Because of the sig- nificant fraction of low coordinated surface atoms, the cluster is not just a piece of the bulk lattice. The ground state geometry is crucial for the stability of the cluster and will also determine the other properties of the cluster, such as the polarizability [18], frequency of the collective electronic response [25], and the fission patterns [17], just to name a few. Since present experimental tools cannot tell us directly where the nuclei are located in the free clusters, we have no direct information on regarding equilibrium structure of a cluster. Thus, the knowledge of the equilibrium structure of a cluster relies on theoretical calculations. Many investigations have addressed this question mostly using ab initio methods. Considerable progress has been made despite the substantial difficulties associated with this approach [18, 26, 27, 28]. Among the ab initio techniques, one can list the HF—CI calculations (Hartree— Fock plus configuration interaction) [27], pseudopotential DFT (Density Functional Figure 1.2: Equilibrium geometries of small sodium clusters [From Bonacic et al, Phys. Rev. B 37, 4369 (1988) and Moullet et al, Phys. Rev. Lett. 65, 476 (1990)]. Theory) calculations based on the LDA (Local Density Approximation) [29], LSD (Local Spin Density) calculations [18], and calculations based on the Car—Parrinello method (unified density—functional theory and molecular dynamics) [28]. It has been found that there are generally many different geometries with a very similar total energy. Therefore each cluster has possibly several coexisting isomers. In general, most of the calculations agree with each other as far as the equilibrium geometry is concerned. Minor disagreements are found typically in some details such as the exact bond length. In Figure 1.2, geometries of different small sodium clusters are shown. From Nag to N as, the clusters have a planar structure. N at has a rhombic equilibrium geometry. Nas has an almost trapezoidal shape. N36 is a relatively flat pentagonal pyramid, as stable as a planar structure with the D3}, symmetry. Na7 exhibits the first 3—D structure in the sequence, a pentagonal bipyramid. For N as, the energetical lowest geometry is the T4 geometry with four caps attached to the four faces of an inner tetrahedron. 1.1.3 Electronic shell structure The mass abundance spectrum of sodium clusters, shown in Figure 1.3(a), revealed interesting new physics. Clusters with a certain number of atoms (N =2, 8, 20, 40, 58, 92, )..., are much more abundant than their neighbours, indicating a remarkable stability of these clusters. The above mentioned numbers are called “magic numbers”, in analogy to the magic numbers of nucleus associated with very stable nuclei. This observation shows that one is dealing with finite spherical Fermi systems, similar to nuclei. The magic numbers are determined mainly by the number of valence electrons in the clusters [11], corresponding to the closure of an electronic shell. This is con- (a rll wumwi . . 11.13.”: [m 92 Counting rate 1.5 .1 1° (61‘- 1.2- > 9, . . 2: 503- < ' . I 0'" 1t ' as E. ' td a < o. i -0.4LZ111...1LLJ.1L1..iLLL.1.L.LLL1.Ln1 8 20 34 58 70 Numberotsodunetomsperohsterm Figure 1.3: Mass abundance spectrum of N aN clusters. (a)Mass abundance spectrum of N aN clusters, N =4—75. The inset corresponds to N =75—100. (b) The calculated sec- ond derivative A2(N) of the total energy E(N) of jellium clusters, defined in Eq. (1.1), as a function of cluster size [From Knight et al, Phys. Rev. Lett. 52, 2141 (1984)]. 10 firmed in the abundance spectrum of the positively charged potassium clusters [30]. In all alkali clusters, the magic numbers N =2, 8, 20, 40, 58, 92, ..., are common. For other elements, the magic numbers can be substantially different. The shell structure underlying the abundancies in the mass spectra, results from quantum size effect in these systems. To understand the shell structure of clusters, detailed quantum calculations are necessary. Calculations of the electronic structure of a cluster are a challenging subject. The quantum chemistry approach typically starts with the Hartree—Fock (HF) method, which is based on the independent elec- tron approximations and neglects the correlation between spin—up and spin-down electrons. A more sophisticated approach is based on configuration interaction (CI) calculations, which is exact and includes all correlation effects if the full CI expansion is included. HF can be done routinely for small clusters, but CI calculations can only be performed only for few fixed geometries due to the large number of configurations necessary for a converged calculation. HF—CI calculations are feasible only for small clusters (less than 10 atoms with the nowadays available computational resources). On the other hand, physicists developed the Local Density Approximation (LDA) which is based on the Density Functional Theory (DF T) (the details will be discussed in Chapter 2). The Local Density Approximation to the DF T involves a local potential which mimics correlation effects in a mean-field framework, i. e., it is superior to HF. Compared to CI calculations, LDA calculations are easier but still give accurate ground state geometries. LDA is a powerful tool to calculate the total energy of a quantum system with a given atomic geometry, but is computationally intensive. The simplest theoretical approach is the so—called jellium model, which has been used in solid state physics and nuclear physics for a long time. In the jellium model, the ions are smeared out to form a jellium of positive charges. The electrons move in 11 a potential of spherical symmetry generated by the jellium background. The jellium model is especially useful for large clusters, where the symmetry lowering due to the ions makes full ab initio calculations computationally prohibitively demanding. This simple model provides a good semi—quantitative understanding of different aspects of the physics governing the stability and electronic excitation spectra [11]. The requirements for a meaningful applicability of the jellium model are that the electrons be delocalized and the positions of the ions do not play an important role. It is generally agreed that this is the case in most alkali metal clusters [6]. Jellium calculations assume that electrons are moving in a spherical potential well. All the energy levels are obtained by solving the Kohn—Sham equations (see Chapter 2) self—consistently. The electrons fill the levels in ascending order with the constraint of satisfying the Pauli principle. Clusters with fully occupied shells of electrons will have a lower total energy than neighbouring clusters, and are therefore more stable. A quantity A2 has been defined to measure the relative stability of the clusters, A2=E(N+1)+E(N—1)—2E(N) (1.1) where E(N) is the total energy of electrons in spherical jellium with N electrons. A2(N) is the second derivative of the total energy E(N), and represents the relative binding energy change for clusters with N atoms compared to clusters with N +1 and N -— 1 atoms. A peak in A2(N) indicate that the cluster with N atoms is relatively stable than its neighbors. In Figure 1.3(b), A2 is shown for sodium clusters as a function of the cluster size. As we can see, the trends clearly match the observed mass abundance spectrum of sodium clusters, shown in Figure 1.3(a). 12 1.1.4 Optical response The response of a system to an external time dependent electromagnetic field is called optical response. There are essentially two experimental techniques to measure the optical response of free clusters. Namely the depletion technique and the resonant two—photon ionization technique. In addition to the setup for the generation and detection of clusters, shown in Figure 1.1, the depletion technique uses a pulsed laser which propagates collinearly against the molecular beam. At resonance, the cluster beam is depleted due to the dissociation of the excited clusters which have absorbed a photon from the pulsed laser beam. The resonant two—photon ionization uses the first laser (pump) to pump the cluster and the second laser (probe) to further ionize it, before the cluster beam enters the time-of-flight mass spectrometer (in a typical experimental setup, only one laser is used to pump and probe the clusters at the same energy). The mass spectrum provides a direct measure of the cross section as a function of the laser beam energy. In Figure 1.4, I show the observed optical response of the Nag cluster using the depletion technique [31]. In this spectrum we can see a single prominent peak at 490 nm (2.53 eV). This peak is due to a collective excitation of the valence electrons. Its oscillator strength exhausts almost the entire Thomas—Reiche—Kuhn (TRK) sum rule ( f —sum rule), which indicates that essentially all the valence electrons participate in the resonance. As I will discuss below, the physics of this transition is very similar to the dielectric response of a classical metal sphere. In 1908, Mie [32] studied the optical response of small systems. In particular, he studied the optical response of a metal sphere to an external electric field with a wavelength much larger than the sphere radius. Under the influence of an external field, the negative charges move back and forth with respect to the positive charge 13 12.00 8 ALAAALAAAJJJLAJL cross section (sq. A) .e 8 lnLALLAALALLLA '8 500.0 500.0 706.0 000.0 wavelength (nm) E Figure 1.4: Optical response of the N as cluster using the depletion technique. [From Wang et al, Chem. Phys. Lett. 166, 26 (1990)] 14 background, as illustrated in Figure 1.5. Mie established that the collective motion of the electrons, the so—called “Mie plasmon”, solely depends on the density of the electrons n, as 2 41rne2 w . = —— M3: (1.2) 3mc Here, me is the mass of the electrons and e the electron charge. The factor 3 in the denominator occurs due to the spherical geometry. Since the rigid displacement of the electronic charge on a constant spherical ionic background results in a nonvanishing charge density only close to the surface, this collective motion is sometimes called the surface plasmon of the cluster. In applying this classical model to clusters, two questions arise immediately. The first one is how large should the cluster be to support a collective electronic motion. As I will discuss later, even very small clusters can exhibit collective excitations. The second one arises from the fact that in this simple classical description the energy of the plasmon is independent of the cluster size and is totally determined by the free electron density of the cluster. Will the optical response of a cluster be peaked at the same energy, independent of cluster size? The answer is no. The energy of the col- lective excitation has a size dependence. In Figure 1.6, I show the experimental data showing the dependence of the excitation energy on the size of sodium clusters [33]. The frequency shifts from 2.4 eV in small clusters of about 10 atoms to the value 3.4 eV near the bulk limit. The energy of the collective mode obtained from the Mie formula overestimates the plasmon energy in small clusters. The discrepancy cannot be explained without a detailed calculation. Its origin is speculated to be partly due to the inadequacy of the spherical jellium model, partly due to the quantum effects, such as the spill-out of electrons which is neglected in the Mie theory, and finally the electronic structure of the cluster which can be significantly different from jellium Figure 1.5: A classical picture of the Mie plasmon. (a) If no electrical field is applied, the positive and negative charge background coincide with each other. (b) Under an external electrical field with a particular frequency, the negative charge background will move back and forth with respect to the positive background. Since the collective motion results in a nonvanishing total charge density only near the surface, it is often called the surface plasmon of the cluster. 16 spheres. The first improvement over the classical Mie description are RPA calculations for spherical jellium [34, 35, 36, 37]. Assuming that the position of the ions can be neglected, the electrons are assumed to move in a spherical potential well. Particle— hole interactions in the excited states are considered in RPA. LDA—RPA results for spherical jellium show a weak size dependence of the excitation energy [36]. Within the ab initio approach, CI calculations of electronic excitations have been carried out for small sodium clusters [38, 39]. Due to the large number of electrons, the calculations have been done with empirical core potentials. Since the computa- tional load is large, the CI calculations have been performed only for several selected geometries, and the expansion space has been limited to low energy states. Therefore, the results can only give semiquantitive answers [38]. To overcome these difficulties and to get a quantitative understanding of the ex- perimental results, we adopted the ab initio LDA—RPA approach. We describe the ground state and the single particle spectrum using LDA, and calculate the electronic excitations using RPA. As I have mentioned in Section 1.1.2, LDA has the advantage of a local exchange—correlation potential, hence it includes some correlation effects. It has been shown that LDA gives reliable results for the structure of sodium clus- ters [18]. RPA is a linear response theory, and is the small amplitude limit of the time—dependent LDA. It considers the one—particle one—hole interactions and is most suitable for a large number of active electrons. Our ab initio results for electronic excitations in alkali clusters will be presented in Chapter 3. According to the simple Mie classical picture discussed above, the optical response spectrum of a cluster exhibits only one single sharp peak. But due to different damp- ing mechanisms, a broad peak is observed in the experiment. For example, in the 3.0 3.2 A > 3 a. 2.0 3 a4 a0 17 r' r f 1 f T r r ' r r NaBulkLimit. ___[. .. O O O O O I O O O O b ”fiz‘...’ . .e e e_I 0’00” 45" + b .‘. 4 O q N0 Atomic Resonance . L 1 L l 1 l 1 L L 1 A 50 too too 200 200 300 (N) Average‘Number Atoms / Cluster Figure 1.6: Dependence of the collective electronic excitation energy on the cluster size in N aN clusters. Energies derived from reflectivity change spectra (solid circles) and energies calculated via sum rule from experimental static polarizabilities (open circles) are compared with jellium calculations (crosses) [From Parks et al, Phys. Rev. Lett. 02,2301 (1989)]. 18 case of Nag, the observed width of the plasmon peak is about 0.25 eV, as shown in Figure 1.4. There are three different known mechanisms which give rise to a fragmentation of the plasmon, namely: fragmentation due to a static shape deformation, Landau damping, and the coupling of electronic excitations to the vibration modes. The static fragmentation due to shape deformations, illustrated in Figure 1.7(a), occurs simply due to the fact that the cluster is asymmetric. For a cluster with a magic number of electrons, the shape is spherical in the jellium model. Non—magic clusters are likely to be prolate or even oblate ellipsoids in the jellium approximation in analogy to nuclei. According to the Mie theory, for a non—spherical geometry, a single resonance frequency will be split into two or three distinct peaks [25]. The oscillator strengths will also be divided into the corresponding parts. If the splitting is small and the levels are indistinguishable, the absorption will be observed as a single broad peak. If there is a large density of electronic states near the highest occupied molec- ular orbital (HOMO) (close to the Fermi energy), damping can also be caused by particle—hole excitations which can build up a collective state. This will smear out the spectrum, as shown in Figure 1.7(b). This mechanism is called Landau damp- ing [40]. Landau damping is not important for the small clusters since the energy levels are well separated in energy and particle-hole excitations unlikely. For large clusters, however, Landau damping is the major plasmon fragmentation mechanism, since the energy levels are very dense. There is also damping due to the coupling of electronic excitations to vibration modes. This mechanism is very similar to the Frank—Condon effect in molecular physics. This effect is illustrated in Figure 1.7(c), where I plot the electronic levels 11 in (b)/'\D(E} IKE) [-1 _ j l (C) Al E Excited State. If E) d. (a) Static Figure 1.7: Damping mechanisms for collective electronic excitations. fragmentation due to aspherical shape. (b) Landau damping. (c) Electron—vibration coupling. 20 as a function of the generalized coordinate. The lowest curve represents the ground state total energy EM, which can be obtained by LDA. The other curves are given by Eta: + (RPA, where (RPA is the electronic excitation energy. In the Frank—Condon model, the configuration coordinate remains unchanged during the transition, since the electronic excitation is much faster than the motion of the ions; we can speak of an adiabatic process. The probability density distribution of the nuclear coordinates due to the ionic zero point motion will spread the energies of the electronic excitations. This latter mechanism can also give rise to thermal line broadening of the plasmon. As we discussed above, the vibrational zero point motion couples to the electronic excitations and hence broadens a given transition even at T = 0 K. At higher temper- atures, the thermal energy can activate higher vibration modes, which increase the shape fluctuations of the cluster. This is the so—called thermal line broadening. The line width resulting from this mechanism has been estimated for the jellium model, and the temperature dependence of the broadening has been found to be propor- tional to \/T [25], where T is the temperature of the cluster. At zero temperature, the linewidth will be due to zero—point motion only. In Chapter 2, I will address these mechanisms again for the interpretation of the results for the equilibrium structure, collective electronic excitations, and their damping in N32 and Nag, 1.2 Carbon Clusters In the last years, carbon clusters have been investigated extensively both experimen— tally and theoretically, by chemists and physicists alike [8]. The importance of carbon clusters in chemistry is evident, since the whole organic chemistry is based on carbon compounds. A good understanding of carbon composites is a prerequisite for gain- 21 ing insight into chemical reactions involving organic compounds. In physics, one of the original motivation for an investigation of carbon clusters is the still unanswered question regarding the absorption spectrum of the interstellar dust. An unexplained absorption line is believed to be caused by carbon and its clusters [50]. The study of carbon clusters and their interaction with other elements in the laboratory could provide us with a unique tool to gain further understanding of the universe. One family of carbon clusters is represented by the fullerenes [41]. Fullerenes designate a group of carbon clusters with a hollow cage structure and only pentagons and hexagons on the surface. From Euler’s rule for closed polyhedra (C — E + F =2, where C, E, and F are the number of corners, edges, and faces, respectively), one can deduce that the number of pentagons in any fullerene is 12. The best known member of this family, which has been identified so far, is the C60 cluster [1], the so—called buckminsterfullerene. This name was inspired by the well known architect Buckminster Fuller, who designed geodesic dome structures. The study of C60 clusters lead also to the discovery of a novel crystalline material fullerite [13], a crystal composed of C60 clusters. F ullerite is the third pure crystalline form of carbon, in addition to diamond and graphite. One of the aims of this work is to study in detail the structural and electronic properties of the C60 cluster and the C60 solid. In the following sections 1 will re- view the basic properties of the C60 molecule and of fullerite; these concepts will be necessary for the discussion that follows in Chapters 4 and 5. 1.2.1 Synthesis of C60 and the C60 crystal C60 was first synthesized using laser vaporization of graphite [l]. The experimental setup is shown in Figure 1.8. The method can be summarized as follows: graphite is 22 vaporized by a laser, the generated carbon plasma is carried to a nozzle by helium gas which provides the environment to cool the gas. The cooled carbon gas aggregates into clusters, which are ejected from the nozzle in a supersonic molecular beam. The molecular beam is analyzed using the time—of—flight mass spectrometer. The outstanding characteristic of the mass abundance spectrum, shown in Figure 1.9, was that a single cluster with 60 atoms was observed to be far more abundant than any other cluster size. The high stability of this cluster with 60 atoms has been postulated to result from a closed cage structure, shown in Figure 1.10. In the mass abundance spectrum, the C70 cluster turned out to be very abundant as well, hence probably more stable than the neighboring clusters. The above graphite vaporization method produces only free C60 clusters which are hard to analyze in any detail. In 1990, however, a new method [13] was developed, which is able not only to produce C60 clusters in bulk quantities but can also produce the crystalline fullerite solid. The experimental setup is depicted in Figure 1.11. Graphite rods are butted together and a high current passes through. Carbon vaporizes in the vicinity of the contact and in the helium gas the carbon plasma condenses into soot. The soot contains a large fraction of C60 clusters and also a sizable amount of C70 clusters. Since C60 and C70 can be dissolved in benzene, they can be separated from the soot. C60 can be separated from C70 by high—pressure liquid chromatography. Drying the solvent, one obtains the C60 fullerite crystal. 1.2.2 Structural properties of C60 and the C60 crystal Since a single crystal consisting of pure C60 clusters can be produced, crystalgraphic techniques can be used to determine not only the lattice structure but also the struc- 23 Vaporization laser Integration cup Figure 1.8: Schematic diagram of the pulsed supersonic nozzle used to generate carbon cluster beams [Horn Kroto et al, Nature 318, 162 (1985)]. 24 W 44 52 60 68 76 84 N0. 01 carbon atoms per cluster Figure 1.9: Experimental mass spectrum of carbon clusters. [From Kroto et al, Nature 318, 162 (1985)] 25 Figure 1.10: Structure of the C30 “buckyball” cluster. 26 Figure 1.11: Mass production technique for fullerite. [From Huffman, Physics Today, 44, 22 (1991)] 27 ture of the cluster itself. Based on X-ray diffraction data [13], fullerite has been shown to be a close-packed molecular solid with a face-centered cubic structure and with a nearest neighbor distance D = 10.04 A. Raman and infrared (IR) spectroscopy data [42, 43] confirm that the “soccer ball” structure of C60 is preserved in the solid. The C60 cluster has an uncommon hollow cage structure, resembling a soccer ball, composed of 12 pentagons and 20 hexagons. It is a truncated icosahedron and hence has I h symmetry. N uclear—magnetic—resonance (N MR) experiments [44] show a single peak in the absorption spectrum at room temperature, which indicates that all the carbon atoms in the cluster are equivalent and that the clusters rotate in the lattice. C60 is a low strain structure; it is nearly spherical, and all pentagons are separated by hexagons. Because of this geometry, carbon atoms have neither pure sp2 nor sp3 bonding, which are typical of graphite and diamond, respectively. In the hexagonal bonds the p; electrons, which are locally perpendicular to the surface, form a resonant 7r bond, while in the pentagons, an anti—resonant 7r bond is formed. The result is that the bond length in the hexagons is shorter than that in the pentagons. The chemical bonds in the pentagons are called “single” bonds and those remaining in the hexagons “double” bonds. Extended-X-Ray-Absorption-Fine-Structure (EXAF S) data [42] indicate that the single bond length is 1.45 A and the double bond length is 1.40 A. The average carbon-carbon nearest-neighbour distance in C60 Clo—c = 1.42 A, the same as in graphite. This bond length corresponds to a radius R = 3.55 A of the C60 fullerene cluster. Because of the large lattice constant of the fullerite crystal, the closest distance between the two surfaces of adjacent clusters is d = 2.9 A at zero pressure, somewhat smaller than the interlayer spacing in graphite, d = 3.35 A. Due to this large equilib- rium separation between C60 clusters, their mutual interaction is mainly via Van der 28 Waals forces, which are weak. However, the C60 cluster itself is a rigid object, due to the covalent bonding between the atoms. Pressure dependent X-ray diffraction data, obtained in a diamond—anvil cell, indicate a large change of the bulk modulus of ful- lerite as a function of pressure [45]. Speculations that the stiffness of fullerite might exceed that of diamond, the hardest material known, will be addressed in Chapter 4. Perhaps the most exciting property of C60 fullerite is superconductivity which occurs when the crystal is doped. Following the discovery of superconductivity in K3C60 with a critical temperature Tc = 18 K [46], new compounds have been synthe- sized using a large variety of intercalants [47], yielding critical temperatures as high as 33 K in C82RbC60 [48]. While Tc values of doped fullerite are still below those found in high-Tc cuprate perovskite superconductors, intercalated fullerite shows su- perior material properties and hence bears a higher potential for applicability. The intercalation process and the rigid—band behavior of intercalated fullerite resembles in many ways the extensively studied graphite intercalation compounds. The crucial property of fullerite intercalation compounds Ame is their stability against decom- position into the components in the standard state, i.e. C60(solid) and A(solid). The formation enthalpy is of interest not only for the donor compounds mentioned above, but also for potential acceptor compounds. The stability of intercalated fullerite will be addressed in Section 4.2. 1.2.3 Electronic properties of C60 and the C60 crystal In Figure 1.12, I show the energy spectrum of the C60 cluster. A large energy gap of 2.2 eV contributes to the extraordinary stability of the C60 cluster. The highest occupied molecular orbital (HOMO) is five fold degenerate and the lowest unoccupied molecular orbital (LUMO) is three fold degenerate. The levels near the Fermi level are N CD (a) _ lo - — — = = — — 5 - — — 9 o i. = (LUMO) Cal -0 - — -10 - —_,, ~15 ' = :2 -20 L — — — l t .., A E(eV) g - ---- ----- - """ ——Du(HOMO) -3 1' -..ccmccc— h. -4 r 1 9“ -5 i- II Figure 1.12: (a) Single—particle energy level spectrum of a C60 cluster, as obtained using the tight—binding Hamiltonian described in Section 2.3. The levels have been sorted by symmetry. (b) Expanded region of the energy level spectrum near the Fermi level. Allowed dipole transitions between states with gerade (g) and ungerade (u) parity are shown by arrows [From G. F.Bertsch, A. Bulgac, D. Tomanek, and Y. Wang, Phys. Rev. Lett. 67, 2690 (1991)]. 30 all formed by the 1r system of electrons. The electronic states of C60 have a definite parity, and optical excitations follow dipole selection rules. As a consequence, the HOMO to LUMO dipole transition is parity forbidden. An interesting feature of the energy level spectrum is that the levels can be grouped into rotational bands characterized by the angular momentum L. Due to the 1;, symmetry of the cluster, the 2L + 1 fold degeneracy of energy levels in the rotational band with angular momentum L is broken. The representations of the 1;, group give as possible degeneracies 1(a), 3(t), 4(g), and 5(h). A recent photoemission experiment [49] shows that the spectrum of the C60 solid contains remarkably sharp lines, indicating a small band dispersion. This small band dispersion results from the large separation between the clusters. The energy bands are derived from the C60 orbitals. Their isolated molecular character is most pro- nounced for the deeply bound states. Solid C60 is a semiconductor with an indirect energy gap of 1.9 eV. The cubic symmetry of the crystal field splits the molecular orbitals of isolated C60, For example, the hu HOMO level is split into a three—fold and a doubly degenerate band. Because of the three dimensional resonant 7r system of the C60 cluster, which is very similar to that of conjugated polymers, a nonlinear behavior of the electromag- netic response is expected. This topic is the subject of Section 5.1. 1.2.4 Optical response of C50 and the C60 crystal The interest in the optical response of C50 originates in certain unexplained features of the absorption and emission spectra of interstellar dust [50]. The spectrum can be used as a unique tool to identify the components of the interstellar matter, which have not been accounted for so far; this will greatly improve our knowledge of the 31 optical density x20 zo'bmm'éb'o' ' Xbb'f'm 565 ' ' ' "560m ' ' ' ‘7'60' ' ' ' ' “0'60 wavelength (nm) Figure 1.13: Optical response of the C60 cluster [From H. Ajie et al, J. Phys. Chem. 94,8630 (1990)]. 32 outer space. After the hollow cage structures for C60 was proposed, many theoretical studies predicted the energy spectrum and optical response of C60, The HOMO to LUMO dipole transition is forbidden by symmetry and the lowest dipole allowed transition is a HOMO to LUMO+1 excitation. This has been first suggested in a semiempir- ical CI calculation [51], which has shown that the first allowed transition occurs at 3.6 eV, with an oscillator strength of 0.08. This motivated the experimental work of Heath et al [52]. They found the lowest transition to occur at 3.22 eV with a much smaller oscillator strength 0.004, more than one order of magnitude smaller than the theoretical results. This discrepancy is one of the subjects of our study presented in Chapter 5. C60 is a resonant 7r bonding system, in analogy to graphite. Each atom has three neighbouring atoms. In Figure 1.13, we show the low—frequency optical response of C60 in solution [53]. 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Chapter 2 Computational Techniques In the past twenty years, the rapid development of computational facilities has pro- vided us with the ability to predict the equilibrium structure and electronic proper- ties of materials. Eventually it will be possible to design novel materials with the desired properties. This progress is made possible not only by the development of computer technology, but especially by new computational methods. Many compu- tational techniques, such as Hartree—Fock configuration interaction (HF —CI), Local Density Approximation (LDA) [1], the Car-Parrinello method [2], the embedded— atom method [3], and the quantum Monte Carlo technique [4] have been developed and have the power to accurately predict structural, electronic, and other properties of clusters. Ab initio HF —CI calculations with Gaussian orbitals (such as implemented in the GAUSSIAN code) have been used extensively, and have become a basic research tool in chemistry. Due to the large amount of computer resources required by ab initio cal- culations, many semi-empirical methods appropriate for different purposes have been developed in the meantime, such as CNDO (complete neglect of differential overlap), IN DO (intermediate neglect of differential overlap), N DDO (neglect of diatomic dif- ferential overlap) and MN DO (minimum neglect of differential overlap). Today, these 38 39 names have become common jargon in the chemistry literature. Physicists also developed many computational methods to obtain a clear physical picture of different phenomena. In this Chapter I will review the ab initio and other computational methods used in our calculations. 2.1 Density Functional Theory (DFT) In 1964, Hohenberg and Kohn [5] showed that the total energy of a system in a nondegenerate ground state is a unique functional of the total charge density p(r). This theorem forms the basis of the Density Functional Theory (DFT). Moreover, from the point view of the variational principle, the correct charge density has been shown to minimize the total energy of the system in the ground state, i. e., Elpl = Tlpl + f d? Milled?) + EH19] + Exclp] = min- (2-1) Here, T[p] stands for the kinetic energy functional, V6341") stays for the external field (e. g., the Coulomb field of the ion cores). E'H[p] gives the Hartree energy, which is the electrostatic energy of the electrons, and Exc[p] is the exchange and correlation energy functional. Since especially the general form of Ezc[p] is unknown, Kohn and Sham proposed the Local Density Approximation (LDA) to DFT as a workable scheme to determine p and the total energy [1]. 2.1.1 Local Density Approximation (LDA) Starting from the variational principle of the DFT, given in Equation (2.1), Kohn and Sham [1] developed a set of self—consistent equations which use the concept of the q‘ where In the is the IS the solved lS g1 V81 Int Placed 1 The fOI'I gas [5] '. 40 the quasiparticles in the spirit of mean field theory: {—é- v2 +v...ipi + anpl + min. = ..«i , (2.2) where the total charge density is given by m = f Ida-(Fll’. (2.3) In the first Kohn—Sham Equation (2.2), (r__l_)d W1?) /_ Ir — rlldr (2.4) is the Hartree potential, and 6Exc [Pl V..(r‘) = 6pm (2.5) is the exchange-correlation “potential”. The Kohn—Sham equations are typically solved iteratively, until self—consistency is achieved. The total energy of the system is given by Etot— — Z: 65— ép—|:Pf(;|)drdrl + Exc[p]- / ch(p p()(r"')dr. (2.6) In the Local Density Approximation (LDA), the nonlocal functional Exc[p] is re— placed by a local function Em(p), given by E..ipi as / exc(P(7"))P("')d7"- (2.7) The form of Exc(p) is assumed to be universal, given by the homogeneous electron gas [6]. The local exchange—correlation potential is then given by V..(p(i"))= ;;(exc(p (FD/’07)) - (2-8) p05 311C qua and H ere 2.1.1 A soc of Na with g detem (1116 to Valencf It is betweer TngOH o: It W011] d witV’e fl) 1] 41 Several parametrized forms of the exchange—correlation energy have been pro- posed [6]. A simple expression for the local exchange has been given by Slater [7] and the correlation energy, as parametrized by Ceperley and Alder [4, 8] based on a quantum Monte Carlo calculation, reads in atomic units 0.916 Ee:r:ch.a'n_qe = '" 1‘ (2'9) and E 1 _ —0.2846/(1 +1.0529r;/2 + 0.3334r,) for r. > 1 (210) °°"°’“"°" ‘ 0.0622 in(r,) — 0.096 + (0.004 ln(r,) — 0.0232)r, for r, < 1 . ' Here, 1', is the Wigner—Seitz radius given by 3-1‘, = p'l. (2.11) 2.1.2 Norm conserving pseudopotentials A sodium atom has eleven electrons, ten of them in the core. A full calculation of Nag, therefore, has to consider 22 electrons, and a calculation of Nag will deal with 88 electrons. This represents a very large configurational space. Moreover, the determination of multi—center integrals between the valence wave functions is difficult due to the many nodes in the core region, imposed by the orthogonality between the valence and the core wave functions. It is well known that the localized core electrons do not contribute to bonding between the atoms in a cluster. Cohesion is mainly due to the valence electrons in the region of large valence wave function overlap, which is smooth outside the core region. It would appear as desirable to discuss cohesion in atomic clusters by smooth pseudo wave functions, the eigenstates of a conveniently defined atomic pseudopotential . Thec funct insidi the t typic Tran: config syste: In abin (1 for a 42 The optimum pseudopotential yields wave functions which resemble the valence wave functions in the interesting region outside the core, and which are smooth and nodeless inside the core. A further requirement on the pseudo wave function is that it has the the same eigenvalue as the all—electron wave function. A pseudopotential is typically generated to describe best a specific electronic configuration of an atom. Transferability of a pseudopotential, i.e., its ability to describe different electronic configurations with adequate accuracy, is crucial when addressing inhomogeneous systems such as atomic clusters. In this work, I use the so—called Hamann—Schliiter—Chiang (HSC) norm—conserving ab initio pseudopotential [9]. It satisfies the following four criteria: (1) Eigenvalues associated with the all—electron and pseudo wave functions agree for a chosen “prototype” electronic configuration. (2) All—electron and pseudo wave functions agree beyond a chosen core radius re. (3) The integrated total charge between r = 0 and a given radius r for the all— electron and the pseudo wave functions agree for r > 1°C for each valence state. This property is called norm conservation, and will guarantee that the electrostatic poten- tial produced outside rc is identical for the true and the pseudo charge distributions. (4) The logarithmic spatial derivatives of the all—electron and the pseudo wave function and their first energy derivatives agree for r > rc. This will make the scattering properties of the true ion cores to be reproduced by the pseudopotential with minimum error. Additional shifts of the eigenstates with respect to the atomic situation occur due to orbital hybridization mainly outside the core region, which is described correctly by the pseudopotential . In this formalism the interaction between the valence electrons and the ion cores 43 0.5 i r . Wavefunctions ¢(r) O O "0-5 1 L 1 S 0.5 ' . 0.0 Potential (Ry) i .0 0 I 2" o I t" at Radius (a.u.) Figure 2.1: The 3s, 3p and 3d pseudo wave functions and the corresponding all- electron wave functions of the sodium atom. Outside the core radius re, the pseudo wave functions and the all—electron wave functions are the same. Inside the core, the pseudo wave functions are nodeless and smooth. At re, the spatial derivative of the all—electron and the pseudo wave functions, as well as their first energy derivatives, agree with each other. The eigenvalues associated with the pseudo wave function agree with those of the all—electron wave function. The bottom panel shows the pseudopotentials for the s, p and of states. Inside the core radius, the pseudopotentials are finite, and at large radii, the pseudopotentials approach -—e2 / r. 44 is represented by a pseudopotential. The pseudopotentials can also be angular mo- mentum dependent, and this type of pseudopotentials are called “nonlocal pseudopo- tentials”. The residual interaction between the electrons is described by the Hartree term plus a local density-dependent interaction which accounts for exchange and correlation effects. The latter are described using the parameterization of Ref. [8], which is based on electron-gas correlation energies calculated in Ref. [4]. The pseudo wave functions in a cluster are obtained using these potentials in the Kohn—Sham Equations (2.2). Then, the total energy of the atomic cluster is given by Eq. (2.6). Figure 2.1 displays the radial part of the all-electron wave functions (from an all- electron LDA calculation) and the pseudo wave functions of the sodium atom, along with the nonlocal pseudopotentials. For 1' larger than the core radius rc, both wave function are identical. For 1' smaller than re, the pseudo wave function is nodeless and smooth. The bottom panel of Figure 2.1 displays the pseudopotential. Inside the core radius re, the pseudopotential is finite and always less attractive than the full LDA potential. Far away from the core, the ionic pseudopotential takes on the —ez/r form. 2.2 Jellium model As I mentioned in Chapter 1, the simplest theoretical approach to describe the elec- tronic structure of a metallic cluster is the jellium model, which has been used in solid state physics and nuclear physics for a long time. The spherical jellium back- ground model addresses the behavior of a free electron gas in a finite system. The basic assumption is that the ionic positions do not play an important role. In the jellium model, the positive ions are smeared out as a homogeneous jelly across a finite volume, and the electrons adjust to the confining potential [10]. 45 V(eV) r(A) Figure 2. 2: Energy spectrum and self—consistent potential of the N as cluster obtained from the spherical jellium model. 46 The experimentally observed mass abundancies associated with the shell structure of sodium clusters by Knight et al. [11] can be explained by the jellium model [10]. The jellium model is especially suitable for large clusters, for which the computations are beyond the capacity of quantum chemistry methods. In this case, the jellium model proved to be very powerful and has provided us with a good understanding of quantum finite size effects [10]. Conventionally, we take the density of the jellium background as the density p of the valence electrons in the corresponding bulk material. The radius R of the spherical jellium background, similar to the nuclear radius, is given by R = r,Z‘/3 . (2.12) Here, 1', is the Wigner—Seitz radius of the bulk, given by Eq. (2.11), and Z is the total number of valence electrons in the cluster. In the case of neutral (monovalent) alkali metals, Z is equal to the number of atoms. A self—consistent solution of the Kohn—Sham equations yields the electronic structure of the cluster. The first closed—shell configuration in Na,, clusters occurs for N32. Nag has the second closed shell configuration, which also means that it is more stable than its neighbouring clusters. Since N as has a closed shell structure, we do not expect energy lowering upon symmetry reduction, due to Jahn—Teller effect. The cluster keeps spherical symmetry, so the spherical jellium model will be appropriate to use. In Figure 2.2, I show the self consistent potential and the energy spectrum of the Nag cluster, based on the spherical jellium background model [12]. The self consistent potential is very similar to the Woods-Saxon potential of nuclei. The ls and 1p levels are fully occupied. The first unoccupied level is the 1d state, separated from the 1p level by 1.52 eV. Clusters with fully occupied electron shells frhc are Sect clus lian' beco such initit adeqi In use a stabil Consnj Here, , or b1Ital Parame (LDA) and 13111 beend0 47 have a higher binding energy per atom than neighbouring clusters. They are relatively more stable and occur more abundantly in the mass spectra. 2.3 Tight—binding Hamiltonian The jellium model is best applicable to alkali metal clusters, where the electrons are delocalized. This assumption does not apply in nonmetallic clusters. In this Section, I will discuss the tight—binding Hamiltonian, which is suitable for nonmetal clusters with strong covalent, directional bonds. The usage of this semiempirical Hamiltonian for carbon clusters is motivated by the fact that ab initio techniques become enormously cumbersome when applied to anything but very small systems, such as small molecules, or infinitely large crystals. It is far from trivial to apply ab initio methods to very large but finite systems, such as the carbon fullerenes, with an adequate accuracy. In order to determine the single—particle wavefunctions and energy levels in C60, I use a tight—binding model which has been recently developed [13] to study the relative stability of different carbon cluster structures. The tight—binding Hamiltonian, which considers only the s and p valence electrons of C, is given by H = Seong,-am; + E tag(r;j)a;,,-a3,j . (2.13) at a,e,t',j Here, i labels the atomic sites and a = 3, p3, p,,, p, labels the atomic orbitals. co, is the orbital energy, and tag are the hopping matrix elements between different sites. The parameters have been obtained from a global fit to Local Density Approximation (LDA) [1, 5] calculations of the electronic structure of C2, a graphite monolayer, and bulk diamond, for different nearest—neighbor distances [13], similar to what had been done previously to describe silicon clusters [14]. The diagonal elements of this The 48 Hamiltonian are the energy levels 6, = —7.3 eV and e, = 0.0 eV. The off—diagonal matrix elements tag(r) are assumed to have a distance dependence ~ r’z. Their values for r = 1.546 A, which is the equilibrium nearest—neighbor distance in diamond, are V.” = —3.63 eV, Vm = 4.20 eV, V,” = 5.38 eV, and Vm = —2.24 eV in the Slater—Koster parametrization [15]. In this Hamiltonian, we consider those atoms as nearest neighbors which are closer than the cutoff distance rc = 1.67 A. This is the average of the nearest— and second nearest—neighbor distances in bulk diamond, and hence near the minimum of the radial distribution function. The total energy of a carbon cluster has been given as a sum of four terms, n E... = 2n... + 2: as.) + 22(2) + U 2301.- - (1?)? (2.14) «1' i=1 The first term in Eq. (2.14) is the one-electron energy of the cluster, obtained using the tight—binding Hamiltonian. The second term consists of pairwise repulsive energies E.(d) arising from nuclear repulsion and the overlap of the ion cores. E, is defined as the difference of the “exact” calculated ab initio binding energy and the tight—binding one—electron energy of C2. The third term represents corrections to the binding energy during the transition to higher (bulk—like) coordination numbers Z,-. The parameters in this term are chosen to reproduce binding energies of selected bulk-like structures. The final fourth term is an intra—atomic Coulomb repulsion arising from possible charge transfers between inequivalent sites. Zero point vibrational energies are neglected. A more detailed discussion of this Hamiltonian is given in Ref. [13]. The binding energy of a Cu cluster (with respect to isolated atoms) is then given by Beck = n Etot(C atom) -" Etot(Cn)- (2.15) 49 This tight—binding energy formula contains the essential physics which governs bond- ing in carbon structures. It addresses directionality of bonding and the electronic shell structure. It is easily applicable to very large carbon clusters, which are inaccessible to ab initio techniques, at the expense of avoiding an explicit treatment of multi- center integrals which can be important in small structures. It is expected to give a reasonably accurate interpolation between the dimer and selected bulk structures [13]. In Chapters 4 and 5, I will use this Hamiltonian to study carbon clusters. This is done in two steps. First, the equilibrium geometry of the free C60 cluster and C60 under “hydrostatic pressure” is determined. Once the geometry is given, I address the nonlinear optical properties of the C60 clusters and collectivity of electronic excitations in this system. 2.4 Random Phase Approximation (RPA) In this Section I will outline our approach to study the optical response spectrum of clusters. This approach will be used in Chapters 3 and 5. The single particle energy spectrum of a system is the basis for calculating its optical response to an external electromagnetic field. In the crudest approximation, the allowed excitations can be estimated as the energy differences in the single—particle spectrum, and their transition strength from the Fermi’s golden rule, using the dipole matrix element between the initial state and the final state. In this approximation, the configuration of all but the one electron to be excited is frozen, and only this electron is allowed to change its quantum state. This is the so—called free response of a system, and only considers single—particle transitions. This approximation completely ignores the important effect of collective dynamical screening of electronic excitations by the other electrons in the system. This collective 50 response modifies the transition energy, transition strength, hence the entire response function as compared to that obtained assuming free response. I will use the linear response theory to address the response of clusters to small perturbations. In order to describe the dynamical response to external fields, I use the Random Phase Approximation (RPA) which has been developed by Bohm and Pines [16]. RPA is equivalent to the time—dependent Hartree-Fock theory in the limit of small amplitudes. It considers one particle—one hole interactions, and hence presents more realistic results than free response. The independent-particle polarization propagator (particle—hole Green’s function) 11° defined as ace 1 1 11°02 aw) = 22¢?‘(Fx H We) . (2.16) — c,- — w H — c,- + w where Ibo is the independent—particle wave function, e,- is the single particle energy, and w is the energy of the photon (h = 1). The RPA polarization propagator is given by [12] 6 .0 nRPA(r,.=i) = n°(r,1=i)+ / / dfng3H°(F,F2)-61;z(;—:)ZHRPA(F3,W), (2.17) where V is composed of a Coulomb and an exchange—correlation term, and it is given by a local potential, and p is the charge density. v = 6V/6p is called the residual interaction. In matrix form, Equation (2.17) can be expressed as 11““ = (1 + mfg—$411". (2.18) The free response spectrum to an external potential Vex. is relative to the imaginary part of the free polarization propagator and is given by shah...) = Z < av...” >2 6(w — 13,-— 13,). (2.19) f 51 The RPA response spectrum can be obtained using the imaginary part of the RPA polarization propagator from 504....) = 1 / dFdFIVCflU’,w)I/;$¢(Fl,w)lml'l(i’,717,02). (2.20) S (Vance) obeys the energy weighted sum rule [12] /5(w)wdw= /d“(——VK“)p , (2.21) 2mc where mc stands for the electron mass and p0 stands for the ground state charge density. In the case of an external electromagnetic dipole field with the electric field com- ponent aligned along the z axis, V“. = —eEz. The sum rule for the dipole operator is the Thomas—Reiche—Kuhn (TRK) (or f—) sum rule [17], 2 Z2w=N , 2mc (2.22) which relates the integrated oscillator strength to the number of active electrons. The dynamic polarizability of the system is given by a(w) = —e2 / dram zn(i=',n,w)zi, (2.23) which can be also applied to the static case 02 = 0. If we assume that all the oscillator strength is collected in a single mode, namely the Mie plasmon, the f—sum rule can be used to relate the plasmon frequency w to the static polarizability a by 2 Ne2 w _ _ . (2.24) 1" mea 52 The above equations give the RPA formulation in coordinate space. If the inter- action is taken as a local function, the above outlined formulation is the best choice for calculating optical response. This is the case in the jellium model, where the single—particle spectrum and the exchange interaction is obtained using LDA [12]. In the case of non—local potentials, the AB matrix representation of RPA is more appropriate. In matrix RPA, the system response to perturbations is given by [18] (134* If") (1)2431) (2.25) A and B are (sub-) matrices, given by Aphmth: = (e,, — eh)6,,,pt6h,h' + vphv,hpi, (2.26) and Bphm’h’ = vpp'Jih’ - (2.27) The matrix elements vmm of the residual interaction 1) can be expressed as ...,... = / wanes/>220:mamas/idem. (2.28) using the convention that m,n represent the unoccupied (particle) states and i, j the occupied (hole) states. Xpi, stands for particle—hole amplitudes and th for hole- particle amplitudes. Eq. (2.25) is a nonhermitian eigenvalue equation, but it can be replaced by the following Hermitian problem: wgua = (A — B)1/2(A + B)(A — 3W2.a (2.29) 53 where u“ E X a + Y“ is the eigenvector. In the case of local potentials, ”ph’.hp’ = ”pp'.hh' (2-30) and the equation can be expressed in terms of the single—particle energy matrix ephm’h' = (6? " eh)6p.p'6h.h’ as 02211“ = 61/2(€ + 2v)61/2u°‘. (2.31) or where the vector and matrix indices have been omitted for simplicity and us), repre- sents the amplitude of the particle-hole configuration Iph"1 > in the collective mode 0. Finally, the transition strength associated with an external field F (r) is given by 123.1113. < PlFlh > 12 < a F 0 >2: 2.32 I I we th 1113112 ( ) Thus, the oscillator strength associated with the vibration is °' < F h > 2 we, < aIFIO >2: '2’)” up” P] I I (2.33) :ph lustful2 From this it may be seen that the total (or integrated) oscillator strength is the same as for the free response. In other words, the f-sum rule is automatically satisfied. Bibliography [1] w. Kohn and L.J. Sham, Phys. Rev. 140, A1133 (1965). [2] R. Car and M. Parrinello, Phys. Rev. Lett. 55, 2471 (1985). [3] M. S. Daw and M. I. Baskes, Phys. Rev. Lett. 50, 1285 (1983). [4] D. M. Ceperley and B. J. Alder, Phys. Rev. Lett. 45, 566 (1980). [5] P. Hohenberg and W. Kohn, Phys. Rev. 136, 3864-871 (1964). [6] There are many different correlation function, for instance, L. Hedin and B.J. Lundqvist, J. Phys. C 4, 2064 (1971); E. Wigner, Phys. Rev. 46, 1002 (1934); O. Gunnarsson and B. I. Lundqvist, Phys. Rev. B 13, 4274 (1976). [7] N. W. Ashcroft and N. D. Mermin, Solid State Physics, lst edition (Saunders College, 1976), page 337. [8] J. P. Perdew and A. Zunger, Phys. Rev. B 23, 5048 (1981). [9] DR. Hamann, M. Schliiter, and C. Chiang, Phys. Rev. Lett. 43, 1494 (1979). [10] W. Ekardt, Phys. Rev. Lett. 52, 1925 (1984). [11] W. D. Knight, K. Clemenger, W. A. de Heer, W. A. Saunders, M. Y. Chou, and M. L. Cohen, Phys. Rev. Lett. 52, 2141 (1984). 54 55 [12] G.F. Bertsch, Comput. Phys. Commun. 60, 247 (1990). [13] D. Tomanek and Michael A. Schliiter, Phys. Rev. Lett. 67, 2331 (1991). [14] D. Tomanek and M.A. Schliiter, Phys. Rev. Lett. 56, 1055 (1986). [15] J.C. Slater and G.F. Koster, Phys. Rev. 94, 1498 (1954). [16] D. Bohm and D. Pines, Phys. Rev. 92 609 (1953). [17] G. F. Bertsch and W. Ekardt, Phys. Rev. B 32, 7659 (1985). [18] D. J. Rowe, Nuclear Collective Motion: Models and Theory, lst edition (Methuen and Co. LTD., 1970), Chapter 14. Chapter 3 Alkali Clusters Alkali metal clusters are very interesting systems which bridge the gap between iso- lated atoms and bulk metals. The large degree of delocalization of valence electrons, characteristic of the simple metals, is an important property of these clusters. This delocalization shows up most dramatically in the appearance of magic numbers cor- responding to shell orbitals encompassing the entire cluster [1, 2]. The electronic response of these systems is particularly interesting in that it shows one of the char- acteristics of a macroscopic plasmon, namely a large fraction of the oscillator strength concentrated in a narrow frequency range [3]. In the following, we will address this collective mode as the cluster plasmon mode [3]. In contrast to the prediction of the classical Mie theory as applied to metal spheres, this collective mode is typically 30% lower in frequency for small clusters, and does show a pronounced size dependence [4]. The results of this study will also be published elsewhere [5]. Of especial interest to us is the width of the collective excitation. The observed width of the plasmon peak is 0.11 eV in N32 [6] and 0.25 eV in Nas [7], much larger than the natural line width for the photoexcitation which is due to the life time of the excitation. In large clusters with a high density of states near the highest occupied (HOMO) and lowest unoccupied molecular orbital (LUMO), particle—hole excitations 56 57 (Landau damping) and multiparticle—hole excitations will dominate the fragmenta- tion of the plasmon [8, 9]. In the small clusters with n S 8 — 40 atoms, however, Landau damping is considered to be negligible due to the low level density near the HOMO. As we will show below, the dominant plasmon damping mechanism in these systems results from the coupling of electronic excitations to cluster vibrations. The quantum motion of nuclei gives a broadening which is described more precisely as a distribution of states with a variable number of vibrational quanta excited together with the plasmon. An increased nuclear motion in the ground state has been esti- mated to account for a line broadening with increasing temperature [10, 11]. Because of the large number of vibrational modes and the small value of the vibrational energy, it hardly makes sense to study this effect by explicit calculation of the vibrational wave functions and the associated Franck-Condom factors for the transitions. Instead, we shall apply an approximation that directly gives the width of the strength distribution irrespective of the quantization of the vibrational final state. We determine the ground state structural and electronic properties of small alkali clusters using the Local Density Approximation (LDA) [12]. Electronic excitations are calculated using the Random Phase Approximation (RPA) [13], using LDA single— particle wave functions and energies. In this Chapter, we apply this formalism to Liz, Na; and Lia, Nag clusters. The complete spectroscopy of the dimers is well estab- lished [14]. Consequently, these systems are ideally suitable for testing the accuracy of our methods in the ground state, and the power of our predictions in the excited state. For the larger clusters, the scenario is not so clear, since even their exact ground state geometry is uncertain [15, 16, 17, 18]. We proceed as follows. We first demonstrate the precision of our method by calculating the ground state properties 58 of the bulk metals and comparing them to experimental data. We do the same for the ground state of the addressed clusters. Next, we use the LDA-RPA to determine the excitation spectra. Finally, we determine the coupling of electronic excitations to cluster vibrations and compare the results with experiment. 3.1 Equilibrium geometry of small alkali clusters In our LDA calculations, we consider the valence electrons only, and describe the ef- fect of the core electrons by ab initio norm—conserving nonlocal pseudopotentials. Our pseudopotentials have been generated using the Hamann—Schlfiter—Chiang scheme [19]. The electronic configurations which we used to generate the pseudopotentials, Li 2.30'82p0'l [with the core radii rc(2s) = 0.915 A and rc(2p) = 0.788 A] and Na 330'73p0'13d0'05 [rc(3s) = 1.005 A, rc(3p) = 1.323 A, rc(3d) = 1.746 A], provide very good transferability especially towards the excited states. A partial core correction has been used in the LDA calculations [20]. We use the Ceperley—Alder parametriza- tion of the exchange—correlation potential [21] in the Kohn—Sham equations. In order to minimize the influence of a finite basis on our results, we decide to place our clusters on a face—centered cubic superlattice with a large lattice constant. This minimizes the volume associated with each cluster for a constant inter-cluster separation. Plane waves are the natural basis in this case which can be improved systematically. We find this approach more reliable for alkali clusters with delocalized electron states than an atom—centered Gaussian basis. Treatment of an isolated cluster in real space on a radial grid turns out numerically as involved as our approach. We used an energy cutoff in the Fourier expansion of the charge density Em” = 6.9 Ry for the solid and Ema, = 4.0 Ry for the clusters. Symmetry has been used to reduce the computational effort. The lattice constant a = 15 A for the superlattice guarantees 59 negligible overlap between the clusters and hence vanishing crystal field splitting, as verified by comparing the band structure at different points in the Brillouin zone. The calculated ground state properties of bulk Li are the lattice constant a LD A = 3.42 A (amp, = 3.49 A [22]), the bulk cohesive energy (with respect to an isolated coh spin—polarized atom) E L D A = 1.64 eV (E222, = 1.63 eV [22]), and the bulk modulus BLDA = 0.112x10‘1Pa (Bap; = 0.116x10uPa [22]). The corresponding values for Na are aw, = 4.0411 (am. = 4.23/1 [22]), Egg, = 1.23 eV (Egg, = 1.11 eV [22]), and BLDA = 0.089x1011 Pa (Bap. = 0.068x1011 Pa [22]). As expected from converged LDA calculations, the bulk is somewhat overbound. The larger difference between the calculated and the observed bulk moduli is presumably due to the pseudopotential approximation which suppresses exchange and correlation between valence and core orbitals. This effect is expected to be much smaller in atomic clusters where long— range exchange and correlation is absent. The smallest system we aim to describe are the dimers, the first closed—shell system within the spherical jellium background model. The large stability of alkali dimers is explained within the jellium model by a large separation between the fully occupied ls state and the empty 1p state of the cluster. The dimer geometry deforms the charge density along the molecular axis, and splits the threefold degenerate 1p level into one a and two 1r states. The dimer has only one nuclear degree of freedom, the dimer stretch mode, which simplifies the calculation of electron-vibration coupling significantly. In Figure 3.1, we show the total energy of the system as a function of the inter- atomic distance. LDA results for dissociation energies, bond lengths and vibrational energies of L12 and N32, shown in Table 3.1, are in striking agreement with the ex- perimental values [14]. 60 2.5 1 1 2.5 (a) Na.2 (12)) III“ 2.0 l 4 2.0 - 121-1- 3 "5 ' S; 1.5 - 1 1 s s E 2?? a : 3 - re 1 l “r i If ‘5 J l I” -0.5 t j 1 + -0.5 » j v} E —l.0 r 1 -t.0 - . ] —1.5 ‘ ‘ -t.5 - ‘ 2.5 3.0 3.0 4.0 2.0 2.5 3.0 3.5 (1(3) dd) Figure 3.1: Franck-Condon broadening of the collective electronic excitations in (a) Na; and (b) Liz. The lowest levels are the LDA dissociation energies D(d) of the dimers as a function of the bond length d. The higher levels give the excitations energies, which are presented as D(d) + ERpA(d) [From Y. Wang et al, (submitted for publication)]. 61 Table 3.1: Ground state properties of sodium and lithium dimers: Equilibrium bond length dc, dissociation energy 0,, and vibration frequency we. System d.(A) Dc(eV) hw,(meV) expt.“ theory expt.“ theory expt.“ theory L12 2.672 2.730 1.03 1.01 43.572 46.0 Nag 3.078 3.032 0.72 0.91 19.742 20.0 “ See G. Herzberg, “Molecular Spectra and Molecular Structure. I Spectra of Diatomic Molecules”, second edition, (D. Van Nostrand Company, Inc.), New York, 1950. The next closed shell configuration in alkali clusters occurs for 8 atoms. The physics of these systems is much more complex due to their 18 nuclear degrees of freedom and many different isomers which lie close in energy. As mentioned before, not even the equilibrium geometry is well established [15, 16, 17, 18], although cal- culations [16, 17, 18] suggest the Ta symmetry for the ground state. Consequently, we base our calculations on this geometry. The LDA superlattice calculations are essentially the same as for the dimers, but we increase the fcc lattice constant to a = 50 A in order to minimize the interaction between clusters. The latter was checked by observing the calculated band dispersion Ac across the Brillouin zone. Our value Ac z 0.01 eV, gives an estimate for the upper bound of cluster-cluster interaction. We use again an energy cutoff of 4.0 Ry, corresponding to a plane wave basis with 4279 components. The equilibrium structure of these clusters in the T4 geometry is uniquely defined by the radial distance d,- of the “inner tetrahedron” atoms from the cluster center, and the corresponding distance do of the outer tetrahedron atoms. The calculated atomization energy per atom for the Nag cluster in equilibrium geometry with d.- = 2.11 A and do = 3.51 A is 0.77 eV, in reasonable agreement with the value of 0.86 eV, obtained in a previous LSDA calculation [15]. 62 3.2 Collective electronic excitations (Mie plasmon) in small alkali clusters Once the equilibrium geometries are known, we proceed to calculate the response to external electric fields. The static response is a ground state property of the system and can be obtained directly from LDA. We use the above described superlattice geometry1 to determine the static dielectric response of these systems to a field which is parallel or perpendicular to the dimer axis. For an isolated Na atom we find aLDA(Na) = 22.0 A3, in good agreement with the experimental value of aczpt(Na) = 23.6 A3 [23]. The polarizability of a negatively charged sodium ion aLDA(Na") is 63.0 A3, much larger than that of the atom, caused by the weak binding of the outermost electron. The polarizability of the N32 along the axis is 091D A(Nag) = 63.5 A3, while the value perpendicular to the axis is aiDA(Na2) = 22.1 A3. The 3 average over all directions of the polarizability gives (aLDA(Na2)) = ézifl a.- = 35.9 A3, which can be measured experimentally. This value agrees well with Local Spin Density Approximation (LSDA) calculations of Moullet et al. [18], who obtained a"(Na2) = 53 A3 and 011(Na2) = 30 A3, leading to (aLDA(Na2)) = 37.7 A3. Once the static dielectric response is established, we proceed to calculate the electronic excitation spectrum within the linear response framework. We use the RPA which is based on an electronic ground state described by LDA. RPA automati- cally satisfies energy-weighted sum rules, and has the correct physical limits, namely independent-particle transitions at high momentum transfer, where the interaction is weak, and strong collective excitations at low momentum transfer, where the in- 1In a cluster superlattice, the external field is generally modified by the field of the induced dipoles on the other sites. Since our system has inversion symmetry, the corresponding correction vanishes exactly at each lattice point, and is very small over the cluster volume. The polarizabilities of atoms and dimers can then be obtained directly using second order perturbation theory. 63 teraction is strong. It is customary in condensed matter physics to implement RPA by choosing the potential field as the basic object of computation. In this case, the wave function enters indirectly via the dynamic polarizability. However, if only a few electron states participate in the excitation, the most efficient approach is to set up the RPA equations for the wave function directly [24]. We shall use this method in the present work. We start with the single-electron wave functions 45,-(r) and energies 6;, obtained from the LDA calculation. We shall need both occupied and unoccupied orbitals, from which we construct the particle—hole states. We designate the particle-hole state as [i j ’1), where i designates an unoccupied (particle) state and j an occupied (hole) state. The Hamiltonian matrix may be separated into a diagonal part that gives the energy of the particle—hole state, and an off—diagonal part that describes the coupling to other particle—hole excitations. The diagonal part includes the kinetic energy operator and the self-consistent Hartree and exchange—correlation field. We write this part of the Hamiltonian matrix as (ii-116127”) = 6ii'6jj'(€i - 6.1- (3-1) The residual interaction contributes matrix elements of the form (ij‘llvli’jH) = / drdr’¢2‘(r)¢j(r')v(r,r’)¢ti(r)¢3’t(r')a (3.2) where 1) includes the residual Coulomb interaction and exchange correlation. The RPA eigenvectors u“ and their associated frequencies too, are given by the eigenvalue Equation (2.31). Let us first discuss the application of the above formalism to the dimers, Liz and N a2. The results of our calculations for these systems, obtained using different approximations, are summarized in Figure 3.1 and Table 3.2. 64 Table 3.2: Collective electronic excitations in small sodium and lithium clusters. Our results for the plasmon frequency hoopla...” and its width F are listed together with results based on spherical jellium [25], thELLYRpA, and results of the classical Mie theory, thie- System hwpzamon (CV) F (6V) thELLYRPA (CV) thgc (8V) expt. theory expt. theory theory theory Liz 2.23 0.063 3.6 4.6 N32 1.92“ 2.43 0.11“ 0.095 2.8 3.5 L13 3.6 4.6 Nag 2.535 3.10 0.25" 0.03 2.8 3.5 ‘ See A. Herrmann et al, Chem. Phys. Lett. 52, 418 (1977). b See C. R. C. Wang et al, Chem. Phys. Lett. 166, 26 (1990). In Figure 3.1, we plot the energy of the dimers for a given electronic configuration as a function of the bond length d. The lowest curve gives LDA results for the 12: ground state. The 12: curve is obtained by adding the RPA excitation energy to the energy of the 12; state. In the adiabatic approximation, we determine the transitions from the energy difference between the vibrational ground state and the excited state in the same geometry, as indicated by arrows in Figure 3.1. From Figure 3.1 we notice that the potential energy surface and equilibrium geometry of the excited state are different from those of the ground state. The equilibrium bond length (12,313 A(N a2) = 3.50 A compares well with the experimental value d:,.zpt(Nao) = 3.63 A [14]. The corresponding value for Liz is d:RPA(Li2) = 3.17 A, which again compares well with the observed value d:,cxp,(Li2) = 3.10 A [14]. Experimental data [14] indicate that the energy difference between the 12: ground state at de and the 121' excited state at d; is 1.76 eV for L12 and 1.82 eV for N a2. These energies compare reasonably well with our LDA—RPA results of 2.20 eV for Li; and 2.33 eV for N a2. However, a comparison between calculated and observed adiabatic (vertical) excitation energies in Table 3.2 65 shows that the calculated plasmon energy is blue—shifted by 0.5 eV (see Figure 3.2) with respect to the observed value. This blue shift is characteristic of LDA—RPA calculations, and reflects the incorrect asymptotic behavior of the effective potential which is exponential decay instead of } potential [9]. As we will discuss below, the difference between the potential energy surfaces in the ground and the excited state is responsible for vibrational broadening of electronic excitations. Of particular importance in this respect is the shape of the excited potential energy surface. We find our calculated values for the vibrational frequencies in the 12: state w;(Li2) = 31.7 meV and w;(Na2) = 17.8 meV to compare very well with the experimental data [14] w:,cxp,(Li2) = 31.7 meV and w" c’c,,,,,,,(Na2) = 14.6 meV. As expected and discussed in the following, these results are superior to calcu- lations for spherical jellium representing Na; and Liz clusters. Our corresponding LDA—RPA results, obtained using the JELLY—RPA program [25], are shown in Ta- ble 3.2. In the jellium model scenario, the single-particle ground state has a 13 character, and the lowest unoccupied states have 1p, 1d and the 23 character. The spherical potential clearly cannot describe the splitting of the first two excited states which is substantial in the dimers. Among the above jellium states, there is only one dipole—allowed transition from the ground state, namely the Is --2 1p transition. Other allowed transitions have a much larger excitation energy, and are essentially single-particle transitions. Our numerical results, shown in Table 3.2, yield values for the collective excitations in jellium which lie up to 60% above the LDA—RPA results for the realistic geometry, mainly due to the spherical approximation in the jellium model. Another important disadvantage of spherical jellium is that it cannot address vibrational damping of electronic excitations, which we shall discuss below. Next, we turn to the L13 and N as clusters. The results for the collective excitation 66 100 " r“ - :1 I “ .9. 1 ~. -i-> ’ \ o. ,’ \ in i ‘ o l ‘ In I “ .0 I" \ (U . o 50 7 ' ‘ "‘ ‘0’ ' ‘ .c: . , D-i ,’ \ I’ , ‘ " ll * # ‘ 1.8 1.9 2.0 2.1 Energy (eV) Figure 3.2: Calculated spectral function of Na; (in arbitrary units) and its broad- ening due to nuclear zero—point motion (dashed line), as compared to the observed photoionization spectrum of Ref. [6] (solid line). The width of the Gaussian envelop is 0.10 eV and the displayed theoretical data are red—shifted by 0.5 eV with respect to the calculated results [From Y. Wang et al, (submitted for publication)]. 67 energies in these systems, obtained using different approximations, are summarized in Table 3.2. The LDA calculation for spherical jellium gives the occupied ground state levels at C(18) = —4.46 eV and ((11)) = —3.19 eV. The lowest unoccupied states lie at c(1d) = —1.65 eV and C(28) = —1.15 eV. The relatively large HOMO— LUMO gap of 1.54 eV contributes substantially to the stabilization of this magic cluster size. The LDA—RPA calculations for this system predicts a plasmon energy of thELLYRPA = 2.8 eV. Our LDA calculation for N as in Td geometry shows that the lowest nondegenerate unoccupied level, corresponding to the jellium 23 level, lies at c = —0.82 eV and is lower in energy than the manifold of levels originating from the jellium 1d level. This manifold results from symmetry breaking of the fivefold degenerate 1d type level of the spherical jellium into a threefold degenerate level at e = —0.73 eV (consisting of orbitals with my,yz, 23: character) and a doubly degenerate level at at e = —0.35 eV (consisting of orbitals with 222 — 3:2 — y”, 3:2 — y2 character). The RPA spectrum of N as in the realistic geometry discussed above is given in Figure 3.3. The spectrum shows three distinct peaks, but is dominated by a sin- gle resonance at hwpiamon = 3.1 eV. This is in agreement with experimental results which show three distinctive resonance [7], but disagrees with previously calculated photoabsorption spectra which has five excitations [26]. The strong resonance ex- hausts 87.4% of the f—sum rule, which is indicative of its strong collective charac- ter. As in N32, this value is blue-shifted with respect to the experimental value thlasmon,expt : 253 CV [7] 68 8 l l 7- .. .1: ”6’0 6- C: 3 +2 5" ‘ m x... 4*- . O 4.3 ,9. 3' - {a a, 21- . O 1- - o—I—il~l 1 . Energy(eV) Figure 3.3: Calculated oscillator strength distribution in the excitation spectrum of N as [From Y. Wang et al, (submitted for publication)]. 69 3.3 Damping of the Mie plasmon in small alkali clusters Our above results indicate that the present scheme is able to determine collective electronic excitations reasonably well, especially when compared to the jellium model. We now turn to the fragmentation of this collective mode. Due to the low level density near the HOMO, Landau damping is improbable in these systems [9]. The dominant broadening mechanism at T = 0 K is the coupling of the electronic excitations to nuclear zero—point motion, as described by the F ranck—Condon effect. With increasing temperature, higher vibration modes and possibly transformations between different isomers are likely to further broaden the plasmon linewidth. This thermal broadening mechanism is expected to play a more pronounced role in the larger n = 8 atom clusters with soft vibrational modes. A Hamiltonian which describes the coupling between the electronic excited state so and the vibrational normal modes ,1: with energy kw“ is [27] H = clc (cc + Z Mu(a’l‘ + a») + z hwpala,‘ . (3.3) u u Here, cl and at are the creation operators for electronic and vibrational states, re- spectively. In the case of dimers, Eq.(3.3) is simplified due to the presence of a single ground state vibration mode with energy hwo. The coupling of electrons to nuclear motion is described by the term M (a’r +a) E F 2:, where F is the slope of the potential energy surface for the excited state at the transition point. As a result, the exact solution for the spectral density distribution A(hw) at zero temperature is given by the Poisson distribution [27] A(hw) = 21re’9 Z 1176(hw -—- cc + ghwo — hwln) , (3.4) n=0 ' 70 where wl is the vibrational frequency of the excited state. In this equation, g is related to the slope F and the ground state vibration energy Two by g = F 2/ (thwg), and n gives the corresponding quantum level. A(hw) is hence a sum of equally spaced delta functions (as shown in Figure 3.2), with separation energy hwo and a Poisson peak height distribution. In the limit of large g, the Poisson distribution can be approximated by a Gaussian distribution, as °° l (rt-92’ A h = ’ 2n 6 h — c h — h . 3.5 ( no) ”game ( w 6 +9 (.00 com) ( ) The resulting line shape has a full—width at half maximum (F WHM) I‘, which is given by han mwo r = 2F (3.6) for the vibrational ground state corresponding to T = 0 K. An intuitive way to understand this formula is the following. The probability distribution for a harmonic oscillator in the ground state is a Gaussian with a width (A23)2 = h/(mwo). Assuming that the dependency of the excitation energy on a: is given by AB = F Ax, one obtains for the distribution of excitation energies f (E) = exp{—mwo(E' — E0)2 / (15172)}. This is essentially the same result as in Eq. (3.6). Our results for the plasmon damping in N a2 and Liz are summarized in Table 3.2 and Figure 3.2. For Liz, no such experimental data are available to the best of our knowledge. For Nag, we obtain for the 123' —-+ 12: transition a FWHM of 0.10 eV, in very good agreement with the experimental value F = 0.11 eV [6]. The perfect agreement of the envelope functions in Figure 3.2 indicates that in this case, the coupling between electronic and vibrational degrees of freedom dominates the plasmon fragmentation. For the 12: —> 1Ilu transition, we predict a line width of 0.06 eV. No experimental data for the line width are presently available for this transition. 71 Investigations of the vibrational broadening of collective electronic excitations are in progress for Lls and Nag. The calculation is more complex not only due to the larger cluster size, but also due to the significantly larger number of nuclear degrees of freedom in these systems. In order to obtain a rough estimate of the plasmon line broadening in these systems, we proceed as follows. We assume that the broadening is dominated by a single low-frequency mode with a large quadrupolar component. We restrict our calculations of the clusters with assumed Td geometry to the lowest vibration mode with ng symmetry, which is obtained using the parametrized Many— Body Alloy Hamiltonian [28]. The LDA—RPA calculation for this distortion indicates only a very small line broadening in Nag of I‘ z 0.03 eV, much smaller than the observed value 1‘th = 0.25 eV [7]. The discrepancy between the calculated and the observed value may be due to our neglect of the other vibrational degrees of freedom, or a large temperature of the observed clusters. For a thermally excited cluster, I‘ can be estimated in analogy to Ref. [11] as I‘ = 2F{(k3Tln 2)/(mwg)}1/2. Using this expression, and relying on the validity of the harmonic approximation, we find a line broadening of 0.25 eV to correspond to a temperature T z 4000 K for the vibrational mode above. Even though this temperature is likely to be overestimated by the harmonic approximation, our result suggests that other vibrational modes contribute significantly to the line broadening. Moreover, substantial line broadening could result from structural transitions between different isomers of the n = 8 atom structures which are very close in energy [15, 16, 17, 18]. 3.4 Conclusions In conclusion, I have presented results for the equilibrium structure and collective electronic excitations and their damping in small Na, and Lin clusters. In the calcu- 72 lations, we have used the Local Density Approximation to describe the ground state properties of these systems, and the Random Phase Approximation for the electronic excitations. We have discussed the collective excitations in the first two closed—shell clusters with n = 2, 8 atoms in detail. Our results indicate that the coupling of elec- tronic levels to vibrational degrees of freedom accounts quantitatively for the observed width of the collective electronic excitations in alkali dimers. More calculations are necessary to address the damping mechanism of the collective electronic excitations in Lig and Nag. Bibliography [1] W. de Heer, W. Knight, M. Chou and M. Cohen, Solid State Physics 40, 93 (1987). [2] S. Bjornholm et al., Phys. Rev. Lett. 65, 1627 (1990). [3] W. A. de Heer, K. Selby, V. Kresin, J. Masui, M. Vollmer, A. Chatelain, and W. D. Knight, Phys. Rev. Lett. 59, 1805 (1987). [4] J. H. Parks and S. A. McDonald, Phys. Rev. Lett. 62, 2301 (1989). [5] Yang Wang, Caio Lewenkopf, David Tomanek, George F. Bertsch, and Susumu Saito, (submitted for publication). [6] A. Herrmann, S. Leutwyler, E. Schumacher, and L. woste, Chem. Phys. Lett. 52,418 (1977). [7] C.R.C. Wang, S. Pollack, and M.M. Kappes, Chem. Phys. Lett. 166, 26 (1990). [8] C. Yannouleas, R. Broglia, M. 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Li, and D. Tomanek, Phys. Rev. B 44, 13053 (1991). Chapter 4 Structural Properties of C60 and SOIid 060 The carbon atom has two 23 electrons and two 2p electrons in the valence shell. These four valence electrons can hybridize in various ways, leading to sp, sp2 and sp3 bonding in carbon. Carbon crystallines either in the layered graphite structure with sp2 bonding, or in the diamond structure with 3123 bonding, the hardest material in nature. Recently [I], a third form of solid carbon has been synthesized: fullerite, a crystal based on the C60 clusters. In this Chapter, I will investigate the structural properties of the C60 crystal and the stability of fullerite intercalation compounds. The results pertinent to the elastic behaviors of solid C60 have been published in the meantime as Ref. [2] and those related to C60 intercalation compounds as Ref. [3]. 4.1 Structural and elastic properties of the C50 based solid When crystallized, C60 clusters [4] form a solid with a facecentered cubic structure which has been given the name “fullerite” [1]. In spite of the considerable effort 76 77 invested in understanding the equilibrium properties of C60 clusters [5, 6, 7, 8], many questions regarding the material properties of fullerite remain open. So far, based on X-ray diffraction data [1], fullerite has been shown to be a close-packed molecular solid with a face-centered cubic structure and a nearest distance between neighboring C60 clusters D = 10.04 A. Extended-X-Ray-Absorption-Fine-Structure (EXAFS) data [9] indicate an average carbon-carbon nearest-neighbour distance dc-c = 1.42 A which is the same as in graphite. This bond length corresponds to a radius R = 3.55 A of the fullerene cluster. In other words, the closest distance between two surfaces of adjacent clusters is d = 2.9 A at zero pressure. Raman and infrared (IR) spectroscopy data [9, 10, 11, 12] confirm that the “soccer ball” structure of C60 is preserved in the solid. Pressure dependent X-ray diffraction data, obtained in a diamond anvil cell, indicate a large change of the bulk modulus of fullerite as a function of pressure [13]. In this Section, I will address the static and elastic properties of fullerite as a function of pressure. I will present a physical model, based on first principles calcula- tions, for the cohesion of the solid. In this investigation, I will focus on the interesting question, whether fullerite can become less compressible than diamond. When fullerite is compressed, part of the volume reduction comes from squeezing the clusters closer together, and part from the compression of the clusters themselves. Our model exploits this distinction to make a tractable calculation without the full- scale apparatus of the Local Density Approximation (LDA) theory [14]. While the LDA is computationally feasible with present computers [5], it does not provide the insight possible with a simplified treatment. Also, the LDA does not have any funda- mental significance for purely Van der Waals forces which dominate the interaction at separations between the C60 clusters which are much larger than their equilibrium distance. A schematic picture of the model is shown in Figure 4.1. Effectively, there are tV assoc? [I ical 5 tion t icosa' dral: (per sphei bond equfl nuun betw inter is th bet“ the ( bori: for (j 78 are two spring constants, c1 associated with the interaction between clusters and c2 associated with the compression of the clusters themselves. Under external pressure, we expect C60 clusters not to deviate much from a spher- ical shape due to the close—packed structure of the crystal with twelve-fold coordina- tion of the lattice sites. Also, the symmetry of the lattice is incompatible with the icosahedral symmetry of C60, which would make distortions towards a cubooctahe- dral shape difficult. This is supported by the low activation energy of only z 0.15 eV (per cluster) for molecular rotation [15]. We shall therefore treat the clusters in the spherical approximation. The C60 surface is similar to a curved piece of graphite, with predominantly sp2 bonding and a nearest neighbor distance of dc_c = 1.42 A. Owing to the large equilibrium separation of d = 2.9A between C60 clusters, their mutual interaction is mainly due to a Van der Waals force which should be very similar to the interaction between layers of graphite. We shall base this part of our model on the LDA results for interactions between graphite layers [16]. Our expectation, confirmed by the analysis, is that the individual clusters are highly incompressible compared to the interaction between clusters. Thus, for low pressures at least, there is a close relationship between the compressibility of fullerite and the c-axis compressibility of graphite. We shall model the Van der Waals interaction assuming that atoms in the neigh- boring graphite layers interact pairwise [17], as E = 2211035) . (4.1) The pair interaction is constructed to reproduce the ab initio LDA calculations for the binding energy of graphite [16]. We use a modified Morse potential of the 79 Figure 4.1: Schematic drawing of the the elastic parameters describing the interac- tion between neighboring C30 clusters in fullerite. The weak Van der Waals bond between these clusters can be mapped onto an anharmonic soft spring (spring con- stant CI). The compressibility of the stiff C60 fullerene cluster itself can be'described by a stiff anharmonic spring (spring constant 0,») [From Y. Wang, D. Tomanek, and G. F. Bertsch, Phys. Rev. BR 44, 6562 (1991)]. f0' Para; 2.619 80 form U(r) -_—. D.[(1 — e-W-M)2 — 11+ 13.64" . (4.2) Here, i, j denote atoms in adjacent graphite layers, Dc is the equilibrium binding energy of these atoms, rc is the equilibrium distance between these atoms, and E, de- scribes an additional hard-core repulsion. fl and [3’ describe the distance dependence of these interactions. The binding energy of graphite as a function of the interlayer distance d, obtained using Eqs. (4.1) and (4.2), is shown in Figure 4.2(a) together with the LDA data. The corresponding interlayer force is shown in Figure 4.2(b). The parameters used in Eq. (4.2) are listed below‘. Since the Van der Waals interaction is long-ranged, it is convenient to replace the double sum in Eq. (4.1) by a double integral which averages over the atomic sites. The interatomic binding energy U is then replaced by the energy (7 corresponding to the interaction energy between two small areas AA in adjacent graphite layers. Then, E = /A1 drl [42 dr2U(|r1 — rzl) , (4.3) U(r) = De[(1 — (”t-W)? — 1] + Ere-‘3" . (4.4) We use Eqs. (4.3) and (4.4) to determine the interaction energy Ede between neighboring C60 clusters and note that the double integral extends over the surface areas of both clusters. In case that the direct line connecting the area elements at r1 and r2 contains a part of any cluster, we neglect the corresponding contribution to the double integral due to screening. The resulting pairwise interaction energy 1In Eq. (4.2), we use D, = 6.50 X‘IO‘3 eV, rc = 4.05 A, E, = 6.94 x10"3 eV, 3 = 1.00 A'1 and ,6’ = 4.0 A”. In Eq. (4.4), we use D8 = 9.47 x 10" eVA“ and E, = 9.915 x 10"" eVA“. The parameters in Eqs. (4.2) and (4.4) are related by the fact that the area per C atom in graphite is 2.619 A“. (a) 4.0 O 0 Energy (eV) Pr'r‘ COO 1.5 2.0 2.5 3.0 3.5 6.0 -d(B)/d(d)(oV O O O O O C O O 8 8 O O I O '8 fl 0 I! p- y- r- 2.0 2.5 5.0 5.5 4.0 4(1) Figure 4.2: (a) Binding energy of hexagonal graphite (with respect to isolated layers, per carbon atom) as a function of the interlayer spacing d. The solid line represents a modified Morse fit [Eqs. (4.1) and (4.2)] to ab initio LDA results of Overney et al, J. Phys. C 4, 4233 (1992). (b) Negative gradient of the energy given in (a), corresponding to the interlayer force [From Y. Wang, D. Tomanek, and G. F. Bertsch, Phys. Rev. BR 44, 6562 (1991)]. 82 between neighboring clusters is shown in Figure 4.3(a). The corresponding force, given in Figure 4.3(b), indicates that at zero pressure, the distance of closest approach between neighboring C60 clusters is d = 2.65 A. We calculate the binding energy E60 of an isolated C60 cluster using a modified tight—binding Hamiltonian [18], which had been tested successfully in previous studies of the equilibrium structure and vibration modes of small Sin clusters [19]. The breathing mode of the C60 cluster is described by the dependence of the binding energy on the cluster radius R, as shown in Figure 4.3(c). The restoring force, shown in Figure 4.3(d), is zero at the equilibrium radius Reg = 3.25 A. With all force constants at hand, we can now proceed to calculate the equation of state of fullerite. The solid can now be viewed as an fcc lattice of fullerenes represented by mass points and connected with strongly anharmonic nearest neighbor springs, shown in Figure 4.1. In compressed fullerite, the equilibrium geometry minimizes the binding energy per C60 cluster in the fee structure Ecoh(D) = GEde(d) + E6002) (4.5) with fixed D = d + 2R, corresponding to a unit cell volume V = D3/\/2. The first term in this equation correctly avoids double counting the nearest neighbor Van der Waals bonds, and the second term is the energy of an isolated C60. The binding energy of fullerite Ecoh(V) is shown in Figure 4.4(a). At T = 0, one obtains the pressure from p = —dEcoh/dV and the bulk modulus from B = —-V(0p/6V). In Figs. 4.4(b) and 4.4(c), we show the dependence of the cell volume and the bulk modulus on the external pressure. From these results it is obvious that the elastic behavior of fullerite resembles closely that of an inert gas solid. At very small pressures, the interactions between clusters are dominated by the compressible Van 83 A a m b v Energy (eV) 2’ Energy (eV) I § w 0 I v .0 O V -1.0 14 ‘- 3°" ' a 13 \ > a I“ f d 3, no x A I ‘3' 0 g 1000 '- >. ' > a ‘ 5 5M " ‘ 1:1 ‘0 I 1 2 o o -2 ‘ ‘ . . -5oo ‘ * 1.5 2.0 2.5 3.0 3.5 4.0 2.5 3.0 3.5 4.0 4(1) - Ra) Figure 4.3: (a) Interaction energy between two C60 fullerene clusters as a function of the closest approach distance d. (b) Negative gradient of the interaction energy in (a), corresponding to the pairwise force between neighboring C30 clusters. (c) Binding energy of an isolated Cm fullerene cluster as a function of the cluster radius R. ((1) Negative gradient of the binding energy given in (c). Note the difference in scales between (b) and (d) [From Y. Wang, D. Tomanek, and G. F. Bertsch, Phys. Rev. BR 44, 6562 (1991)]. aoooouoooouom V(l') (c>;' 5.0 b .0 L Dmong....ocoo A‘.° b a»: 3' 2.0 - Fullerite , 1.0 ’ o.o . ‘ . - 4 A 0 20 40 00 903?.) Figure 4.4: (a) Binding energy of fee-fullerite (per C60 cluster, with respect to iso- lated carbon atoms) as a function of cell volume V. (b) Pressure dependence of the equilibrium cell volume V of fullerite. (c) Pressure dependence of the bulk modu- lus B of fullerite (solid line), as compared to diamond (dashed line, from Yin et al, Phys. Rev. Lett. 50, 2006 (1983)) [From Y. Wang, D. Tomanek, and G. F. Bertsch, Phys. Rev. BR 44, 6562 (1991)]. 1 der exte can the lin tlL p0 we fre int sur sio. Th intc this full SCal graI; lndfi II neig Earh( 85 der Waals bonds causing a very low bulk modulus B a: 0.2 Mbar. With increasing external pressure, the clusters themselves are compressed at a high cost in energy, causing a large increase in the bulk modulus. We found it instructive to compare the bulk modulus of fullerite at high pressures to diamond. The diamond data of Ref. [20], obtained using LDA calculations, are shown in Figure 4.4(c) by a dashed line. From our calculation, we conclude that the compressibility of fullerite exceeds that of diamond only at pressures exceeding z 70 GPa. As discussed earlier, fullerite can be viewed as an fcc solid consisting of heavy mass points representing C60 clusters, with nearest neighbor interactions. In Figure 4.5, we show the phonon band structure of this lattice. The relatively low vibration frequencies result from the heavy mass of the clusters and the weak Van der Waals interactions at p = 0. It should be possible to turn fullerite locally into diamond under very large pres- sures. The mechanism is very similar to that discussed by Fahy et al. for the conver- sion of rhombohedral graphite with sp2 bonding to diamond with sp3 bonding [21]. This transition is initiated in graphite by a strong interlayer coupling occurring when inter- and intralayer carbon nearest-neighbor bonds are comparable. In fullerite, this transition should occur when the distance of closest approach between adjacent fullerene clusters d is close to 1.5 A. This occurs at the upper end of the pressure scale in Figs. 4.3(b) and 4.3(c) and should be more easily achieved in fullerite than in graphite. Following this prediction, a transformation from fullerite to diamond has indeed been achieved experimentally [22]. In this study, we used Eqs. (4.3) and (4.4) to calculate the interaction between neighboring clusters which have been approximated by spherical shells. As mentioned earlier, the atomic granularity of the clusters is averaged out to a large degree. Based 86 Figure 4.5: Phonon dispersion relation V(k) of bulk fullerite with fcc structure [From Y. Wang, D. Tomanek, and G. F. Bertsch, Phys. Rev. BR 44, 6562 (1991)]. CE GI pr by 87 on our expression in Eqs. (4.1) and (4.2), we find a residual activation energy for clus- ter rotation of the order of 0.1 eV (per cluster), in fair agreement with experimental data [15]. 4.2 Stability of donor and acceptor intercalated €60 solid Perhaps the most exciting property of C60 fullerite is superconductivity which occurs in the doped compound. Following the discovery of superconductivity in K3C60 with a transition temperature Tc = 18 K [23], new compounds have been synthesized us- ing a variety of intercalants [24, 25, 26, 27, 28], yielding critical temperatures as high as 33 K in CsszCeo [29]. While Tc values of doped fullerite are still below those found in high—Tc perovskite superconductors [30], intercalated fullerite shows supe- rior materials properties and hence bears the higher potential for applicability. The intercalation process and the rigid-band behavior of intercalated fullerite resembles in many ways the extensively studied graphite intercalation compounds [31]. The crucial property of fullerite intercalation compounds AxCeo is their stability against decomposition into the components in the standard state, i.e. C60(solid) and A(solid). The formation enthalpy is of interest not only for the donor compounds mentioned above, but also for potential acceptor compounds. This quantity is hard to calculate, since cohesion in these ionic compounds is dominated by a large Madelung energy [32, 33]. Still, even a rough estimate of the formation enthalpies across the periodic table is useful when considering the synthesis of novel C60 based materials. The difficulty to obtain a reliable value for the formation enthalpy is best illustrated by the spread of ab initio values for the formation enthalpy of K3C60 from K metal 88 and bulk C60, ranging from AH° = —1.7 eV per K atom 34 to AH° = —6.6 eV 35 2, f f indicative of an extremely exothermic intercalation process. We decompose the formation process of fullerite intercalation compounds into well-defined steps and estimate the energy involved in each step across the periodic table. These steps are combined into a thermodynamic Born—Haber cycle which de- termines the formation enthalpy. The prerequisite for this calculation is a detailed knowledge of the structure, lattice constant and compressibility. Since this infor- mation is not available for most of the systems discussed here, we calculate these properties, together with the phonon structure, for the compounds of interest first. This is interesting information on its own merit and will be presented together with the calculated formation energies. 4.2.1 Born—Haber cycle The formation enthalpy AH? at T = 0 K of AnCso is defined by . . 71A”? . nA(soIid) + 060(solzd) ———» AnCso(solzd) . (4.6) If AH? is negative, the compound AnCso is stable against decomposition into the pure components, namely the intercalant A in its solid form, A(solid), and pure fullerite, C60(solid). We determine AH}J by formally decomposing the formation process of a fullerite intercalation compound into several physically well defined steps and evaluating the energies involved in the individual steps. This procedure, known as the Born—Haber cycle, has been used to determine reaction enthalpies of complex multi—stage reactions. The cycle for the formation of the donor compound A3C60 is illustrated in Figure 4.6(a). 2The value quoted has been obtained using the experimental cohesive energy of metallic K, Ecoh(K) = 0.934 eV, and of fullerite, Ecoh(C50) = 1.6 eV. The latter value is a theoretical result obtained in the same reference. 89 (a) Arcs (b) Aacga E(atom) + "(clinical M‘htom) + Canclusrem 31W- A(Cco)- «Cm-MC}; ) ~3A(A)+ 1(Ca)+l(c,+,)+l(cgg) Mata») + Cdchst§fl mm“) + CdclusreTfl 33...“) + 3““in 33...“) + 19...“; C23) s...(c..) 3.1m.) ' mm» + cam) M) + team) an; an: ‘5 C33 (cannoundl Figure 4.6: Born-Haber cycle used to predict the formation enthalpy AH? of (a) donor and (b) acceptor C60 fullerite intercalation compounds [From Y. Wang, D. Tomanek, G. F. Bertsch, and R. S. Ruoff, Phys. Rev. B (1993)]. 90 The first step involves the separation of the reference system into individual A atoms and C60 molecules, taking into account that 3 A atoms occur in the formula unit of the doped solid. The energy involved in this step is the cohesive energy of A, 3Ecoh(A), and the binding energy of a C60 molecule in C60 (solid), Ecoh(C60 solid). In the next step, we consider the ionization of the three A atoms and the charge transfer to the C60 molecule. Here we have implied that C60 can act as electron acceptor; the electron affinity of the C60 molecule will be discussed later. This step requires the energy 3 I (A), I (A) being the the ionization energy of the A atom. The three electrons are transferred from the donor atoms to the C60 molecule and release the energy Am = A(Cso)+A(C§0)+A(C§;), where A is the electron affinity. In the last step, the A+ and Cg; ions are combined to form the solid, thereby releasing the formation energy Emu/51:03:31 Hence, the total energy gain during the formation of the A§C23 system is AH? = 3E...(A) + 13...].(060) + 31(A) — A(Cso) — A(Cgo) — A(Céo‘) - fled/1:030.) - (4-7) The relatively low ionization potential of C60 makes it a potential electron donor, raising the question about the stability of acceptor intercalated fullerite. The Born— Haber cycle for the formation of the acceptor compound A3Cso is illustrated in Fig- ure 4.6(b). It differs from the former one in the direction of charge transfer between the intercalant and the matrix. The electron affinity A(A) of the intercalant and the ionization potentials of multiply charged C60 clusters, I (C23), are required in this step. The formation energy of the compound from the ions, Ecoh(A§ C23), is defined with respect to the appropriately charged ions, and is given by AH? = 3Ecoh(A) + Ecoh(060) — 3A(A) + [(C60) '1' [(036)-1- I(C620+) _ coh(A;Cgf)+) ° (4.8) 91 When evaluating the formation enthalpy using the Born—Haber cycle, we approx- imated each step by the corresponding energy and hence have neglected the con- tributions of nonzero temperature and pressure to AH?, which we estimate to be of the order of S 0.1 eV. Precise experimental data exist for the cohesive energies Ecoh(A) [36], the ionization potentials I (A) and electron affinities A(A) across the periodic table [37, 38]. Unfortunately, no reliable experimental values exist for the binding energy of a Geo molecule in single crystals of C60, and ab initio techniques tend to underestimate the weak Van der Waals binding between the C60 clusters [5]. Therefore, we estimate this quantity in the close—packed fullerite lattice using a pair bond model as Ecoh(C'60 solid) = 6D(C’60 - 060). The distance dependence of the pair potential D(Ceo — 060) is given by the Morse form D(r) = D. ((1 _ e‘3("'e))2 — 1) , (4.9) where DC is the the dissociation energy of a pair of C60 molecules and r.3 is their nearest—neighbor distance, and where ,6 describes the distance dependence of the 060-060 interaction. We use rc = 10.04 A [6], ,3 = 0.866 A“, and De = 0.8 eV based on a combination of our previous calculation for the C60 solid [39] and experimental data [13]. Recent experimental data suggest a smaller value Ecoh(C'60 solid) = 1.76 eV (at T = 0 K) for polycrystalline C60 films [40]. As we will show later on, an accurate value of the cohesive energy is not crucial for the stability of the compounds since it is partly or mostly compensated in the formation of the compound with the same fee crystal structure. It only has a small influence on the formation energy of the AeCeo phase with a bcc structure, and a small inaccuracy in Ecoh(C6O solid) will not reverse the conclusions we draw. We use the experimental results for the electron affinity of neutral C60, A(Ceo) = 2.74 eV [41], and the ionization potential I (C60) = 7.54 :l: 0.04 eV [42]. We note 92 that the electron affinity of Cm is only slightly smaller than that of the electronega- tive elements in group 7A, which makes the C60 molecule a good electron acceptor. On the other hand, the ionization potential of the C60 molecule lies close to that of electropositive Mg, which makes the C60 molecule a good electron donor as well. When calculating the higher electron affinities and ionization potentials, we modify the above values by the electrostatic energy which occurs during the attachment or de- tachment of electrons from a charged sphere with the C60 molecule radius R = 3.5 A. The calculated total ionization potentials and electron affinities are summarized in Tables 4.1 and 4.2. These estimates are in good general agreement with available ex- perimental data of Refs. [43, 44, 45, 46]. In particular, there is experimental evidence for a linear dependence of the ionization potentials and electron affinities on the final state charge [47]. The formation energy Ecoh(A,f C26” of the intercalation compound from the ions depends strongly on the structure. Here, we consider the ACso and A3C60 solid with the fcc structure, and the A6C60 solid with the bcc structure [32]. Ecoh(A,",’C20‘ ) can be decomposed into three terms, Z —Ecoh(A:Cgo-) : EMadelung 'l' EBM — ED( 610- _ 030-) ' (410) The factor Z in the pair potential term denotes the coordination number of the C60 molecules, which is 12 in the close—packed fee structure and 8 in the bcc structure. The formation of intercalation compounds is driven by a large gain in Madelung energy. We consider a complete charge transfer between the intercalants and the C60 clusters, in agreement with ab initio results of Refs. [34, 35] for the alkali compounds. We express the Madelung energy per unit cell as EMadelung = —0q2/0 - (4.11) 93 Table 4.1: Total ionization energy Ito. corresponding to the process C60 AC3; +n e‘. Final state configuration 0:0 C6201 035* 033L 0%? 1....(eV)a 7.545 19.20 34.96 106.98 362.07 19.006 ‘ The first line contains data used in our calculation. b Experimental value of Hertel et al, Phys. Rev. Lett. 68, 784 (1992), based on photoionization. c Experimental value of Steger et al, Chem. Phys. Lett. ( 1992), based on photoionization. Table 4.2: Total electron affinity Am corresponding to the process Ceo+n e" ———>C30' . Final state configuration Ce], 0:; 0630- 030- C(33- Atot (8V) 2.74“ 3.42 1.09 -—24.65 —173.06 “ Experimental value of S. H. Yang et al, Chem. Phys. Lett. 139, 233 (1987). 94 Table 4.3: Madelung constants a for the structures considered in this work. Structure a A+Cg0 3.4951 .43“ng 22.1220 A303; 56.2670 Here, q is the charge on the intercalant and a is the lattice constant of the conven- tional cubic unit cell. The Madelung constants for the different structures have been evaluated using the Ewald method. Our values are in agreement with previous re- sults [32, 33] and are listed in Table 4.3. Note that the Madelung constants for A3C60 and A6C60 are extremely large when compared to the ACeo compound. This fact is mainly due to the large number of neighboring counter—ions for each C60 cluster. The gain in Madelung energy is only partly compensated by the (mainly) closed— shell repulsion between the A+ and the C35 ions in the lattice. An accurate knowledge of this repulsive interaction is necessary since it affects not only the cohesive energy, but also the equilibrium structure and compressibility of the bulk compounds. We model this closed—shell repulsion energy EBM by pairwise Born-Mayer type repulsive potentials K?M(r) [48] as EBM = Z VijBM(T) . (4.12) i. The prefactor 2 takes care of the spin degeneracy. The third—order nonlinear polarizability is related to the energy change in fourth-order perturbation theory which is given by [13] Vh V IV! :1 uh AE“) = p W M r 222?:ng (Eh — Ep)(Eh — Ep’)(Eh _ E13") Vhp Vpp' nlh' Vh’ h —2 223;; (E), - E,)(E1. " EP'XE” — EP') Vhp Vph" Vh"h' Vh'h + :22: (E. — E.)(E. — E.)(E.~ - E.) P h’ h” EVthph'Vh' pIme {322213. — E.)(E. - E.)(E. - E.) ' (5'5) Pp'h' Diagrams 2 and 3 in the fourth order diagrams are equal, giving the prefactor of 2 in the second term. The fourth term of the Eq. (5.5) is the sum of diagrams 5 and 6. AE“) can also be calculated in perturbation theory using a basis of many—particle 128 :03 / A W16 Figure 5.1: The Graphs for (a) the second order, and (b) the fourth order perturbation theory expressions in Eqs. (5.4) and (5.5). -+— represents a particle state and -+— denotes a hole state. 129 states [14, 15]. However, that formula is more difficult to use numerically since its energy denominators can be small, unlike the particle—hole energies in Eq. (5.5). The expressions in Eqs. (5.4) and (5.5), together with Eq. (5.2), yield directly the optical polarizabilities a and 7. We find that the values for a and 7 obtained using perturbation theory agree with values which we calculate directly by diagonalizing the tight—binding Hamiltonian. In order to determine the reliability of our approach, we first calculate the linear and third-order polarizabilities of the benzene molecule, a system which has been studied extensively both experimentally [4, 16] and theoretically [12]. For this pur- pose, we have to augment our tight—binding Hamiltonian for carbon by parameters suitable for hydrogen. We adjust the difference between the H13 and C2p energies to the difference of the atomic ionization potentials, which gives E(HIS) = —2.3 eV. For the hydrogen—carbon hopping integrals, we use V,” = —3.15 eV and V,,,,, = 1.7 eV at the H—C distance of 1.07 A found in C6H6, obtained by fitting the level spectrum of a CH radical which we calculated using the Local Density Approximation [17]. Our results for CeHe are presented in Table 5.1. The calculated polarizability in the plane of the benzene molecule is a” = 31.1 A3. Assuming the same value of the polarizability along the two principal axes in the molecular plane and zero perpendicular to it, we would predict < 0: >= (2/3)a“ = 20.7 A3. This value is consistent with the experimental result < 0 >= 10.0 A3 obtained for the solution [4] in view of the fact that we have neglected internal screening in the benzene molecule. The third order polarizability turns out to be 7” = 13.5 x 10‘36 esu, giving < 7 >= 9.0 X 10"36 esu. This is again comparable to the ab initio results [12] in the range of < 7 >= 1.3 — 1.7 x 10"36 esu and experimental data of Ref. [4] giving < 7 >= 3.85 x 10’36 esu. Our hyperpolarizability is somewhat larger than the ab initio results 130 Table 5.1: Calculated and observed optical susceptibilities of C60 and C6H6 molecules. < abare > < ascreened > X“) < 7bare > < 7screened > X(3) (A3) (A3) (esu) (10'36 esu) (10"36 esu) (10‘1” esu) G.H., 0* 20.7 9.0 G.H., b 10.02 3.85 0.101 60" 215.0 35.7 0.063 346.2 2.3c 0.05“ 60° 195 56 0.116 044’ 1.07 x 108 Ca,“ 750 C..." 0.2391 313 7 “ Present calculation. 5 Experimental values of Ref. [4]. C This value is obtained using < ab". > and < amemd > of Ref. [18]. d This value is obtained using the experimental value n = 2 of Ref. [6] in the expression n2 = 1 + 470(0), ° Theoretical values of Ref. [18]. ‘ Experimental values of Ref. [3]. 5 Experimental values of Ref. [5]. 11 Experimental values of Ref. [6]. which again is to be expected due to our neglect of intramolecular screening. Screening is even more important in the large C60 cluster than in benzene, and we shall include it in our calculations of this system. For a spherical molecule such as the C60 fullerene, the screened linear and third order nonlinear polarizabilities are given by abare ascreened = abare/(l '1' R3 ) 1 (5'6) and abare 7screened = ”flare/(1 '1" R3 )1l , (5.7) 131 where R is the radius of the fullerene. The matrix surrounding the cluster in a bulk sample also modifies the external field and hence the screening. We determine the bulk linear susceptibility using 4 11' X“) = Nascrccned/(l _ ”'3‘Nascreened) a (5.8) which is equivalent to the Clausius—Mossotti relation. In the same way, we calculate the third—order nonlinear susceptibility using 47r X(3) = NVscreencd/(l _ E'Ivascreencd)4 - (5.9) In these equations, N z 1/ 720 A3 is the density of clusters and amemd and 7,c,ee,,cd are the screened linear and nonlinear polarizabilities of an isolated cluster, respec- tively. Our results for the C60 clusters and the solid are given in Table 5.1. As mentioned above, our perturbation theory calculations are consistent with results obtained by a direct diagonalization of the Hamiltonian in a weak external field. We fit the energy to Eq. (5.2) using 5 = 0 — 0.01 V/A which is much weaker than the field 8 z 0.7 V/A when the first level crossing occurs. We find a large bare polarizability 0154.4 = 215 A3, which is reduced considerably due to the internal depolarization field to amcmd = 35.7 A3, close to R3 = 42.8 A3 the linear polarizability of a metallic sphere. It is also in agreement with the quantum—chemical result [19] a z 300 — 400 a.u., depending on the basis set. The latter value is very close to that for a classical metallic sphere with a radius R = 3.5 A, amemd = R3 = 42.9 A3. Of course, such large screening cannot be expected in the planar benzene molecule. The polarizability of an isolated cluster can be inferred from the linear suscepti- bility or index of refraction of the bulk material using the Clausius—Mossotti equa- tion. Our predicted susceptibility is x“) = 0.06, in relatively poor agreement with 132 the experimental value x“) = 0.24 [6]. Part of the reason for this discrepancy is that Coulomb interaction is overestimated by the approximation of Ref. [2]; a better treatment of the Coulomb interaction with the tight—binding Hamiltonian yields a susceptibility of x“) = 0.116 [18]. Also, the crystal field in the bulk breaks the sym- metry of the C60 molecule, so that direct HOMO—-—>LUMO transitions can occur. The susceptibility of the solid would also be increased by the possibility of a virtual electron transfer between C60 molecules. We find a very large positive value of < 7541-4 >= 3.5 x 10'3“ esu for the bare third order hyperpolarizability, more than one order of magnitude larger than in benzene. Our value is in fact within the range of two of the experiments, Refs. [5] and [6]. However, this value gets screened by the induced dipole field in the C60 which will be much stronger than in the planar C6H6 structure [20]. Using the screening factor ozmwmal/ozg,a,c = 3.5 from Ref. [18], we find < 78mm...) >= 2.3 x 10"36 esu, about the same as in benzene. For the solid, we obtain x(3) = 5 x 10"“ esu using the empirical susceptibility, which is almost two orders of magnitude smaller than the experimental value of Ref. [6]. We do not understand the origin of this large discrepancy. One possible reason is the high laser frequency ha: 3 1.2 eV used in the experiments. We have considered the effect of the frequency dependence in the perturbation calculation, Eq. (5.5). We find that a substantial change in the hyperpolarizability only occurs due to virtual two—photon transitions within a very narrow energy range of the HOMO—LUMO transition. Since the transition is spread out by crystal field effects, we ignore this enhancement in the present paper. Another possibility is that one must go beyond the tight—binding approximation to calculate this quantity, as seems to be the case for the linear susceptibility. 133 In any case, our result is more than seven orders of magnitude below the data quoted in Ref. [3], and we conclude that these data are probably in error. 5.2 Collective electronic excitations of C60 In this Section I will discuss the dynamical response of a C60 cluster to an external electromagnetic field. Motivated by a measurement of the photoabsorption strength in C60 clusters in the low frequency region [21], we have calculated the electromag- netic response of this remarkable system at nonzero frequencies. As I discuss in the following, we obtain quantitative agreement with the experiment for fit.) S 10 eV. Moreover, our calculations predict a giant Mie—type resonance at 710) z 20 eV, which was later observed experimentally [22]. We use linear response theory, which is most appropriate for large systems with mobile electrons where screening can be significant. Within the one—electron theory, for which we shall use mostly a tight—binding model, the dipole operator has two contributions, from a dipole moment due to intersite charge transfer, and from the dipole moment on a site. We write the dipole operator as Dz : D£‘)+D[2) = 2 “1,.an 2(2’) + 61201.9... + al,.a...) , (5.10) a,i where 2(i) is the z—coordinate of the i—th carbon atom and d is the s —+ p. dipole matrix element on a carbon atom. Starting from an independent particle picture, we define the polarization propa- gator for the free dipole response by [23] 11(0)“, = < Dzh> 2 . . D.( ) gl pl l I(Cp-Eh)2_(w+"})2 (5.11) 134 Here, p and h label particle and hole eigenstates of the single—particle Hamiltonian and 6,, and c), are the corresponding particle and hole energies. The full response requires the interaction between electrons, which we shall ap- proximate as a pure Coulomb interaction. For the dipole response we only need to consider the fields generated by D9) and D9). We shall first consider a simple approximation, keeping only the charge operator 09’. Then the electron—electron interaction is e2 D9) Dgll/Re', where R z 3.5 A is the radius of the C60 cluster. The screened response function due to D9) in Eq. (5.1) is given in this approximation by [23] W39) = (1 + n:°’(w);::)-‘n£°)(w) . (5.12) Note that in the present approach the 11’s are ordinary functions and the equation is algebraic and easily computed. We shall later consider a more refined approximation by including the dipole moments on the sites, described by D9). The effect will be to replace Eq. (5.3) by a 2 x 2 matrix equation 11 = (‘1' + fI(°)f/)-‘fl(°) , (5.13) Here, we have separated the charge and the internal dipole operators and write the free response as a 2 X 2 matrix with elements 2(6, —- 5),) IN) = < Nflh> . . .6400) 2:: 1% z I I z b? (c,-e.)2-(a:4-zn)2 mh (5.14) V in Eq. (5.4) is the 2 x 2 matrix of the interaction, given by ~ 62 1 1/2 V = H5( 1/2 R/2d )' (5°15) 135 The HOMO to LUMO transition is forbidden by parity, and the lowest optically allowed transitions are hu -—) tlg, hg —) t1“, and hu —-2 kg, with tight—binding excita- tions energies of 2.8 eV, 3.1 eV, and 4.3 eV. These values compare well with the LDA values 2.9 eV, 3.1 eV and 4.1 eV [24] and are reflected in the free response shown in Figure 5.2(a). As we discuss in the following, the electron interaction changes the excitation energies significantly and is essential for even a qualitative understanding of the transitions strengths. Our results for the screened response, based on the RPA treatment of the tight— binding Hamiltonian and the charge dipole operator D9), are shown in Figure 5.2(b). A comparison to the free response shows that the lowest allowed particle—hole transi- tion is slightly shifted in energy to 2.9 eV and agrees well with the observed [21, 25] value of 3.1 eV. The oscillator strength6 of this transition is drastically reduced by a factor of 400 from the value 3.8 in the free response to 0.010 in the RPA. This brings the transition strength close to the measured [25] oscillator strength of 0.004. An independent calculation of the interacting response has been performed using the quantum-chemical CN DO/ S method [26], but this method yielded an oscillator strength of 0.08, which is considerably less screening than in RPA. Turning to the next few excitations, we find the transitions to be shifted sub- stantially upward in energy as compared to the free response shown in Figure 5.2(a). This brings them into fair agreement with the observed [21, 25] dipole excitations at 3.76 eV, 4.82 eV and 5.85 eV. These transitions are also screened, but the screening factor is only in the range 10—30. They thus appear relatively strong compared to the low transition, in agreement with the experimental data of Ref. [25] [see Figure 5.2(c)]. The results for the low—lying excitations are essentially unaffected when the on— 6The oscillator strength is defined by f = 2mel < f IDzli > |2(E, - E§)/h2. 136 Differential Oscillator Strength (ew') mums 1°1'm080 9010.1331th ' l (c) 3 1 E 8 a 3 1 8' 1 4.44.4444?» E(OV) Figure 5.2: Free response (a) and RPA response (b) of C60 clusters to an external electromagnetic field (solid line). The sharp levels have been broadened by adding an imaginary part hr) = 0.2.eV to the energy. The dashed line indicates the inte- grated oscillator strength. (c) Observed photoabsorption spectrum of C60 in solution [From H. Ajie et al, J. Phys. Chem. 94, 8630 ( 1990) and G. F. Bertsch, A. Bulgac, D. Tomanek, and Y. Wang, Phys. Rev. Lett. 67, 2690 (1991)]. 137 site dipole operator D?) is added to the response7. For example, the positions of the low states are shifted by less then 0.1 eV, and the strength of the lowest transition is changed by only 5%. As we will discuss below, the effect of D?) on the higher excitations is much more pronounced. Turning to the plasmon—like transitions at high energy, we first note that the tight—binding Hamiltonian with the operator D?” has a total oscillator strength of N = 2m/7’t2 EM I < pIDglllh > [2(cp—eh) z 180, which is, of course, the same in both the free response and in RPA. This value is close to the theoretical upper bound of 240, ignoring the core electrons, giving some credibility to the model for the entire energy range. Figure 5.3 displays the excitation spectrum of C60 extending up to plasmon energies, obtained using several approximations. The D9) free response function, shown in Figure 5.3(a), has a broad band of transitions in the “intermediate” energy range hw z 10 - 20 eV. With the electron—electron interaction present, the main effect of the Coulomb field is to collect the strength of these transitions into a single collective excitation, a Mie—type plasmon. The spectrum shown in Figure 5.3(b) has this giant resonance at an unusually high frequency hw z 30 eV, well beyond the typical plasmon range (hw < 10 eV), which has not been observed so far. In contrast to the low energy region, the inclusion of the on—site dipole term D?) has a substantial effect on the high—frequency response. The total integrated oscillator strength is reduced to 71, leaving most of the total strength outside the model space. We find that these extra terms shift the plasmon energy to fit.) z 20 eV and decrease the oscillator strength by a factor of £2 when compared to the results in Figure 5.3(b). A similar plasmon mode at hw z 22 eV has been observed previously [27] in amorphous carbon films. 7The oscillator strength is defined by f = 2me| < fIDzli > [2(E', — E;)/f12. 240 220 200 1 30 160 140 120 100 Differential Oscilator Strength (eV‘l) Figure 5.3: Dipole response of C60 clusters to an external electromagnetic field, shown in an expanded energy region. (a) Free response, (b) RPA response based on the charge term D9), and (c) RPA response based on both the charge and the dipole terms D9) and D?) in Eq. (5.6). (d) Interacting response of a thin jellium‘shell, describing the electron—electron interactions in LDA. The response function is given by the solid line, and the integrated oscillator strength is shown by the dashed line [From G. F. Bertsch, A. Bulgac, D. Tomének, and Y. Wang, Phys. Rev. Lett. 67 , 138 0 510152025303540 E(eV) 2690 (1991)]. 1 ‘ y‘a.‘ 240 220 200 150 150 140 120 1 00 30 60 40 20 10 15 20 25 so 35 40 E(eV) 240 4 220 ‘ zoo - 180 - 160 - 140 ~ 120 100 80 60 4O 20 qfiuans Joiemaso paieJfiawI 139 This high frequency Mie—like plasmon has its origin in the high valence electron density p in the Geo cluster, and can be understood qualitatively by considering a conducting spherical shell with a radius R z 3.5 A and 240 conduction electrons. The Mic frequency for a solid conducting sphere is given by 710: = [41r)0e""/(3m)]1/2 z 25 eV. We have also made a jellium calculation for a charged spherical shell using the program J ELLYRPA [28], and find the strength function shown in Figure 5.3(d). The energy agrees with Figure 5.3(c), but in the jellium model the total oscillator strength is concentrated in the plasmon. An additional plausibility argument for the jellium picture of C60 follows from the static polarizability a. We find the classical conducting sphere value a = R3 = 290 a.u. to be in good agreement with the tight—binding value of 250 a.u. and the quantum—chemical result [19] a z 300 — 400 a.u., depending on the basis set. Our predictions for the Mie plasmon at 710) z 20 eV have been confirmed by photoionization experiments on free C60 clusters [22]. The C3}, photoion yield as a function of photon energy is reproduced in Figure 5.4 and shows a clear maximum in the predicted energy range. Note the low frequency cutoff in the experimental spectrum at the ionization potential of C60 at 7.54 eV. I would like to mention that there are several other effects that should be included in a more refined treatment of the response function. We have neglected higher multipolarities of the Coulomb screening field, as well as the exchange and correlations terms in the residual interaction. These additional terms could alter the very large screening factor for the lowest transitions. However, we feel that the dipolar Coulomb interaction is dominant in the Mie plasmon region, and a refinement of this interaction would not significantly affect the results. This has been confirmed by a subsequent calculation of Bulgac and Ju [18]. 140 1.0 0.6 - 0.4 « ~35 m. .4.) 0'2 T 430 'E 0.0 - :1 r v T P25 '0. 70 7.4 \ 7.8 {3 20- . \ ~20 \ \ ' uo-D 10- \ -10 2 Q) a.) 5. *5 5 O f f ' r V T V I v ' r r O 0 4 12 15 20 24 28 32 36 photon energy / eV Figure 5.4: Observed C3}, photoion yield as a function of photon energy displaying excitation of the giant plasmon resonance [From I. V. Hertel et al, Phys. Rev. Lett. 68, 784 (1992)]. 141 5.3 Conclusions I have presented results for the static and dynamic dielectric response of the C60 cluster, based on a tight—binding Hamiltonian for the single—particle states. Our calculations of the polarizability and hyperpolarizability of the C60 cluster have shown that the valence electrons are quite delocalized. Its linear polarizability is close to that of a metal sphere and the hyperpolarizability is close to that of conjugated polymers which can be understood in the free electron model. Our RPA calculations of the dynamical response indicate strong dynamical screen- ing which results in a collective electronic plasmon mode in C60, Due to the strong Coulomb interaction between the electrons, the lowest allowed excitation at hw z 3 eV is strongly screened by a factor of 400 as compared to the free response, in good agreement with experiment. Other low lying dipole excitations are moderately suppressed. We predict a giant collective resonance at an unusually high energy ha) z 20 eV, which has been recently observed experimentally [22]. We interpret this mode as a Mie plasmon caused by the large delocalization of the carbon valence electrons and the large charge density. Bibliography [1] Yang Wang, George F. Bertsch, and David Tomanek, Z. Phys. D (1993). [2] G.F. Bertsch, A. Bulgac, D. Tomanek and Y. Wang, Phys. Rev. Lett. 67, 2690 (1991). [3] W.J. Blau, H.J. Byrne, D.J. Cardin, T.J. Dennis, J.P. Hare, H.W. Kroto, R. Taylor and D.R.M. Walton, Phys. Rev. Lett. 67, 1423 (1991). [4] G.R. Meredith, B. Buchalter and C. Hanzlik, J. Chem. Phys. 78, 1543 (1983). [5] Ying Wang and Lap—Tak Cheng (submitted to J. Phys. Chem.). [6] Z.H. Kafafi, J.R. Lindle, R.G.S. Pong, F.J. Bartoli, L.J. Lingg and J. Mil- liken, Chem. Phys. Lett. 188, 492 (1992). [7] Z.H. Kafafi, F.J. Bartoli, J.R. Lindle, and R.G.S. Pong, Phys. Rev. Lett. 68, 2705 (1992). [8] KC. Rustagi and J. Ducuing, Optics Communications 10, 258 (1974). [9] R.J. Knize and J.P. Partanen, Phys. Rev. Lett. 68, 2704 (1992). [10] H.W. Kroto, J.R. Heath, S.C. O’Brien, R.F. Curl, and RE. Smalley, Nature 318, 162 (1985). 142 143 [11] W. Kratschmer, L.D. Lamb, K. Fostiropoulos, and D.R. Huffman, Nature 347,354 (1990). [12] E. Perrin, P.N. Prasad, P. Mougenot, and M. Dupuis, J. Chem. Phys. 91, 4728 (1989). [13] RD. Mattuck, A Guide to Feynman Diagrams in the Many—Body Problem, McGraw—Hill Book Company, New York (1967). [14] J.F. Ward, Rev. Mod. Phys. 37, 1 (1965). [15] J.O. Morley, P. Pavlides and D. Pugh in: Organic Molecules for Nonlin- ear Optics and Photonics, pp. 37-52, edited by J. Messier et al., Kluwer Academic Publishers (1991 ). [16] F. Kajzar and J. Messier, Phys. Rev. A 32, 2352 (1985), and references cited therein. [17] P. Hohenberg and W. Kohn, Phys. Rev. 136, 3864 (1964); W. Kohn and L.J. Sham, Phys. Rev. 140, A1133 (1965). [18] A. Bulgac and N. Ju, Phys. Rev. B 46, 4297 (1992). [19] P.W. Fowler, P. Lazzeretti and R. Zanasi, Chem. Phys. Lett. 165, 79 (1990). [20] J. Ducuing, in “Nonlinear Spectroscopy”, edited by N. Bloembergen, pp. 276— 295, North Holland Publ., Amsterdam (1977). [21] H. Ajie, M.M. Alvarez, S.J. Anz, R.D. Beck, F. Diederich, K. Fostiropoulos, D. R. Huffman, W. Kratschmer, Y. Rubin, K.E. Schriver, D. Sensharma, and R.L. Whetten, J. Phys. Chem. 94, 8630 (1990). 144 [22] I. V. Hertel, H. Steger, J. de Vries, B. Weisser, C. Menzel, B. Kamke, and W. Kamke, Phys. Rev. Lett. 68, 784(1992). [23] G.D. Mahan, Many—Particle Physics, (Plenum Press, New York, 1981). [24] Susumu Saito and Atsushi Oshiyama, Phys. Rev. Lett. 66, 2637 (1991). [25] J.R. Heath, R.F. Curl, and RE. Smalley, J. Chem. Phys. 87, 4236 (1987). [26] Sven Larsson, Andrey Volosov and Arne Rosén, Chem. Phys. Lett. 137, 501 (1987). [27] CC. Ahn and O.L. Krivanek, EELS Atlas: A reference guide of electron energy loss spectra covering all stable elements, (HREM Press, Center for Solid Science, Arizona State University). [28] G.F. Bertsch, Comput. Phys. Comm. 60, 247 (1990). Chapter 6 Summary and Conclusions Cluster physics is a fascinating new field of growing importance. Thanks to my advisors, I have been able to actively participate in the research in this field, and to witness its rapid growth. In this Thesis 1 have investigated both metallic clusters and nonmetallic clusters. The main achievements of this Thesis are: (1) predictive calculations for the optical response of small Lin and Nan clusters, specifically the evolution and fragmentation of the plasmon mode; (2) predictions of the stiffness of C60 fullerite as a function of external pressure; (3) prediction of the stability of the fullerite intercalation compounds; (4) investigations of the linear and nonlinear polarizability of the C60 cluster; (5) predictions of the dynamical dielectric response of C60, specifically the high—frequency Mie plasmon mode. In Chapter 1, I presented a brief overview of cluster properties. I have focused on the structural and electronic properties of alkali metal clusters and the C60 cluster and discussed the experimental techniques as well as theoretical concepts pertinent to my Thesis. Chapter 2 has been devoted to the theoretical tools used in the study of clusters in this Thesis. I reviewed the techniques used in this Thesis, namely the Density Functional Formalism and the Local Density Approximation (LDA) as well as the 145 146 tight-binding formalism for ground-state properties of clusters, and the Random Phase Approximation (RPA) for electronic excitations. In Chapter 3, I have presented results for the equilibrium structure of small N an and Lin clusters and collective electronic excitations and their damping in these sys- tems. The results for the collective excitations in the first two closed—shell clusters with n = 2,8 atoms are given in detail. Our results indicate that the coupling of electronic levels to vibrational degrees of freedom accounts quantitatively for the ob- served width of the collective electronic excitations in alkali dimers. The origin of the analogous line broadening in Nag is presently unresolved. In Chapter 4, I have presented calculations of the equilibrium structure of C60 fullerite as a function of external pressure. I found that at zero pressure, carbon atoms in neighboring C60 clusters are no closer than 2.65 A apart and interact by pairwise Van der Waals forces. Consequently, the bulk modulus of C60 should be very low, similar to a molecular solid. With increasing hydrostatic pressure, a gradual transition to a hard-core repulsion between neighboring clusters is predicted. Only at high pressures beyond z70 GPa, the bulk modulus of fullerite is expected to exceed that of diamond, and a transition to diamond is predicted. In order to sort out the likely candidates for C60 based superconductors, I have also calculated T = 0 K formation enthalpies of donor and acceptor based C60 ful- lerite intercalation compounds with the A060, A3C60, and A6C60 stoichiometries, as well as their structural and elastic properties. The results indicate that all alkali and some alkaline earth elements form stable fullerite intercalation compounds. The corresponding calculations for acceptor intercalants indicate that none of the group 6A and 7A based ionic intercalation compounds is stable with respect to solid C60 and the intercalant in the standard form. 147 I have devoted Chapter 5 to the study of the electronic properties of the C60 cluster. Our calculations, based on a parametrized tight—binding Hamiltonian, have proven the valence electrons to be quite delocalized, giving rise to a large polarizabil- ity and hyperpolarizability of the C60 cluster. Our RPA calculations of the dynami- cal response indicate strong dynamical screening resulting in significant screening of low—lying excitations, and a giant collective resonance at an unusually high energy hw z 20 eV. This mode, which has been recently observed experimentally, has been interpreted as a Mie plasmon mode, which results from the large delocalization of the valence electrons and their large charge density in C60, "Illllllllllll'lllllllf