R. (I. as. ”1. 3. max” .wfiu . a... 1... . 43.. “$5.3. a , l )C c. ~‘k . .316 r. 1.... d1 v, -. .1 ~\X 10> .. .. ab), .) ...‘ i rn .- msrw ‘ $1... a 12...: .- N. .éf. :e.,..:.:% ‘1! um v-q‘ \r' . I n r t' ' I 6:039 53c 50 This is to certify that the dissertation entitled STRUCTURAL TRANSITIONS IN NANOSCALE SYSTEMS presented by Mina Yoon has been accepted towards fulfillment of the requirements for the PhD. degree in Ptysics @Qfl/fl awful P Major Professor’s Signature 09/22/qu Date MSU is an Affinnative Action/Equal Opportunity Institution —‘-‘-a-n-p-.---o-u-o-o-o---u--.b-v-o_-o-o-n-o--4. _ —.— —.—.-.- LIBRARY Michigan State University PLACE IN RETURN Box to remove this checkout from your record. TO AVOID FINES return on or before date due. MAY BE RECALLED with earlier due date if requested. DATE DUE DATE DUE DATE DUE 6/01 cJClRCJDateDuo.p65-p.15 I will; [LiibE-f fifihvEu-Uirlrv "bravura; (until... {tafialriu/ was“! infra. ) t 1. t.» .I‘ . Larrfld r, u t . .‘u‘r lbs In.‘l.lil?. .l»....:. STRUCTURAL TRANSITIONS IN NANOSCALE SYSTEMS By Mina Yoon A DISSERTATION Submitted to Michigan State University in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Physics 2004 ABSTRACT STRUCTURAL TRANSITIONS IN NANOSCALE SYSTEMS By Mina Yoon In this work I investigate three different materials: nanoscale carbon systems, fer- rofluid systems, and molecular-electronic devices. In particular, my study is focused on the theoretical understanding of structural changes and the associated electronic, mechanical, and magnetic properties of these materials. To study the equilibrium packing of fullerenes in carbon nanotube peapods Opti- mization techniques were applied. In agreement with experimental measurements, my results for nanotubes containing fullerenes with 60-84 atoms indicate that the axial separation between the fullerenes is smaller than in the bulk crystal. The reduction of the inter-fullerene distance and also the structural relaxation of fullerenes result from a large internal pressure within the peapods. This naturally induced ”static” pressure may qualify nanotubes as nanoscale autoclaves for chemical reactions. Combining total energy calculations with a search of phase space, I investigated the microscopic fusion mechanism of Ca) fullerenes. I show that the (2 + 2) cycload- dition reaction, a necessary precursor for fullerene fusion, can be accelerated inside a nanotube. Fusion occurs along the minimum energy path as a finite sequence of Stone-Wales (SW) transformations. A detailed analysis of the transition states shows that Stone-Wales transformations are multi-step processes. I propose a new microscopic mechanism to explain the unusually fast fusion pro- cess of carbon nanotubes. The detailed pathway for two adjacent (5, 5) nanotubes to gradually merge into a (10,10) tube, and the transition states have been identified. The propagation of the fused region is energetically favorable and proceeds in a mor- phology reminiscent of a Y-junction via a so called zipper mechanism, involving only SW bond rearrangements with low activation barriers. Using density functional theory, the equilibrium structure, stability, and electronic properties of nanostructured, hydrogen terminated diamond fragments have been studied. Such diamondoids can enter spontaneously into carbon nanotubes where polymerization of diamondoids is favourable. I studied the equilibrium structure of large but finite aggregates of magnetic dipoles, modeling a colloidal suspension of magnetite particles in a ferrofluid. With in- creasing system size, the structural motif evolves from chains and rings to multi-chain and multi—ring assemblies. These structural changes depend on external parameters and result from a competition between various energy terms, which can be described analytically within a continuum approximation. I use advanced quaternion molecular dynamics to model a potential application of magnetic nanostructures for targeted medication delivery. Inert microcapsules, containing the active medication and a small number of magnetite nanoparticles, may be transported using an inhomogeneous magnetic field through blood vessels to a desired location in the body. Triggered by an abrupt change in the applied field, structure of magnetite aggregates changes from a ring to a chain, thus puncturing the microcapsule and releasing the medication. The stability of the system under thermal and magnetic fluctuations has also been studied. The investigation of the energetics, electronic structure and electron transport properties of oligo (phenylene ethylene) molecules shows that a net charge transfer to the molecules, induced by applying a bias voltage can result in a transition from the stable planar to a less stable twisted isomer. This structural transition alternates significantly its electric properties from a conducting to an insulating state. Table of Contents LIST OF FIGURES vii 1 Introduction 1 2 Structural Changes in Nanoscale Carbon Materials 9 2.1 Theoretical Techniques .......................... 9 2.2 2.3 2.1.1 Total Energy Calculations: Ab initio Density Functional Theory 10 2.1.2 Total Energy Calculations: Semiemprical Methods ....... 14 Equilibrium Packing Geometry of Fullerenes in Nanotube Peapods . . 19 2.2.1 Introduction ............................ 19 2.2.2 Energetics of Fullerene Encapsulation .............. 20 2.2.3 Packing of Fullerenes in Nanotube Peapods .......... 23 2.2.4 Summary ............................. 28 Fullerene Fusion Mechanism ....................... 30 2.3.1 Introduction ............................ 30 2.3.2 Microscopic Mechanism of F ullerene Fusion ........... 31 2.3.3 Summary ............................. 38 iv 2.4 Nanotube Fusion Mechanisms ...................... 40 2.4.1 Introduction ............................ 40 2.4.2 The Zipper Mechanism of Nanotube Fusion .......... 41 2.4.3 Summary ............................. 49 2.5 Diamondoids as Building Blocks of Functional N anostructures . . . 51 2.5.1 Introduction ............................ 51 2.5.2 Properties of Isolated Diamondoids ............... 54 2.5.3 Interaction of Diamondoids with Carbon Nanotubes ...... 61 2.5.4 Summary ............................. 63 3 Structural Transitions in Ferrofluid Systems 66 3.1 Introduction ................................ 66 3.2 Equilibrium Structure of F errofluid Aggregates ............. 68 3.3 Targeted Medication Delivery Using Magnetic Nanostructures . . . . 79 3.4 Summary ................................. 90 4 Quantum 'D'ansport through Molecules: Effect of Structural Changes 91 4.1 Theoretical Techniques .......................... 91 4.1.1 Conductance Calculations: Green’s function technique . . . . 92 4.1.2 Non-equilibrium Green’s function technique .......... 99 4.2 Microscopic Switching in Molecular Memory Devices ......... 104 4.2.1 Introduction ............................ 104 4.2.2 Electronic Properties of Isolated Molecules ........... 104 4.2.3 Quantum Transport through Molecules ............. 111 4.2.4 Current-Voltage (IV) Characteristics .............. 114 4.2.5 Summary ............................. 116 BIBLIOGRAPHY 120 vi 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 2.10 2.11 List of Figures Energetics of fullerene encapsulation. .................. Equilibrium packing structure of fullerenes in peapods. ........ Fullerene-induced strain in the nanotube wall of peapods for different packing geometries ............................. Microscopic mechanism of fullerene fusion in peapods .......... Energetics of the fullerene fusion in peapods ............... Zipper mechanism of the nanotube fusion in the Y—junction geometry of one of our model morphologies, containing one octagon and four heptagons in the junction area. ..................... Dynamics of partly merged nanotubes. ................. Time sequence of HRTEM images evidencing the zipper process for the coalescence of two SWNTs. ..................... Relaxed structures of lower diamondoids, identified experimentally in Ref. [11] ................................... Electronic structure of diamondoids. .................. Electronic, structural, and cohesive properties of diamondoid isomers considered here as a function of the polymantane order, reflecting the number of adamantane cages. ...................... 21 24 26 33 42 47 49 52 55 57 2.12 Structural and electronic properties of an elongated hexamantane isomer. 59 vii 2.13 Energetics of a diamantane molecule entering (n, n) carbon nanotubes 2.14 3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8 4.1 4.2 4.3 4.4 4.5 4.6 with various diameters ........................... 62 Energetics of diamantane inside carbon nanotubes. .......... 64 Structures of ferromagnetic aggregates under considerations. ..... 70 A comparison between discrete and continuum approach for energy as a function of number of particles for a given structural motif ...... 74 Structural transitions in ferrofluid aggregates. ............. 75 The phase diagrams of ferromagnetic particles as a function of param- eters ..................................... 77 Schematic view of a microcapsule containing a few magnetic tops and fluid in a membrane. ........................... 81 Thermodynamic behavior of magnetic particles. ............ 84 Thermodynamic behavior of magnetic particles under external mag- netic field (H = 1500 Oe). ........................ 85 Time evolution of the microcapsule from its initial equilibration (T=300 K) in zero field at time t = 0. ........................ 88 Illustration of a system for conductance calculation ........... 92 Representation of a semi-infinite electrode by a periodic array of con- ducting units called slices [145]. ..................... 94 Illustration of a reduced system for conductance calculation. ..... 100 Equilibrium structure and potential energy of isomers as a function of the rotation of the central phenyl ring. ................. 106 Schematic geometry of the self-assembled monolayers (SAM) and po- tential energy of of the dipole layer as a function of the dipole orientation. 109 Total energy of oligo molecules as a function of the torsional angle. . 110 viii 4.7 4.8 4.9 Conductance of oligo molecules as a function of bias voltages. Current-Voltage (IV) characteristics of three phenyl ring molecules. IV characteristics of three phenyl ring molecules with introducing side groups in the middle ring. ........................ 112 115 Chapter 1 Introduction I have studied three different nanoscale materials: nanoscale carbon materials, ferrofluid systems, and molecular electronic devices. These seemingly completely different systems share interesting aspects that are both fundamentally interesting, and could be useful for applications in nanotechnology. My work was in particular focused on understanding structural transitions in those materials. It turns out that even minor structural changes in such nanoscale systems may dramatically affect their physical properties. Understanding ways to achieve such structural modifications will thus enable us to change the physical properties of these nanostructures. Hence, theoretical understanding of structural transitions opens a wide range of applications and is fundamentally important for the development of nanotechnology. l. Nanoscale Carbon Materials Carbon-based materials are unique due to the large variety of chemical bonding they exhibit. Depending on whether the interatomic bonds are sp, spz or spa, the morphology ranges from linear chains to planar graphite and the bulk diamond struc- ture. Among these, the 3-dimensional network of sp3 diamond and 2-dimensional sheets of SP2 graphite are prevalent due to their stability. The discovery of low di- mensional carbon materials, such as fullerenes and nanotubes, has attracted great research interest during the past two decades. Their unique mechanical and electrical properties open new possibilities in many areas of nanotechnology. I have focused my studies on selected mechanical, electrical, and magnetic prop— erties of various nanoscale carbon materials such as carbon nanotubes, fullerenes, hybrid structures of nanotubes and fullerenes, carbon foams, and nanoscale diamond. My study of ’peapods’, consisting of fullerenes encapsulated in nanotubes, was inspired by the High-Resolution Transmission Electron Microscopy (HRTEM) obser- vations of these systems. N anotube peapods could be used as diodes, logic circuits based on single fullerene molecules or for quantum computing. Some of the structural aspects of carbon nanotube peapods are intriguing, and even appear counter-intuitive. In particular, the equilibrium structure of fullerenes in peapods is quantitatively different from that in bulk solids. HRTEM images, electron diffraction and Raman measurements suggest that the equilibrium spacing between fullerenes in peapods is smaller by 3—4% than in three-dimensional molecular crystals, but larger than in solids based on polymerized fullerenes. I investigated the energetics and the packing of fullerenes being encapsulated in nanotubes. I found a net energy gain associated with fullerene encapsulation in nanotubes, giving rise to a ”capillary force”. In nanotube peapods, this force compresses encapsulated fullerenes with an effective pressure of the order of GPa, inducing a strain in the nanotube wall. The naturally induced pressure in nanotube peapods could turn them into chemical reactors. The strain in the nanotube wall, associated with closely packed fullerenes, modifies locally the reactivity of the tube wall, which could be used for chemical functionalization. I also identified the optimum geometry of fullerenes and nanotubes, which maximizes the encapsulation energy. Due to the unusual stability of the graphitic sp2 bond, large-scale structural changes in bulk fullerene crystals occur only under extremely high pressures and temperatures. On the other hand, fullerenes in nanotube peapods have been observed to fuse at relatively low temperature near 1, 100°C, far below the melting of fullerenes near 4, 000°C, which seems puzzling. No information is available about the detailed fusion process except the obvious conclusion that strong sp2 bonds should not be broken during structural rearrangements leading to fusion. In view of the fact that even minor structural changes in carbon nanostructures may modify significantly their physical properties, including magnetism, there is additional interest in understanding fusion as a way to control large—scale structural transformations. In a collaborative research effort, I found that large-scale structural changes and also fusion can be achieved by a finite sequence of generalized Stone-Wales transfor- mations, involving only bond rotation and avoid bond breaking. Also, the details of the transition states were identified so that the reaction time could be estimated. It turns out that the Stone-Wales transformations are multi-step processes with lower individual activation barriers. Especially, the 060 (2 + 2) cycloaddition reaction 1, a necessary precursor for fullerene fusion, can be accelerated inside a nanotube. Not only the fusion mechanism of fullerenes, but also that of nanotubes has been studied, to enable the use of nanoscale materials as potential building blocks for nanotechnology applications. I proposed a new microscopic mechanism to explain the unusually efficient fusion process of carbon nanotubes, where the pathway for the fusion process involves a sequence of Stone-Wales type bond rearrangements rather than energy intensive bond breakings. The zipper-like fusion mechanism proceeds by increasing the waist at the expense of the leg segments, and is energetically favorable. This finding oflers a new insight into the observed fast coalescence of single-walled carbon nanotubes under irradiation and heating in a High-Resolution Transmission Electron Microscope. 1(2 + 2) cycloaddition reaction is the reactions of two alkenes to form a cyclic product. In the peapod, the (2 + 2) cycloaddition is initiated by two ’double bonds’ facing each other in adjacent fullerenes. It has been known that the minute changes in the spatial arrangement of car- bon atoms can profoundly alter the electronic properties of materials such as nan- otubes from a semiconductor to a metal or superconductor. One could expect that structural rearrangements might also significantly change the magnetic properties of all-carbon allotropes from their known diamagnetic behaviour. Only recently, a weakly magnetic all-carbon structure has been reported, formed under high-pressure and high-temperature conditions, consisting of rhombohedrally polymerized C60 mo- lecules. Other experimental observations suggest the occurrence of ferromagnetism in a semiconducting nanostructured carbon foam with a low mass density. In a collaborative effort I also investigated the possibility that structural changes in nanoscale carbon materials can significantly change the magnetic properties of all- carbon materials from their known diamagnetic behaviour. The unexpected magnetic behaviour of all-carbon systems can be quantitatively interpreted using spin-polarized ab initio calculations. Our results suggest that unpaired spins are introduced by ster- ically protected carbon radicals, which are immobilised in the non-alternant aromatic system of sp2 bonded carbon with negative Gaussian curvature. This new mecha- nism to generate unpaired spins in semiconductors may find a useful application in the emerging field of spintronics. A new class of carbon nanostructures, hydrogen terminated nanoscale diamond fragments (”diamondoids”), has been successfully isolated experimentally. The iso- lated diamondoids occur in very different shapes, and many of them form molecular crystals. These nanoscale building blocks are very different, but compare well in stability and light weight with fullerenes and nanotubes. These uncommon organic molecules can be chemically functionalized, by substituting carbon or hydrogen atoms at the surface by other atoms or groups, to promote formation of specific polymers. I followed up on the interesting question whether these fragments can be en- capsulated in nanotubes to make functional building blocks. These studies involved investigating the equilibrium structure, stability, and electronic properties of nanos- tructured, hydrogen terminated diamond fragments. The equilibrium atomic arrange- ment and electronic structure of these nanostructures turn out to be very similar to bulk diamond. I found that such diamondoids may enter spontaneously into carbon nanotubes that are wide enough . 2. Ferrofluid Systems Ferrofluids, colloidal suspensions of magnetic particles in liquid carriers, are a unique class of materials, with high potential use in nanotechnology. The magnetic particles of an average size of about 10 nm are coated with a surfactant to prevent particle aggregation. They contain a single magnetic domain because of their small size. Many applications rely on the strong magnetic interaction between individual dipoles in these complex liquids. There already is a well developed ferrofluid tech— nology based on successful applications, enhancing the performance of mechanical or electromechanical devices. Of fundamental interest is the fact that finite dipole aggregates display a plethora of nontrivial equilibrium structures due to the competition between the strongly anisotropic dipole-dipole interaction, favoring open structures, and isotropic inter- particle forces as well as surface tension, which favor compact structures. Experi- mentally, a wide range of ferrofluid morphologies, ranging from compact structures to complex labyrinthine patterns, have been observed. Many studies of complex fluids focus on pattern formation with infinitely many particles. A more detailed under- standing of the equilibrium geometry of ferrofluid systems is essential for developing applications. To gain microscopic insight into the causes of pattern formation in ferrofluids, which is beyond the scope of present experimental observations, I performed total energy and structure optimization calculations for ferrofluid systems with a finite number of magnetic particles, where the surface tension of the aggregate plays a dominating role in its equilibrium structure. I found a structural evolution of sys- tems from chains to multiple rings and multi-wall tubes, as the number of particles increases. I mapped the inter-particle interactions in the colloidal suspension onto a continuum model, which offers an efficient tool to study the structural changes depending on external parameters, such as an external magnetic field or the liquid- particle interaction. I further used advanced quaternion molecular dynamics simulations to model a potential application of magnetic nanostructures for targeted medication delivery. In- ert microcapsules, containing the active medication and a small number of magnetite nanoparticles, may be transported using an inhomogeneous magnetic field through blood vessels to a desired location in the body. Triggered by an abrupt change in the applied field, the structure of magnetite aggregates may change from a ring to a chain, thus puncturing the microcapsule and releasing the medication. I obtained new results for the stability of the magnetic nanostructures under thermal and magnetic fluctuations, an important issue related to preventing an accidental delivery. 3. Molecular Electronic Devices The revolutionary history of electronic devices started with the invention of the transistor in 1947 by John Bardeen, William Shockley, and Walter Brattain. The development of integrated circuits by putting millions of transistors on a single silicon chip brought the foundation of the development of modern electronics. Since then the number of transistors on a chip has doubled ca. every 18 months; this exponential behavior is known as Moore’s Law. Today’s chips based on silicon include often more than 100 million transistors and the typical transistor size is about 100 um”. However, basic theoretical limits of the performance of silicon-based transistors will probably 2In 2003 NEC developed a chip with the record number of 40 billion transistors. be reached within a decade. To shrink the size of transistors, while following Moore’s law, we will rely on new types of transistors, consisting of few atoms or molecules. The advent of the Scanning Probe Microscopy in the 19808 offers a tool to ob— serve individual molecules, which led in the 19905 to the observation of individual molecules as electron conductors. This discovery gave researchers a vision of com- puters containing molecules as circuit components, as suggested by Mark Ratner and Ari Aviram in 1974. Since the late 19908, a series of successful attempts were un- dertaken to build molecular devices, including molecular diodes and molecular fuses. The eflort to go beyond these basic nanodevices has already been made by wiring these devices up into more complex circuits. The preliminary step was achieved by making circuits out of a single semiconducting nanotube, semiconducting nanowires, and organic molecules by the end of 2001. At that time people were very euphoric and molecular-electronics-based computer chips were expected to be in the stores by 2005. But the implementation of molecular electronic turned out to be more problematic than anticipated. I have contributed to a theoretical study of electronic and transport properties of oligo (phenylene ethylene) molecules sandwiched between two gold electrodes. This system is an interesting representative of organic molecules being used in molecular electronic devices. Due to their small size, in comparison with the electron mean free path, individual molecules conduct electrons in the ’ballistic’ regime, without internal Joule heat dissipation. I used the Landauer-Biittiker formalism to calculate the quantum conductance through molecules, such as oligo (phenylene ethylene) as a function of their structure. I found that a net charge transfer to the molecule, possibly correlated with the applied bias voltage, may cause an internal twist within a monolayer of charged mole- cules. Since the twisted molecule acts as an insulator, whereas the planar molecule is a good conductor, this device can be viewed as a molecular switch. When this molecule is entangled in a monolayer, its change from the energetically unfavorable twisted state to the planar ground state may be delayed due to inter-molecular interactions, resulting in a memory effect. Chapter 2 Structural Changes in N anoscale Carbon Materials 2.1 Theoretical Techniques The total energy of fullerene-nanotube systems has been calculated based on an energy functionals using a linear combination of atomic orbitals (LCAO). This func- tional successfully described the formation of peapods [1], multi-wall nanotubes [2], the dynamics of the “bucky-shuttle” [3], and the melting of fullerenes [4]. Our total energy formalism describes accurately not only the covalent bond- ing within the .9102 bonded graphitic substructures, but also the modification of the fullerene-nanotube interaction due to the inter-wall interaction and weak inter- fullerenes interaction. We find the use of an electronic Hamiltonian to be required in this system, as analytical bond-order potentials do not describe the rehybridization during the fullerene or nanotube fusion process with a sufficient precision. Our numer- ical results based on LCAO basis are compared to the first principles pseudopotential method [5, 6] that employs a numerical basis set for localized atomic orbitals [7], and which has been applied successfully to nanotubes and fullerenes [8]. When we consider the energetics and electronic structures of nanoscale diamonds, we employ methods based on the ab initio density functional theory (DFT) within the local density approximation (LDA). We use 'Ifoullier-Martins pseudo-potentials to describe the effect of atomic nuclei plus core electrons on the valence electrons, and the Perdew—Zunger parametrized exchange-correlation potential, as implemented in the SIESTA code [9]. Our Optimized diamondoid structures are in good agreement with those inferred experimentally [10, 11] and theoretically [12, 13]. The magnetic properties of different nanoscale carbon tetrapods were investi— gated using the first principles pseudopotential method [5, 6] within the local spin density approximation (LSDA) [14, 15]. To facilitate the numerical treatment of very large aggregates containing hundreds of atoms within LSDA, we make use of our re- cently developed approach based on an atom-centered numerical basis set [16], proven to describe correctly similar carbon systems [17]. 2.1.1 Total Energy Calculations: Ab initio Density Functional Theory The fundamental Hamiltonian of many-body system is 52 Z1621 62 —;2m\7? +;Ir,—R—T|+ 2:|r,-—rj| ZIZJ€2 ';2M,V’ “L2ZIR,—R—__T| Usually, the nuclear degrees of freedom are frozen in, and our interest focuses on the electronic degrees of freedom. This all-electron Hamiltonian can be decomposed as H =T+Vext+lfinti (2.1) 10 where T is the kinetic energy of electrons, Vex, is the potential energy acting on the electrons due to the nuclei, and Vin, is the many-body electron-electron interaction term. The total energy E can be evaluated by _ < \II|H|\II > E— ’ am where [W > is the all-electron wavefunction. In the electronic ground state, Density Functional Theory (DF T) allows us to replace the complex many-electron wavefunction and the associated Schrfidinger equa- tion by the total electron density n(r) and a total energy functional, which depends only on n(r). This approach is based on the Hohenberg and Kohn theorem [18], which states that the total energy of a system in the ground state E [n] is a unique functional of the total charge density n(r), except for a constant. An important corollary states that the correct density n(r) can be obtained variationally by minimizing this func- tional. In analogy to Eq. 2.1, Hohenberg and Kohn separate the energy functional E[n] as E[n] = T[n] + / der(r)n(r) + E,,,¢[n], (2.3) where the explicit functional E,,,¢[n] is unknown. Kohn and Sham [19] replaced the system of interacting electrons, which estab- lish the total charge density by a system of non-interacting quasi-electrons with the same density. In this way, they were able to replace the many-electron Schr6dinger equation by a set of one-(quasi)electron equations, the Kohn-Sham equations. For- mally equivalent to the one—electron Schr6dinger equation, the set of Kohn-Sham equations describes the behavior of a quasi-electron in an effective potential, which is a functional of the total electron density n(r). The normalization of the Kohn-Sham wavefunctions d),- of all occupied states, which yield n(r), is established by Lagrange 11 multipliers, which enter as Kohn-Sham energy eigenvalues 6,, (Her! ’ 51') (Mr) = 0- (2-4) The Kohn-Sham Hamiltonian acting on the Kohn-Sham states is 5.2 Hm“) = -2177. V72 +Ven(1‘), (2-5) where _ JEI-Iartree (SEXC _ Van“) _ v“t(r)+ 6n(r) +611(r) — = Vext (1') + VHartree[n] + VXC [n]° These different contributions in the effective potential ‘4’] reflect the expression for the total energy, given as E[n] = T, + fdrVezt(r)n(r) + Eyartrceht] + Exc[n]. (2.6) This translates to N 1 i=1 where "’ 1 263' = <¢i|_'2' V2 +Veff ¢i> = T,[n]+/drV¢U(r)n(r). For a given electron density n, the exchange-correlation energy functional E xc is defined by Eq. 2.6 as the total energy difference between the exact total energy of 12 the homogeneous electron gas, and the sum of T [n], Ecx¢[n], and Egartreehz]. There is no simple correlation between the quasi-electron wave functions (f),- and energies 6,, and the corresponding observables in the interacting system. Only the total energy density n(r) is rigorously correct. The DF T calculation largely depends on the appropriate functional form for E xclnl- By definition, the universal exchange-correlation functional, E xcln], and the exchange-correlation potential ch(r), are exact for a homogeneous electron gas. In the Local Density Approximation (LDA), Exc[n] is replaced by Em = / drncxc(n(r)), (2.8) where éxc is energy density for the homogeneous electron gas. The LDA exchange- correlation functional may further be subdivided into an exchange functional E x and a correlation energy EC, Exclnl = Exlnl + Eclnl- (2-9) The exchange functional can be calculated [20] 0.458 E x [n] = — H artree, (2.10) where 1', (gnrf = i) is the average free-electron radius. The correlation term is found by substracting the kinetic and exchange energy from the total energy. The numerical procedure within the DFT scheme is the following: An initial guess for the total electron density is used as input. The effective potential can be calculated based on the input density. Solving the Kohn-Sham equations yields a new electron density as the output of the calculation. This charge density output is used as the input for the next iteration. This procedure is repeated, until self-consistency is achieved. 13 2.1.2 Total Energy Calculations: Semiemprical Methods The time-independent Schrédinger equation of a many-body system consisting of electrons and atomic nuclei is the following: H so = Ecp, where the exact Hamiltonian H within the Born-Oppenheimer approximation reads ’12 2162 ——;2mv’+;|r,—R1|+2Z|r,—r—_j’| which describes the motion of electrons in the field of atomic nuclei. Based on density functional theory the exact ground state energy can be found in principle as discussed in the previous section, yet the calculation is computationally costy. Approximations can be made in the Hamiltonian and the wavefunction to solve the Schr6dinger equa- tion in a computationally cheap way. The most common approximation to make is the independent electron model, ’one-electron model’, in which the explicit interac- tion between electrons is ignored. Instead, each electron is described by the mean field created by other electrons and fixed nuclei. The corresponding one-electron wavefunction is called as a orbital. Hiickel Method In the tight-binding approximation, it is assumed that the full molecular Hamilto- nian can be approximated by the Hamiltonian for a single atom. Also, the localization of the bound levels of this atomic Hamiltonian is assumed. Therefore, it is natural to expand the tight-binding wavefunction |cp > in a linear combination of atomic orbitals (LCAO) Ii >, N Ir) >= Zea-Ii > i=1 14 . This one-electron function contains no fundamental natures of the wave function such as the Pauli exclusion principle and the information about spins. Using a LCAO basis set, we can minimize the energy E with respect to the coefficient c,- by applying variational methods, which results the secular equation, :0: (H3, — Esz‘) = 0, where j = 1, . - - ,N, Hji = (leIz'), and Sji = (jlz’). To find non-trivial solutions, the following determinant needs to be solved, detlng — ESjil = 0, which determines the coefficients c,- and hence the molecular wavefunction and the energies. In the tight-binding method H j,- is zero unless j and z' are nearest neighbor atoms: electrons are tightly bound. Hiickel [21] approximation goes further simplification in the tight-binding method by neglecting the overlap integrals between different atomic orbitals and by treating energy of all atomic orbitals to identical energy, i.e.: H,” = Hjj = a,H,~J- = Hj.‘ = 3, Sit = Sjj = 1,3:‘1‘ = Sji = 0, where a and B are called as the Coulomb integral and the hopping integral, respec- tively. The parameters can be determined based on experiments of other theoretical calculations. The Hiickel hamiltonian can be written down in a matrix form, H=Za|i > 0.65 i 0.70 ‘ 0.75 A 0.80 RNT (nm) C84@(10,10) —. C60@(10.10) " ‘ Cs4@(1 1 .1 1) q." + \ \ J \ ‘.‘:.~.’..'.-’.1-..‘..'.‘-‘..~.fi 1.0 A 0.0 i -1.0 z(nm) Figure 2.1: (3) During the insertion into a nanotube, a fullerene is pulled in by a “capillary” force F, which is linked to the energy gain upon axial displacement A2. (b) The encapsulation energy AE of C60 (0) and C34 (0) in single-wall carbon nanotubes with radii ranging from 0.6 nmSRN7~£O8 nm. Best fitting fullerene/nanotube pairs Show an encapsulation energy of 50.4 eV. (c) Energy change during the fullerene insertion process along the tube axis 2, with z = 0 at the tube end. C60 and C34 are pulled into the best fitting (10, 10) and (11, 11) nanotubes by F 20.1 nN, reflected in a constant slope of AE/Az near the tube end. The high energy cost prevents the entry of the C84 fullerene into the narrow (10, 10) nanotube. 21 Fig. 2.1(b). The results of our atomistic calculations, given by the data points, also reflect the relaxations in the nanotube peapod system. These data indicate how en— ergetically favorable the encapsulation process is for a particular fullerene/nanotube combination. Following our expectations, only nanotubes with a radius RNT beyond a threshold value may encapsulate a particular fullerene with an energy gain. Some- what surprising is our result that the encapsulation energy AE seems to depend only on the nanotube radius and shows only a negligible dependence on the chirality of a particular (n,m) nanotube. I find the minimum of the AE(RN7~) curve, reflect- ing the favorable fullerene/nanotube combination, at 230.4 eV for C60@(10, 10) and C84@(11,11). This value agrees with ab initio calculations [40], but is lower than empirical fits to experimental data [41]. In general, we find the optimum snug fit to occur at RNTRRF +0.3 nrn, where Rp is the fullerene radius. Increasing the nanotube radius reduces the snug fit and the fullerene-nanotube attraction. For very large tube radii, the encapsulation energy should approach the fullerene adsorption energy on planar graphite. As expected, fullerene encapsulation is energetically less favorable and eventually turns endothermic with decreasing tube radius. Close inspection of the structural relaxations in optimized peapods, both in ab- sence and presence of an external force F, reveals that the major modifications occur in the inter-fullerene and fullerene-nanotube distances, with only a minor shape de- formation of the fullerenes and the enclosing nanotube. Furthermore, I have found that the continuum approximation [42], which ignores discrete atomic positions, pro- vides a good estimate of the packing geometry. Indeed, our data in Fig. 2.1(b) for near-spherical C60 and C34 fullerenes in various (n, m) nanotubes lie very close to model predictions for perfect spheres inside smooth tubes, given by the solid lines. As mentioned above and shown in Fig. 2.1(c), the maximum energy gain upon encapsulation is close to 0.4 eV in case of the snug fit of CGO@(10, 10) or C84@(11,11). This energy gain near the tube end at z=0 occurs across the short distance Azszz05 nm, 22 resulting in a typical capillary force of F201 nN. Even though the insertion of C84 inside the narrow (10, 10) nanotube is strongly energetically unfavorable, the poten- tial well near the tube end may be used to manipulate a fullerene, which adheres to a carbon nanotube tip of a Scanning Probe Microscope (SPM) [43, 44]. At nonzero temperature, the static “capillary force” F201 nN is augmented by the force resulting from collisions between the encapsulated fullerenes. Assuming thermal equilibrium, the average collision force amounts to 20.5 nN at room temper- ature. In view of the small cross-section of the nanotube, the effective compressive force in the nN range translates into an effective pressure in the sub-GPa range. This effective pressure modifies the packing geometry, in particular reducing the inter- fullerene distance [35, 38]. In view of this high effective pressure, the nanotube may be considered a nanoscale pressure container or autoclave. 2.2.3 Packing of Fullerenes in Nanotube Peapods The equilibrium geometry of the encapsulated fullerenes, which are subject to an external force F, is discussed in Fig. 2.2. This force can be thought of as being mediated by a “piston”, shown schematically in Fig. 2.2(a). The main effect of the effective pressure is to reduce the axial inter-fullerene distance Dz, and to increase the off-axis displacement A. I focus my investigation on peapods based on 1.4 nm wide nanotubes, which are most abundant among the single-wall nanotubes, and which have been used in Ref. [38]. In the atomistic calculations, I considered the most stable fullerene isomers with 60-84 atoms, selected the (18, 0) nanotube to represent 1.4 nm wide nanotubes of Ref. [38], and performed a global structure optimization for a given applied force. I found that also these results can be reproduced well by a continuum model, which considers rigid spheres contained in a rigid tube. As suggested by our results in Fig. 2.1(b), the packing geometry inside an (n, m) nanotube depends primarily on its radius, given by RNT = 3.92x10‘2 nm (m2 + 23 I1 -_ —_ -_ —_| 8'0 84 Figure 2.2: (a) An external “capillary” force F reduces the axial inter-fullerene dis— tance D,. An off-axis fullerene displacement A is expected especially if the fullerene radius Rp is much smaller than the nanotube radius RNT. (b) Reduction of the ax- ial inter-fullerene distance D, in peapods with respect to the equilibrium separation Do in bulk crystals of C,, fullerenes with 60 — 84 atoms. Observations for different fullerenes inside a 1.4 nm wide nanotube, reported in Ref. [38] and given by the data points with error bars, are compared to our analytical results for various applied forces F, shown by the dashed lines. (c) Predicted off-axis displacement A inside a 1.4 nm wide nanotube as a function of the fullerene size, for various applied forces F. 24 mn + n2)1/2, and not the chiral index. Similarly, I may neglect deviations from sphericity for encapsulated fullerenes, and will assume their mean radius to be given by RF = 4.58x10‘2nm 721/2. I will also assume dvdW = 0.3 nm as the equilibrium separation between the walls of fullerenes and nanotubes in absence of external forces. I find that close to equilibrium, the interaction energy between two Cu fullerenes can be expressed by a harmonic potential with the force constant opp = (0.41n) N /m. In the limit of very wide nanotubes, the fullerene-nanotube interaction resembles the fullerene-graphite interaction, which also can be represented by a harmonic potential with the force constant CFC = (0.361).) N / m. With these values, the optimum packing structure within any peapod, consisting of fullerenes with radius Rp encapsulated inside a nanotube of radius RNT and subject to an external force F can be determined analytically from total energy minimization. My quantitative results for the reduction of the axial separation between Cu fullerenes inside a 1.4 nm wide nanotube are presented in Fig. 2.2(b). Comparison between my predictions and the experimental data of Ref. [38], suggesting an inter- fullerene distance reduction by 3 — 4% and displayed by the data points, suggests that encapsulated fullerenes are likely subject to an axial compressive force in the nN range, in agreement with my estimates above. For peapods containing fullerenes with a radius below the optimum value RF = RNT — dudw, a nonzero off-axis displacement [45, 46] A of the encapsulated fullerenes is expected even for small external forces F —90. Increasing the fullerene radius leads to a more snug fit and reduces A, as seen in Fig. 2.2(c). Furthermore, in presence of an axial compressive force, we also find a significantly larger off-axis displacement, which has been observed by HRTEM [38]. In Fig. 2.3 I depict the strain distribution on the wall of nanotube peapods by dis- playing the reduction of the atomic binding energy. The schematic packing geometry of peapods containing too small and too large fullerenes is shown in Figs. 2.3(a) and 25 a!“ ' g: 5 we: . .3 I .! :13 it: ' '0 ‘e f” 3 1 4 } r 3‘. S fm'sf ",3 3 .3‘ )8”. 3 A}? i: . A. s ‘\ 9 Mewlav '\ a? («A 0’. e' I 3 3': Q 2": a .2 .5 3' ii a :' _ '3: "a. sf, 5? I. ‘ 3 4r 3.. a l'l'i" Irv-N's '— 1 £039.... gr‘v‘f‘o ~ 'c..‘vi Figure 2.3: (a) In peapods containing fullerenes equal to or smaller than the optimum size, strain in the nanotube wall is induced by an axial force. (b) Strain distribution on the wall of a (10, 10) nanotube containing C60 fullerenes, subject to the axial force of 0.5 nN. (c) In peapods containing fullerenes exceeding the optimum size, strain is induced even in absence of an axial force. ((1) Strain distribution on the wall of a (10,10) nanotube containing 034 fullerenes. The strain energy is represented by the reduction of the atomic binding energy on a grey scale in (b) and (d). 26 2.3(c), respectively. Our results in Fig. 2.3(b) suggest that the strain on the (10,10) nanotube, induced by C60 molecules, is localized near the fullerenes. When subject to an axial compressive force of 0.5 nN, the encapsulated fullerenes press towards the nanotubes wall, thus locally reducing the atomic binding energy by as much as 1 meV from the initial value of z? eV/ atom. In Fig. 2.3(d) I display the strain on the (10, 10) nanotube wall containing C84 molecules. According to my results presented in Fig. 2.1, insertion of this large fullerene into the (10, 10) nanotube is energetically highly unfavorable. In this case, the fullerene is centered on the nanotube axis. Even in absence of an external force, the larger C84 molecules locally reduce the binding energy of atoms on the nanotube wall by as much as ~50 meV. The resulting bulge on the wall is still very small, and preserves the cylindrical symmetry of the initial nanotube. My results suggest two unusual applications of nanotube peapods. The first application is a possible way to separate nanotubes by diameter, due to the energetic preference of particular fullerenes to enter nanotubes within a narrow diameter range. All currently known synthesis techniques produce fullerenes and nanotubes in a wide diameter range. Whereas separation of fullerenes by isomer is possible using high- pressure liquid chromatography, there is no analogous technique allowing to separate nanotubes by diameter. Exposing nanotubes with a wide diameter distribution to a particular fullerene should lead to a preferential formation of peapods with an optimum nanotube diameter. The fact that nanotube peapods should have a higher gravimetric density than their empty nanotube counterparts could be utilized for a physical separation of peapods with a specific diameter from other nanotubes in a sample. A second possible application is related to the high effective pressure inside the nanotube, caused by the motion of the encapsulated fullerenes. In view of the small nanotube cross-section, even forces in the nN range give rise to GPa pressures, sug- 27 gesting a possible use of nanotubes as nanoscale autoclaves to facilitate chemical reactions. As a matter of fact, HRTEM observations of peapods subject to elec- tron irradiation [47] or elevated temperatures [48] suggest a Spontaneous fusion of fullerenes to long nanocapsules, in contrast to the more inert 2D and 3D fullerene structures [49, 50]. Also other molecules besides fullerenes, which can be encapsulated at an energy gain, should exhibit a similar behavior. As an example, selected dia- mondoid molecules [11] are expected to enter carbon nanotubes spontaneously [51]. Taking advantage of the physical confinement within the nanotube template, these diamondoids may fuse to one-dimensional diamond wires at nominal pressure. 2.2.4 Summary In summary, I have studied the energetics and equilibrium packing geometry of fullerenes encapsulated in nanotubes. I found that each fullerene has an energetic preference for a narrow range of nanotube diameters for peapod formation. Result- ing selective filling of particular nanotubes could be utilized to separate nanotubes according to diameter. N anotubes, which are too narrow to encapsulate a particular fullerene, may still bind it at the open end and manipulate it, when attached to a Scanning Probe Microscope tip. We found that insertion of a fullerene inside an op- timum nanotube host is associated with an energy gain of 20.4 eV. The “capillary” force produced by the entering fullerene may be augmented by an average force caused by inter—fullerene collisions at nonzero temperatures to a value in the nN range. In view of the small nanotube cross-section, this force should be equivalent to a pressure of the order of GPa. I find the observed reduction of the axial inter-fullerene distance to evidence this effective pressure. This large nominal pressure may become benefi- cial when using nanotube peapods as nanoscale pressure containers. The equilibrium packing geometry of smaller-than-optimum fullerenes inside nanotubes is a zig-zag arrangement, with an expected increase in the off-axis displacement with increasing 28 pressure. The strain in the nanotube wall, associated with closely packed fullerenes, may locally modify the reactivity of the tube wall, which could be used for chemical functionalization. 29 2.3 Fullerene Fusion Mechanism The following discussion on the equilibrium packing geometry of fullerenes and nanotube peapods follows that presented in Reference [52]. Combining total energy calculations with a search of phase space, I investigate the microscopic fusion mechanism of C60 fullerenes. I find that the (2 + 2) cycload- dition reaction, a necessary precursor for fullerene fusion, may be accelerated inside a nanotube. Fusion occurs along the minimum energy path as a finite sequence of Stone-Wales transformations, determined by a graphical search program. Search of the phase space using the ‘string method’ indicates that Stone-Wales transformations are multi-step processes, and provides detailed information about the transition states and activation barriers associated with fusion. 2.3.1 Introduction The discovery of fullerenes [32] and nanotubes [33] has ignited strong interest in these and related carbon nanostructures. Due to the unusual stability of the graphitic .9192 bond, large-scale structural changes in bulk fullerene crystals occur only under extremely high pressures and temperatures [49, 50]. On the other hand, fullerenes in nanotube peapods [34] have been observed to fuse [47, 48] at relatively low temperatures near 1,100°C, significantly below the decomposition temperature of fullerenes [4] or graphite [53] near 4, 000°C. No information is available about the detailed fusion process except the obvious conclusion that strong 3;?2 bonds should not be broken during structural rearrangements leading to fusion. In view of the fact that even minor structural changes in carbon nanostructures may modify significantly their physical properties, including magnetism [54, 55], there is additional interest in understanding fusion as a way to control large-scale structural transformations. Here I study the microscopic fusion mechanism of fullerenes. I show that large- 30 scale structural changes, including fusion, can be achieved by a finite sequence of gen- eralized Stone-Wales transformations, which involve only bond rotations and avoid bond breaking. Using a graphical search program, we determine the optimum re- action pathway for thermal fusion of fullerenes. Search of the phase space by the ‘string method’ provides detailed information about the optimum pathway, including the identification of activation barriers and transition-state geometries. I find the fu- sion process to be exothermic. The fusion dynamics is fast in spite of the formidable total activation barrier close to 5 eV, associated with each Stone-Wales transforma- tion, since bond rotations turn out to be multi-step processes with lower individual activation barriers. 2.3.2 Microscopic Mechanism of Fullerene Fusion The fusion of two C60 molecules to a C120 capsule, which has been observed in peapods [47, 48], is driven by the energy gain associated with reducing the local cur- vature in the system. Still, this reaction involves a large-scale morphological change and will only occur, if the required activation barrier is small. A previous study [57], based on minimizing the classical action, suggests that the fusion reaction should involve multiple steps with relatively high activation barriers of m8 eV. This value may be considered an upper bound for the true activation barrier, since the authors had to guess the one-to-one atomic mapping between the initial and the final structure. Also, this study encountered computational limitations in the unconstrained search of a contiguous minimum-energy path in the 360-dimensional configurational space of the system. It appears that the most likely fusion path may involve a sequence of bond rota- tions, called generalized Stone-Wales (GSW) transformations. GSW transformations are known to require much lower activation energies than processes involving bond breaking, and have been studied extensively in 3122 bonded carbon structures [58]. A 31 possible GSW pathway for fusion has been suggested based on a ‘qualitative reason- ing assisted search’ for structures along the minimum-energy path [59]. The initial step in that study, however, is a reaction between two pentagons facing each other, which is energetically unaccessible. In order to obtain microscopic insight into the fusion reaction, avoiding the above shortcomings, I investigated the optimum reaction path for the the 2C60—2C120 fusion. It is well established that polymerization [60] and subsequent fusion [61, 62] of adja- cent fullerenes starts by the (2 + 2) cycloaddition reaction. This reaction, depicted as the 0—>1 transition in Fig. 2.4(a), requires two “double bonds”, which connect adjacent hexagons in the C60 molecule, to face each other at the contact point of adjacent fullerenes. With the (2+2) cycloaddition reaction completed, we investigated the possibility to complete the ZCw—iCm fusion by generalized Stone-Wales transformations only. We searched all topologically possible pathways for the reaction with the aid of a graphical search program [56, 62]. Among these, I identified the shortest pathway, which is likely associated with the fastest fusion mechanism. This pathway involves only 23 GSW transformations and is depicted in Fig. 2.4(a). Tracing the atomic positions during this structural rearrangement, I found that the diffusion range of individual carbon atoms is limited to about three atomic bond lengths in the struc- ture. Snapshots of intermediate state geometries along the optimum fusion pathway are shown in Fig. 2.4(b). The energetics of the 2C60—»C120 fusion process along the optimum path is de- picted in Fig. 2.5(a). The energy results for the Optimized metastable states of Fig. 2.4(a) are given by the data points. I conclude that the ab initio density func- tional and our parametrized total energy functional give consistent results for the relative energies of the intermediate states, and also for the large net energy gain of z13 eV associated with the fusion. 32 Figure 2.4: (a) Optimum pathway for the 2060—0120 fusion reaction, involving the smallest number of generalized Stone-Wales bond rotations, determined by a graphical search of all possible bond rotation sequences. Polygons other than hexagons are emphasized by color and shading. (b) Snapshots of the optimized initial and final structures, and the metastable structures “2” and “3”, depicted in (a). Also shown are two intermediate structures along the optimum fusion pathway between “2” and “3”, resulting from the phase space search by the ‘string method’. The bond involved in the 2—r3 Stone-Wales transformation is emphasized by color. Images in this thesis dissertation are presented in color. 33 M 1'3 reaction coordinate Figure 2.5: (a) Energy change along the optimum reaction path, given by the solid line. Energy results for the 25 intermediate structures, shown in Fig. 2.4, based on our total energy functional (0), are compared to ab initio Density Functional results (+). The contiguous minimum energy path in configurational space was identified using a ‘string’ technique. (b) Details of the energy change along the optimum path between structures “12” and “13” of Fig. 2.4, showing several local minima and implying a multi—step nature of this Stone-Wales transformation. The activation barrier limiting the reaction rate is denoted by AE]. (c) Energetics of the (2 + 2) cycloaddition reaction, corresponding to the 0—>1 transition in Fig. 2.4(a), which is a necessary prerequisite for the fusion process. reaction coordinate To evaluate the reaction barriers of individual GSW transformations between the 24 intermediate states, we searched for the minimum energy path using the re- cently developed ‘string’ method [63]. This method represents the reaction pathway connecting the initial and final 120—atom geometry in the 360-dimensional atomic configuration space by a string line. In practice, the string is subdivided into finite segments of equal length, connecting structural replicas. The suitability of an energy path is determined by investigating the atomic force acting on each replica. Along the minimum energy path, the force component normal to the string must vanish. We employ 60 - 100 replicas for each GSW step and relax the atomic positions, until the normal component of the atomic force becomes less than 0.05 eV / A in magnitude. Close inspection of the reaction energy along the contiguous optimum fusion path in Fig. 2.5(a) indicates a sequence of 23 activated processes connecting the 24 metastable states. I find the activation energy barriers AEGswz5 eV of these GSW transformations to be significantly lower than in graphite [64], as expected for Stone- Wales processes in non-planar structures due to the deviation from spz-bonding. In presence of extra carbon atoms, the activation barriers for GSW transformations may be lowered further to below 4 eV by autocatalytic reactions [65, 66]. Also, under electron irradiation, this process can proceed relatively fast in view of the high rate of sub-threshold energy transfer to the structure [67]. In extended fullerene systems, moreover, the energy release during the fusion process should heat up the structure locally, thus further promoting activated processes in the local vicinity. Maybe the most significant finding of our study is the occurrence of multiple shallow local energy minima in the course of each GSW transformation. Details for the energy landscape, associated with the l2—>13 reaction, are shown in Fig. 2.5(b). This implies that GSW transformations are multi-step rather than single-step [68] or two-step [69] processes, as postulated earlier. The local minima originate from local stress release during the bond rotation, which can be viewed as breaking of two C-C 35 bonds at the same time as new bonds are being formed. The barriers separating these local minima are very small, suggesting a short lifetime with little effect on the overall reaction rate. I notice that GSW transformations are multi-step processes only in non- planar structures, as no such local minima occur during Stone-Wales transformations in a graphene layer due to the absence of tensile stress in that system. Subdividing the GSW process into discrete steps with lower activation barriers AE; < AEGSW, shown in Fig. 2.5(b), also speeds up the transformation reaction. To estimate the overall reaction time at the temperature of 1, 100°C, where the onset of fusion has been observed [47, 48], I considered the fusion process as a sequence of 23 GSW transformations. Assuming the attempt frequency of 3x 1013 Hz for the GSW transfomations [66] and a limiting activation barrier AE; = 4.5 eV in the Arrhenius formula [70], I find that the fusion reaction should be completed in 7 hours. Reduction of the activation barrier by 0.5 eV should reduce the total fusion time to 6 minutes. In view of the fact that fusion is generally more complex than an optimum sequence of GSW transformations, these values agree well with the observed fusion time of several hours [47, 48]. In spite of its lower activation barrier in comparison to the GSW steps, the initial (2 + 2) cycloaddition reaction between the structures “0” and “1” , the energetics of which is depicted in Fig. 2.5(c), may play an important role, and possibly even limit the rate of the fusion process. Fusion can only be initiated in the optimum geometry, where two double bonds in adjacent fullerenes face each other at the contact point. The probability of this configuration will multiply the attempt frequency v of the 0->1 reaction in the Arrhenius formula [70] and thus reduce the reaction rate, since the low activation barrier of ~07 eV only applies to attempts with the optimum fullerene orientation. At low temperatures, polygons rather than double bonds should preferentially face each other in adjacent fullerenes, effectively preventing the fusion. Only at high 36 enough temperatures, when unhindered fullerene rotation is activated [71], will the probability of double bonds facing each other increase, while each fullerene probes the configurational space. At that moment, the (2 + 2) cycloaddition reaction should stop the rotation [72], and may initiate fusion. To estimate the probability of the configuration required for the (2 + 2) cycload- dition to occur, I first consider the phase space describing the motion of two rigid fullerenes at constant equilibrium distance (structure “0” in Fig. 2.4), which are freely rotating in space. The 8—dimensional configurational space, spanned by the three Eu- ler angles defining the orientation of each fullerene and the two—dimensional vector defining the orientation of the inter-fullerene connection, is explored uniformly by the rotating fullerenes. Next, we assume that the diflerence between a “correct” and an “incorrect” fullerene alignment corresponds to a misorientation exceeding A, and seamlessly reinserted. The resulting average twist of A / L in the segment comes at a cost in the total torsion energy that is inversely proportional to L. As the constituent tubes continue to merge, thus reducing L while approaching the anchor point, the total strain energy in the twisted tube increases [93]. Independent of the tube stiffness or the initial twist A/ L, the strain energy eventually outweighs the energy gain caused by tube fusion. At this point, the fusion process turns endothermic and stops. In order to study whether nanotubes may merge completely via the zipper mech- anism, I have examined two adjacent tubes lying in a plane under electron irradiation at 800°C. Figure 2.8(a-c) shows a sequence of the zipping process in the ‘pants’ ge- ometry. In Fig. 2.8(a) we can distinguish two nanotubes, lying close and parallel to each other. Few minutes later, the tubes were Observed to approach even more and started overlapping, as seen in Fig. 2.8(b). Subsequently, the nanotubes formed a 48 local connection and started merging fast. Figure 2.8(c) depicts the final stage of this process. a single wide nanotube. The Observed fusion process of two tubes into one tubule with a larger diameter, including the intermediate stages and the speed of interconversion, is consistent with our zipper mechanism. 2.4.3 Summary In summary, I introduced a microscopic mechanism to explain the efficient and fast fusion process of two adjacent nanotubes in the Y-junction geometry, reminiscent of pants. I found the fusion process, which proceeds by increasing the waist at the ex- pense of the leg section, to be energetically favorable. I proposed a microscopic zipper mechanism that consists Of a sequence of Generalized Stone-Wales transformations corresponding to bond rotations. In two representative pathways, distinguished by the local bonding geometry in the junction region, only nine or twelve bond rotations are needed to complete a full transformation cycle. I investigated the detailed ener- getics Of the zipper process, including the transition states, using the string technique. Our atomic-level optimization calculations and molecular dynamics simulations re- veal that the energy barriers associated with the Stone-Wales transformations are low, consistent with a fast fusion process. The suggested propagation of the zipper- like motion and the vibration modes of the nanotube ‘pants’ were observed in a time sequence of transmission electron micrographs. 49 Figure 2.8: (a) Tubes prior to coalescence at 800°C. (b) As electron irradiation contin- ues, the nanotubes get closer, overlap, and start merging into one. The fused region is emphasized by the white arrow. (c) The coalescence process is completed, as the fusion region moves down, completing the zipper closure. The white scale bar is 5 mm long. 50 2.5 Diamondoids as Building Blocks Of Functional Nanostructures The following discussion on the diamondoids as building blocks of functional nanostructures follows that presented in Reference [51]. We investigate the equilibrium structure, stability, and electronic properties Of nanostructured, hydrogen terminated diamond fragments. The equilibrium atomic arrangement and electronic structure of these nanostructures turn out to be very similar to bulk diamond. We find that such diamondoids may enter spontaneously into carbon nanotubes. 2.5.1 Introduction In his inspiring presentation entitled “There is Plenty of Room at the Bottom” [94] in 1959, Richard Feynman pointed out the untapped potential of functional nanos- tructures, assembled with atomic precision, to influence our every-day life. In the meantime, technological progress has been associated with a continuous drive towards miniaturization. Recent progress in nanO-scale electromechanical systems (NEMS) suggests that complex functional nanostructures, invisible to the naked human eye, may soon follow suit.[95] Large-scale production of such complex nanostructures is likely to occur by hierarchical self-assembly from well-defined structural building blocks, [96, 97] which can be thought of as “nano-LEGO”. To fulfill their mission, such nanostructures must be strong and chemically inert in their environment. Carbon fullerenes [32] and nanotubes [33] have emerged as promising candidates for such building blocks, providing a variety of functionalities. Here, we investigate a new class of carbon nanostructures, the diamondoids. These nanoscale building blocks are very different, but compare well in stability and light weight with fullerenes and nanotubes. Beings essentially hydrogen-terminated nanosized diamond fragments, 51 (a) Adamantane Figure 2.9: Hydrogen atoms, terminating the carbon skeleton, are omitted from the diagrams for clarity. (a) The smallest diamondoid, adamantane, consisting of a single diamond cage. (b) Diamantane with two diamond cages. (c) Tetramantane with four diamond cages. (d) Decamantane with ten diamond cages. 52 they may occur in a large variety of shapes, as shown in Fig. 2.9. The smallest possible diamondoid is adamantane, [98] consisting of ten carbon atoms arranged as a single diamond cage, [99] surrounded by sixteen hydrogen atoms, as shown in Fig. 2.9(a). Larger diamondoids [100, 101, 102] are created by connecting more diamond cages and are categorized according to the number of diamond cages they contain. The diamondoids shown in Fig. 2.9 contain up to tens of carbon atoms and are only a few nanometers in diameter. 'Ifaditionally, diamondoids have been known in the Oil industry, [103, 104] where they occur naturally dissolved in oil and its distilled by-products. At low tempera tures and low pressures, diamondoids may precipitate from the solution and act as nucleation sites for the formation of sludge, which often blocks pipelines [105, 106]. Only recently, isomerically pure diamondoids with up to eleven adamantane cages have been extracted from the sludge [11] and are now being considered for nano- technology applications. The isolated diamondoids occur in very different shapes, and many of them form molecular crystals. These uncommon organic molecules can be chemically functionalized, by substituting carbon or hydrogen atoms at the surface by other atoms or groups, to promote formation of specific polymers. Given that diamondoids are essentially hydrogen terminated diamond fragments, we anticipate that in absence of significant structural changes, they share the tough- ness and insulating properties with their bulk diamond counterpart. Our study con- firms this anticipation. In the following, we carry out a theoretical investigation of diamondoids using ab initio density functional calculations. We determine their equi- librium geometry, in particular the dependence of the bond lengths and bond angles on the system size, and their electronic density of states (DOS). Beyond identifying the properties Of unmodified isolated diamondoids, we also explore ways to manipulate them and to connect them together. We find that the interaction between unmodified diamondoids is too weak and does not favor poly- 53 merization to larger structures. Hence, we explore the possibility of creating specific binding sites by atomic substitution. We then explore the interaction between unmod- ified and functionalized diamondoids. In particular cases, we conclude, a nanotube attached to the tip of a Scanning Probe Microscope could also be used to manipulate diamondoids into position on a substrate, similar to the way this was achieved for adsorbed rare gas atoms [43]. Even more intriguing is the possibility to use a nanotube as a container for un- modified and functionalized diamondoids, very much the same it acts for fullerenes [34] and polymers [107]. The nanotube surrounding the encapsulated diamondoids may act as a template to connect these molecules in a well-defined way. In particular, we could envisage the polymerization of quasi-linear polymantanes to an infinite di- amondoid wire, enclosed in a nanotube. 2.5.2 Properties Of Isolated Diamondoids Our calculations are based on the ab initio density functional theory (DFT) within the local density approximation (LDA). We use 'I‘l'oullier-Martins pseudo— potentials to describe the effect of atomic nuclei plus core electrons on the valence electrons, and the Perdew-Zunger parametrized exchange-correlation potential, as implemented in the SIESTA code [9]. We used a double-zeta basis [108], augmented by ghost orbitals, and 50 Ry as energy cutofl in plane-wave expansions of the electron density and potential, which is sufficient to achieve a total energy convergence of 31 meV per atom during the self-consistency iterations. We performed a full Optimization of diamondoid structures CnHm containing up to ten diamond cages. Our optimized geometries, some of which are reproduced in Fig. 2.9, are in good agreement with those inferred experimentally [10, 11] and theoret- ically [12, 13]. Also, the equilibrium C-C bond length of 1.540 A in the Optimized bulk diamond structure was found to agree with the experimental value [109]. Constrained 54 o 20 -15 -10 -20 -15 -10 —5 0 5 20 - —5 0 5 10 Energy(eV) Energy(eV) f l A l A l . l . I . I l g M I l I I A l l I (C ‘ 6.815 (d) 6.226 c. “—fi 0. F—fi “‘ 5 : a)“ : : 8 : : O ‘ ' : ‘ ll ,3. mlilui I“ “ii l ’ Iii] Ii —Z0 —15 — 20 —20 —15 -10 1O 15 20 -5 O 5 Energy(eV) o —5 o 5 Energy(eV) Figure 2.10: Electronic structure of an (a) adamantane, (b) diamantane, (c) tetra- mantane, and (d) decamantane, depicted in Fig. 2.9. E = 0 corresponds to the Fermi level. The fundamental band gaps are indicated in eV. geometries were used when determining the interaction between diamondoids [110]. The lower diamondoids we considered here include adamantane, diamantane, triamantane, all four isomers of tetramantane, all nine isomers of pentamantane with a molecular weight of 344 a.m.u., six isomers of hexamantane, two isomers of hepta- mantane and one isomer each of octamantane, nonamantane and decamantane. With increasing adamantane size, the number of isomers increases geometrically, making an exhaustive study of all isomers impractical|11]. In our selection of isomers, we fo— cused on those identified experimentally in Ref. [11]. For each of these diamondoids, we have determined the equilibrium structure, the binding energy per carbon atom, and the electronic DOS. The electronic density of states of the diamondoids presented in Fig. 2.9 is shown in Fig. 2.10. The spectrum consists of discrete eigenvalues associated with the finite size of the molecules. With increasing size, the gap between the Highest Occupied Molecular Orbital (HOMO) and the Lowest Unoccupied Molecular Orbital (LUMO) 55 will converge to the fundamental gap of diamond as the bulk counterpart of the finite molecules. In comparison to other atomic clusters, we find the HOMO-LUMO gap to lie very close to that of bulk diamond even for the smallest diamondoids, including adamantane, as shown in Fig. 2.11(a). In conjunction with the fact that LDA usu- ally underestimates the HOMO—LUMO gap, its large value of several electron-volts suggests that diamondoids—similar to bulk diamond[111]—should appear transparent in visible light, and act as electrical insulators. This is indeed in agreement with observations in diamondoid-based molecular crystals.[11] Figure 2.11 summarizes our results for the equilibrium CO bond length, the HOMO-LUMO gap, and the formation energy per carbon atom in all the diamondoid isomers we have considered. The equilibrium C-C bond lengths in the diamondoids, depicted in Fig. 2.11(b), lie very close to the value found in bulk diamond. Since also the bond angles in these structures lie close to those found in diamond, we conclude that even the smallest diamondoids show sp3 type bonding, which is very different from graphite-like 3p2 bonding found in fullerenes and nanotubes. Only in the smaller diamondoids, including adamantane, diamantane and triamantane, the GO bond length is somewhat smaller than in bulk diamond, reflecting the small difference between the C-H and the C-C(sp3) bond in hydrogen-terminated carbon atoms. The error bars shown reflect mainly the differences between the inequivalent sites in the lower diamondoids, and an estimated uncertainty resulting from our optimization procedure in the larger structures. In agreement with related theoretical studies,[12, 13] we find the hydrogen-terminated small diamondoids, investigated here, stable with respect to hydrogen desorption and surface reconstruction. The structural resemblance between even the smaller diamondoids and diamond suggests that these nanostructures will also share the desirable toughness and insu- lating properties with bulk diamond. The latter is indeed the case, as suggested by the large HOMO-LUMO gap we find in all the diamondoids investigated here, shown 56 A G) v (I) O a 9 . o - 9’ 7o ' . - C "' «‘6' - ' I g . . . . a) 0 ‘g 6.0 - .. <6 :.Qi§'11999 ______________ - m 1 1 1 1 "1M 50 l l L l l O 2 4 6 8 1O A U- v :5: 16 I l i ] I I I I i j *3, :PléTQDQ--- _ _ _ - cc, 9 In G I T i § § 5 _§ ; 1.5 - - § 0 Eileen“? ................................ ‘ L.) 14 1 l 1 l 1 l 1 I 1 ll 0 2 -4 6 8 1O ( ) Polymantane order C A T I I 1‘! l I I 3 -1.5 1- ' 1"! SE} ' . 0 e e ' CU - _ _ 0 16 . I ! 6 - - - Q '17 " d LEI . J. l L J 1 J 1 J 1 l 0 2 4 6 8 1O Polymantane order Figure 2.11: (a) HOMO-LUMO gaps in the density of states (DOS). (b) CO bond lengths, with the error bar reflecting the spread of the values within each system. (0) Formation energy per carbon atom AE / n of the CnHm diamondoids. 57 in Fig. 2.11(a). In agreement with former studies,[12, 13] we find the HOMO-LUMO gap to decrease monotonically with increasing diamondoid size. Among the different isomers of a particular polymantane, we can observe a subtle trend, which correlates the more elongated structures with somewhat larger HOMO-LUMO gaps than more compact structures. Figure 2.11(c) shows the formation energbI per carbon atom AE/n of the CnHm diamondoids as a function of the diamondoid size. We define the formation energy by AE/n(CnHm)= [Eto¢(CnHm)-nE¢ot(C)-(m/2)E¢o¢(H2)]/n, where Eto¢(C) is the total energy of diamond per atom, and Etot(H2) is the total energy of a hydrogen molecule. [112] According to Fig. 2.11(c), we find all diamondoids, as well as bulk diamond, over- bound due to underestimating the total energy of isolated atoms, a well-documented shortcoming of LDA. As suggested by the results in Fig. 2.11(c), the stability of the lower polymantanes is inferior to bulk diamond, since a significant fraction of carbon atoms is connected to fewer than four carbon neighbors. An intriguing observation is that the stability is nearly independent of the stacking arrangement of adamantane cages, suggesting an isomer-independent binding energy. With increasing size, we observe a monotonic increase in stability towards the bulk diamond value. The properties of an elongated hexamantane isomer, and an infinite diamondoid chain as its quasi—1D crystalline counterpart, are discussed in Fig. 2.12. Such struc- tural elements could find use in NEMS devices and, as we discuss in the following sections, could be self-assembled inside carbon nanotubes. Figures 2.12(a) and (d) show the close structural relationship between the finite molecule and the infinite chain, obtained by periodically repeating the unit cell shown in Fig. 2.12(d). The average C-C bond length in the infinite diamondoid chain is 1.530:l:0.008 A, close to the values found for the other diamondoids, given in Fig. 2.11(b). The electronic density of states of hexamantane is shown in Fig. 2.12(b). In all 58 Hexamantane (d) Diamond Chain (unit cell) (a) 2‘3— . 6.278 m __ _ l O 03 l 9: - .5 -2.5 ) 2:5 I (b) Energy(eV) 8 q . U) (f) “ 8 g ,1“. . E 3 - Conduction band - o—zo —15 -10 -5 5 10 g E,. Energy(eV) E 3 Valence band c - w J. 1 - (C) ’ r k x Valence band Conduction band Extended states Figure 2.12: Comparison between structural and electronic properties of an elongated hexamantane isomer (left panels) and an infinite diamondoid chain (right panels). The equilibrium structure is shown in (a) and (d), with the large spheres denoting carbon and the small spheres hydrogen atoms. The electronic density of states is presented in (b) and (e), with the Fermi level at E = 0. The one—dimensional band structure of the infinite diamondoid chain is shown in (f) The charge distribution of electrons in the highest occupied molecular orbital (HOMO) and the lowest unoccupied molecular orbital (LUMO) of hexamantane is shown in (c). The corresponding charge distri— bution in the top valence and bottom conduction band of the diamondoid chain is shown in (g). 59 respects, including its large HOMO-LUMO gap, this DOS lies close to the results found in the other diamondoids, depicted in Fig. 2.10. The electronic spectrum of the infinite quasi-1D counterpart of hexamantane, shown in Fig. 2.12(e), is similar to that of hexamantane, and dominated by a series of van Hove singularities in the valence and conduction region. The dense sequence of these singularities reflects the dense spectrum of relatively flat bands, depicted in Fig. 2.12(f), which makes this system a wide-gap insulator with an indirect gap. In view of the negative electron affinity of diamond,[113] combined with a large length-to-diameter aspect ratio, diamondoid chains may even surpass carbon nanotubes in applications such as cold cathodes or low-voltage electron emitters in flat panel displays. In spite of its wide fundamental gap, the infinite diamondoid chain could possibly acquire conductivity by electron or hole doping. Close inspection of the electronic dispersion relations in Fig. 2.12(f) reveals that the bands close to the Fermi level show an energy dispersion of below 2 eV. To investigate the charge delocalization as a prerequisite for conduction, we display the charge distribution of the topmost valence and bottom conduction band in Fig. 2.12(g), and compare it to that of the HOMO and LUMO of hexamantane in Fig. 2.12(c). Results for the infinite chain suggest that the charge associated with the va- lence band is localized in pockets near inter-atomic bonds, thus hindering hole trans- port. This charge distribution is very similar to that of the HOMO of hexamantane. The conduction band, on the other hand, consists of four strongly delocalized states located on the outer perimeter of the structure, which could be used to conduct electrons. These extended conduction states find their counterpart in the LUMO of hexamantane, which is similarly delocalized across the middle section of this molecule. The bottom of the conduction band is very flat, suggesting that electrons in a lightly electron-doped system should have a low mobility. Due to the wide fundamental gap, chemical doping is unlikely to provide free 60 carriers, since it would require introduction of impurities with a very low ionization potential, likely to introduce trap sites. One possibility of electron doping would involve gating a supported diamondoid chain. A more likely scenario of n-doping would involve enclosing the diamondoid chain inside a carbon nanotube, as discussed below, and n-doping the surrounding nanotube. 2.5.3 Interaction of Diamondoids with Carbon Nanotubes Since the cross-section of carbon nanotubes is compatible with the size of diamon- doids, nanotubes could be used to manipulate and assemble these molecules to larger structures. Precise deposition and positioning of diamondoids could be achieved with the help of a carbon nanotube attached to the tip of a Scanning Probe Microscope (SPM),[43, 44] which may combine its functionality as a storage, deposition, and ma- nipulation device. Combination of diamondoids with nanotubes may, moreover, lead to functional nanostructures for particular applications. Key to all these applications is to understand the interaction of diamondoids with carbon nanotubes. In the following, we describe the interaction of diamantane as a prototype di- amondoid with (n,n) armchair carbon nanotubes of different diameter. Aspects of particular interest to us are the energy change associated with the entry of a dia- mondoid inside a nanotube, and the optimum nanotube diameter to maximize the encapsulation energy. Our results for the entry of diamantane into armchair carbon nanotubes ranging from (4,4) to (7,7) are presented in Fig. 2.13. Due to computer limitations, we considered finite length nanotube segments, which have been hydrogen terminated at both ends. The encapsulation geometry is depicted in end-on and side view in the left panels of Fig. 2.13. The dependence of the diamantane-nanotube interaction energy on the diamantane position 2 along the tube axis is presented in the right panels of Fig. 2.13. 61 1 _ I I I I I l I I A F (a) (4,4) nanotube .. 9 .- . l- ., . 3 g . . . >. n - p 2’ . . . “c’ - Lu : Binding energy = 0.9 eV 0 ~ r l I -7 :5 :5 _'¢ :3 -'.2 —’1 5 I 2 2(A) 8 . . 9 8- ~ 3 3 9 - . Comer or a ' ' : nanotube 1: 0‘ Binding energy UJ “ = 95 eV 0 . -5 .1. —2 D .5 4 6 s 10 z(A) 2 1 6 ‘ S‘ a) o- 3’. 2’ m c o 1 LU '.’ é . —5 -4 -2 0 2 4 6 8 ID z(A) 2 A A 9 a) n * ’ ‘5.” 9 ‘D .A A Bindlf‘g energy . ,5 ° .. - w = 4.34 eV 1’ , . . -5 —4 -2 ( 4 s a 10 z(A) Figure 2.13: The end-on and side view of the insertion geometry is depicted in the left panels. The axial separation between the closest end of the diamantane and the nanotube is denoted by z, with z < 0 corresponding to the diamantane outside and z > 0 to the diamantane inside the carbon nanotube. 62 These energy results suggest that entry of diamantane into a (4,4), (5, 5) and (6, 6) nanotube is energetically unfavorable. Nevertheless, the dip in the total energy at zzO suggests the possibility of attaching this molecule to the hydrogen terminated nanotube end, and thus a possibility of manipulating diamondoids with a nanotube attached to an SPM tip. As seen in Fig. 2.13, only (11, n) nanotubes with n27 are wide enough to accom- modate diamantane endohedrally without energy investment. The energy of diaman- tane encapsulated inside wider (11,12) nanotubes is shown in Fig. 2.14 as a function of its off-axis displacement y. We find that the (7, 7) nanotube has an ideal diameter to contain a single diamantane with optimal encapsulation energy. Containment in the (6, 6) nanotube is moderately endothermic. Containment in the (8, 8) nanotube is exothermic, but the minimum in the energy curve is rather flat, suggesting that the diamondoid would have some degree of lateral freedom and that the (8,8) nanotube is too wide. Nevertheless, the enclosing nanotube should suppress free rotation of the diamondoid, thus facilitating specific reactions. Results very similar to those for dia- mantane are also expected for all higher, linear diamondoids, as well as the diamond chain, addressed in Fig. 2.12. The encapsulation of the smaller adamantane should proceed very similar to that of diamantane, with the exception that orientational alignment of adamantane with no obvious “long” axis cannot be achieved. 2.5.4 Summary In summary, we performed ab initio density functional calculations to study structural and electronic properties of unmodified and chemically functionalized poly- mantanes, as well as their reactivity and interaction with carbon nanotubes. Our results support the conjecture that the lower polymantanes are essentially hydrogen terminated diamond fragments with diamond-like properties. Hence, these systems may be used as molecular building blocks of complex functional nanostructures, to be 63 (a) (6,6) nanotube 3. ; . ; ' "‘ as Ag- : : '- ' > - . . so 38‘ I ' _ > I I 9g- - . _ a) ' ' C I I LU 9- INT wall NT wall. ' 4%Q5é5qi'iiéu WM 3. : : 9,; I I 3" I I ' >‘O I I 9—- . . - w * : : c m .NT wall NT wall. - Lu , . \I O I I n 1 I, -4.34eV . '_s'_'s'.'4—'3-'2—1¢.I'5 5 1'5 1. y(A) g.‘ : : L 522- I I :0 : : g" : : ,‘ LE“: iNTwall 296W NTwaIII I . j/ \ \' ..‘ : use! i I '-e.'s'—'4-3—'2—1¢I53I§I W30 Figure 2.14: Energetics of diamantane, enclosed inside a (6, 6), (7, 7), and (8, 8) carbon nanotube, as a function of an off-axis displacement y. The geometry in end-on view is shown in the left panels. found in future N EMS devices. Within computational uncertainty, we find the bond lengths and angles in the carbon skeletal structure of the diamondoids essentially the same as in bulk diamond. Similarity in bonding between diamondoids and diamond extends also to bond strength and resistance to deformations. The large HOMO- LUMO gaps in diamondoids are the molecular counterpart of a large fundamantal gap in diamond, which is responsible for its optical transparency in the visible range and its insulating properties. We also find the properties of hydrogen-terminated diamondoids to approach those of bulk diamond, as their size increases. Unmodified diamondoids are non-reactive and interact only weakly with each other. We found, however, that substituting carbon by boron and nitrogen atoms may create localized binding sites in these modified diamondoids, and provide the possibility of connecting diamondoids by much stronger, directional bonds. Selective substitution at more than one carbon site per diamondoid may provide the possibility to connect even larger diamondoids into well-defined nanostructures for particular applications. We have found that diamondoids should enter spontaneously into carbon nan- otubes with a wide enough diameter, similar to the spontaneous encapsulation of fullerenes, leading to peapods.[34] Being orientationally constrained in these nano- sized containers, they may fuse into long structures, including a “diamond wire”. Carbon nanotubes can be used not only to store diamondoids, but also to manipulate them sterically, preferably in conjunction with a Scanning Probe Microscope. Even nanotubes that are too narrow to contain a diamondoid may be used for such a pur- pose, since diamondoids preferentially bind to the reactive open end of a nanotube, which may be attached to the tip of a Scanning Probe Microscope. 65 Chapter 3 Structural Transitions in Ferrofluid Systems 3.1 Introduction Complex fluids, consisting of a colloidal suspension of particles carrying elec- tric or magnetic dipole moments [116], are intriguing systems with a wide range of technological applications [117]. Finite dipole aggregates are expected to display a plethora of nontrivial equilibrium structures due to the competition between the strongly anisotropic dipole-dipole interaction, favoring open structures, and isotropic inter-particle forces as well as surface tension, which favor compact structures. In ferrofluids, consisting of a colloidal suspension of magnetite particles, reported obser- vations range from compact and branched macroscopic structures [118] to complex labyrinthine patterns [119, 120]. Similar, but less complex structures have been ob- served in electro-rheological fluids, where the electric dipoles are induced by inter- particle interactions [124]. Except for aggregates with few particles [125, 126], structural studies of complex fluids have focused on pattern formation in systems with infinitely many particles. 66 Here I present results in the interesting finite-size regime, where the surface tension of the aggregate plays a dominating role in its equilibrium structure. We find that with increasing number of particles, as the role of the surface diminishes, the struc- tural motif evolves from chains and rings to multi-chain and multi-ring assemblies, single- and multi-wall coils, tubes and scrolls. I map the inter-particle interactions in the colloidal suspension onto a continuum model and show how changes in external parameters, such as external magnetic field or the liquid-particle interaction, affect the relative stability of these structures and induce structural transitions. To gain microscopic insight into the causes of pattern formation in ferrofluids, which is beyond the scope of present experimental observations [118, 120], I perform total energy and structure optimization calculations for finite aggregates of magnetic particles suspended in a viscous liquid. Our model system is designed to represent a typical ferrofluid that contains magnetite particles, which are covered with surfactants such as oleic acid, which cause short-range entropic inter-particle repulsion to keep the particles in suspension and to prevent a structural collapse. Not only the dynamics of the ferrofluid, but also the effective inter-particle interaction is affected by the viscous liquid used in the colloidal suspension, which is usually n—eicosane or kerosene. 67 3.2 Equilibrium Structure of Ferrofiuid Aggregates The following discussion on the Equilibrium Structure of Ferrofluid Aggregates follows that presented in Reference [121]. I study the equilibrium structure of large but finite aggregates of magnetic dipoles, modeling a colloidal suspension of magnetite particles in a ferrofluid. With increasing system size, the structural motif evolves from chains and rings to multi- chain and multi-ring assemblies. Very large systems form single- and multi-wall coils, tubes and scrolls. These structural changes result from a competition between various energy terms, which can be described analytically within a continuum approximation. I also study the effect of external parameters on the relative stability of these struc- tures. The potential energy Uta, of magnetic particles with magnetic moment 11,- = 11011,- [122] in an external magnetic field H is given by the interaction between each particle and the field, and pairwise interaction between the particles, as Utotz—mZfi, H+Z( uff +1137" . (3.1) j>i The dipole-dipole interaction uff between two identical particles, separated by 13,-: 7‘, — 73-, has the classical form [123] ”I,“ = (Hg/7‘3) [111 'fij — 3011' ' 7:6)011 “fell - (3-2) Following previous work [127, 125], I have described the nonmagnetic part of the inter-particle interaction 111-’3'" = u"’"(r,j) by an isotropic potential with a soft-core short-range repulsion and a weak, long-range attraction with the functional form rig-m = 6 [exp (0 gm) — exp (3%)] . (3.3) 68 To model the inter-particle interaction in a typical ferrofluid, I will consider particles with a magnetic dipole of 110 = 2.1x 104 113. To describe the nonmagnetic interaction between the particles, I chose p1 = 2.5 A, p2 = 5.0 A, a = 100.0 A, and c = 8 meV. In absence of the magnetic interaction, the particle aggregates minimize their surface tension by forming compact, spherical clusters with a near-constant equilib- rium inter-particle spacing Loza = 100.0 A. The dipole-dipole interaction, on the other hand, favors straight chains of aligned dipoles with the same separation L0. Independent of the aggregate size, the equilibrium geometry should be a compact arrangement of deformed chains. In the following, I will analyze complex geometries in terms of the particular arrangement of deformed chains of dipoles. I use a straight, N -—> oo membered chain, which is aligned with an external field H, as a reference structure. In this system, the potential energy per particle is given by U c = Ufa/N“ - 2C (113/ L3) + U"'"(Lo) - uoH , (3-4) where C = 22:172‘3z120206. To compare total energies of finite systems with very many particles and to make universal conclusions about structural transitions, I found it useful to map chains of aligned dipoles onto continuous magnetic rods of diameter dz(\/§/2)Lo = 86.6 A and total length L, illustrated in Fig. 3.1(a). In the continuum description, one particle corresponds to a rod segment of length Lo. I define the rod diameter d as the axial separation between adjacent chains in the optimum staggered geometry depicted in the middle panel of Fig. 3.1(a). The total energy of a system of interacting chains of dipoles, modeled by deformed magnetic rods, has three major contributions. The energy required to bend a straight chain segment of length L0 to a circle of radius R, as shown in the left panel Fig. 3.1(a), 18 AW“ = +aLo/R2 . (3.5) 69 -+- ring coil assembly chain tube scroll assembly Figure 3.1: (a) Discrete and continuum models illustrating major energy terms as- sociated with structural changes in an infinite chain of aligned dipoles, defined in Eqs. (3.5), (3.6), and (3.7), which govern the stability of dipole aggregates. (b) Two- dimensional assemblies of chains in planar layers, ring assemblies and coils forming single-wall tubes. (0) Three-dimensional assemblies of linear chains, multi-wall tubes, and scrolls. The dipole orientation of the individual particles, depicted as spheres, is visualized by the north (black) and south (white) hemispheres. 70 The inter-chain interaction energy gain, associated with the formation of chain pairs, is depicted in the middle panel of Fig. 3.1(a). This energ is maximized, when adjacent chains are offset axially by half a unit cell, and is given by AUic = —BLO . (3.6) Finally, the energy investment to cleave a straight, infinite chain, illustrated in the right panel of Fig. 3.1(a), is AU“It = +7 . (3.7) Using the parameterized interaction within the ferrofluid, described above, I find a = a0 = 0.711 eVA, a = a, = 1.06x10-4 eV/A, and 7 = 70 = 5.70x10-2 eV. The equilibrium arrangement of the ferromagnetic particles results from the com- peting tendencies to minimize strain and to maximize the inter-particle attraction. A single, straight chain contains no strain, but contains two unstable ends. Linear chain assemblies, shown in the left panels of Fig. 3.1(b) and (0), experience further stabilization by the pairwise inter-chain interaction. Even though this geometry is also unstrained, the number of unstable ends is larger than in a single chain. Struc- tures with unterminated chain ends may be stabilized by connecting these ends, which occurs at the expense of increasing strain energy. As shown in the middle panels of Figs. 3.1(b) and (c), a chain may thus form a ring, and rings may stack up to form a single—wall assembly or a multi-wall tube. In very large systems, where optimizing the inter-chain attraction is more important than avoiding a limited number of chain ends, we may find coils and multi-wall scrolls, shown in the right panels of Figs. 3.1(b) and (c), to compete favorably with ring assemblies and multi-wall tubes. To check the accuracy of the continuum approach, I compared the results of the continuum and the discrete description for the two-dimensional structures depicted in Fig. 3.1(b). In the continuum approximation, the total energI of a two-dimensional 71 multi-chain assembly of N = L/Lo particles, distributed over NC parallel chains of equal length, is UmC = +Nc'y —- flL(Nc — 1)/Nc (3.8) with respect to the reference structure of same length L. The corresponding expres- sions for the energy of a multi—ring assembly and of a coil of radius R are, respectively, Umr = —fi(L — 2M) + (IL/R2 , (3.9) U60“ = Umr+7. (3.10) For a given number of particles N, corresponding to a total chain length L, the optimum number of chain segments Nc in the multi-chain assembly is determined by minimizing U ”(L, NC) with respect to NC, yielding N?” = (BM/2L1” . (3.11) In a multi-ring assembly or a coil, the number of ring turns N, is given by N, = L / (27rR), and its optimum value is obtained by minimizing the expression in Eq. (3.9) with respect to N,, yielding B 1/3 N°”‘ = (T) L273 . (3.12) " «20 With the Optimum number of rings given by Eq. (3.12), the optimum radius R0" increases monotonically with L as a 1/3 mac?) L1/3. (3.13) A comparison between the continuum results, energies per particles in given structural motives are compared, which are in quite good agreements as shown in 72 Fig. 3.2(a) and (b). Also the results based on Eqs. (3.8) and (3.9), and those based on the discrete model, described by Eq. (3.1), are presented in Fig. 3.3(a) and (b). In view of the fact that no restrictions were placed on N, or N, being an integer in the continuum approach, I find the agreement between the two sets of results very satisfactory. Since a similar agreement between the discrete and the continuum model is achieved also for the three-dimensional aggregates, depicted in Fig. 3.1(c), I will base the following discussions on the continuum approach. The structural transitions within the suggested motifs (two-dimensional multi- chain, multi-ring, and coil) are observed. A single-chain to double-chain transition occurs as the the number of particles (N) is larger than 24 (L > 0.24pm) and a triple-chain structure becomes a possible ground state when 83 < N < 144, which is presented in Fig. 3.3(a). Considering multi-ring (coil) structures, a double-ring (double-winded coil) structure becomes a possible equilibrium structure when N > 12. Finally, a six-fold ring (six-fold-winded coil) structure is the most stable structure for up to 100 particles as shown in Fig. 3.3(b). With the optimum structural parameters giving by Eqs. (3.11), (3.12), and (3.13), the energy of the optimized two-dimensional chain and ring assemblies, as well as coils, is given by Eqs. (3.8), (3.9), and (3.10). A structural phase transition between the motifs are observed (Fig. 3.3 (c)): a chain to ring transition occurs for L > 0.03pm (N > 3) and a chain to coil transition occurs for L > 0.18pm (N > 18). As an extension of my approach, I consider single-walled and multi-walled tube structures and scroll as next possible equilibrium structures (Fig. 3.1 (0)). A l x w (length l and width 112) sheet consisting of a ferrofluid chain (total length L) was considered to create (single and multi-walled) tube and scroll structures. There are three interactions to describe energy of multi-walled tubes: strain energy (AU 3), inter-ring (chain) interaction (AU i’), inter-wall interaction (AU‘u’). 73 *r fir T r f‘v’ chain assembly 1 0 60 1207 A180 N=lllo (b)0.02--.. -- ring assembly ; 'I'f4 V] .o O .a flu»- 0 60 120 180 Figure 3.2: A comparison between discrete and continuum approach for energy for a chain assembly (a), a ring assembly (b), and a coil structure as a function of number of particles. Lines in each curves are the results from continuum approach and points are from numerical approach. 74 discrete— 6 discrete— 4continuum-u continuum--- . -0.01 , “0'02 7 ‘5‘0‘ ” 150 Figure 3.3: The optimum number of chains, NC, (a) and rings, N,, (b) as a function of system size (N). Numerical calculations by structural optimization are shown as solid lines. The results from the continuum approach are shown as dashed lines. For numerical calculations, I performed optimization up to 100 particles ( L > 1.0um ). The results indicate the structural phase transitions within given motifs. (c) Dependence of the optimum interior radius of a scroll (dashed line) and a multi-wall tube (solid line) on number of particles N = w/Lo. Abrupt changes of R," occur in multi—wall structures, when the optimum number of walls changes. ((1) Total energy of multi-wall tubes and scrolls with respect to a reference strip of the same length and the width 10. For sufficiently large values of w, tubular structures are preferred to the planar strip and approach the energy energy of infinite chain structure. The higher stability of multi-wall tubes over scrolls results from the absence of exposed edges. 75 The energy cost from bending l x w ferrofluid plane to make Nw-walled tube is 1 AU “ 27m‘ARZR,.,+( (n— 1)AR (3.14) where R," is the inner-most tube radius, and the optimized inter—tube distance AR is a\/3/ 2. The inter-ring (inter-chain) interaction is given by AU"- — 2—1rfi(Nl):(—(Rm+(n — 1) A)R) (3.15) Here, N, is the number of rings in each tube. This interaction is identical to —,Bw(N,- 1). With total contact length 1, (=1 - n(Rin - RW¢)N,), inter-tube interaction length, the inter-wall interaction Ain is given by —6 x 1,. The energy of tube U “be is obtained by the sum of AU 3, AU", and AU f‘”. Hence, 2 1 Utah/l: 7rd _2 R1"+ (n — 1)AR _A'BR(2'w—1T(R4n 'l‘ Rout». (3.16) For a scroll structure, the additional term AUe, penalty from exposed edge, should be considered. For a given number of chains N, (= l/AR) in a ferrofluid sheet, AU“3 is 7N,. Since radius of coil R(0), depend on winding angle (6’) and is expressed as R," + OAR/ 21r, the strain energy of a coil structure is 2’” 5s), (3.17) AUS = Whfifl Rm is the out-most scroll radius. The inter-ring (chain) and inter-wall interactions are same as those of multi-walled tube. The resultant energy formula for scroll U ”“1 per unit length l is 7m R UscrOu/l= (£237); ln(—L:5) — A—flR-(Z'w — 1r(R4-n + Road) + (3.18) .1 AR' 76 (3)200 l l . 1 1 Z 100 : ring ,: Z ’ [ ring I | I ’ l I d (C)200 .. -, ( )200, ............. -, s ? i . 3 2100} a 2100’- Chem -1 ring z ‘ z 1 0 2 or) 112“ii4“116 r/ro H/Ho Figure 3.4: The ”phase” diagram as a function of parameters: a (a), B (b), 7 (c), and external magnetic field ((1). The change of B and '7 corresponds to the effect of liquid which covers magnetite in a ferrofluid. All parameters are rescaled: do = 0.711 eVA, R0 = 1.0566x10-4 eV/A, and 70 = 0.05698eV. H0 = 10-2 eV/uo To generalize my approach of describing system using continuum approximation I investigate the effects of surfactant and external magnetic field on the structural phase transitions. Figures 3.4 (a)-(c) show the ”phase” diagram as a function of parameters (a, 6, and '7), which is used in continuum approach. Especially, 6 and 7 corresponds to the effect of surface tension from liquid and magnetite interface in a ferrofluid. As I change the interaction between liquid and particles, the change of ”phases” are predicted as shown in Fig. 3.4 (b) and (c). All parameters are rescaled by the value from my present model system, meaning kerosene based ferrofluid. 77 To properly describe the effect of magnetic field in my scheme 1 consider the special case: no internal structure change can be induced by external magnetic field. Under this special condition, no external magnetic field contribution is expected for ring assembly and coil structures due to zero total magnetic moment. However, the energy gain is expected for the case of chain assembly under external magnetic field. Therefore, the Eq. 3.8 becomes U” = +Nc7 — Rwy, — 1) /N, — Haul/a. (3.19) The effect of external magnetic field on ” phase” diagram is displayed in Fig. 3.4(d). For the strong magnetic field chain assembly becomes to be the only possible phase. However, ring assembly is still most possible structure for the large system under not strong magnetic field. 78 3.3 Targeted Medication Delivery Using Magnetic Nanostructures The following discussion on the targeted medication delivery using magnetic nanostructures follows that presented in Reference [128]. I use advanced quaternion molecular dynamics to model a potential application of magnetic nanostructures for targeted medication delivery. Inert microcapsules, containing the active medication and a small number of magnetite nanoparticles, may be transported using an inhomogeneous magnetic field through blood vessels to a desired location in the body. Triggered by an abrupt change in the applied field, structure of magnetite aggregates changes from a ring to a chain, thus puncturing the microcapsule and releasing the medication. The stability of the magnetic nanos- tructures under thermal and magnetic fluctuations has been also studied to prevent an accidental delivery. The magnetic particles, typically consisting of magnetite, have a typical di- ameter of few hundred Angstroms, carry a large permanent magnetic moment of the order of magnitude 104 — 105 p3, and are covered by an approximately 20 A thick surfactant layer which prevents them from coalescing at room temperature in a viscous suspension. Spontaneous formation of complex labyrinthine [131, 120] and branched [129] macroscopic structures has been observed and theoretically ad- dressed [116, 132, 133, 134, 135, 136] in these systems at low temperatures and in applied magnetic fields. Of particular interest in this study is the fact that aggregates of 4311314 mag- netic tops are a classically tunable two-level system [137] Their most stable structure in zero field is a ring, but they open to a chain when exposed to a large nonzero magnetic field [130]. In this contribution, we describe a possible application of this structural transition as a one-way valve causing liquid-filled microcapsules to burst. 79 Microcapsules have been used extensively in medicine as micro-containers that transport and deliver an active substance to a specific site in the human body. Their typical diameter of 0.1 pm is small enough to allow the microcapsules to pass through all capillary blood vessels. The most significant application of this technique is in the chemotherapy of cancer, since the most potent drugs are indiscriminately toxic to all tissue. Such substances should not come into contact with healthy tissue, and only be locally delivered in the tumor region. The standard solution to this problem has been to use albumin, polyalkyl- cyanoacrylate, ethylcellulose or polyglutaraldehyde for the membrane, that would safely contain the drug, yet biodegrade over time. Here we describe an alternate local drug delivery mechanism, based on the structural transition of an aggregate of magnetic tops, that allows to move the microcapsules to a particular location and to deliver the active substance in a planned fashion using a time-dependent magnetic field [138]. The principle of the mechanism proposed is illustrated in Fig. 3.5(a) and (b). We propose to enclose several magnetic nanoparticles together with the active drug in the microcapsule. In zero or very low applied magnetic field these particles will aggregate to a ring that fits snugly in the microcapsule (see Fig. 3.5(a)). A low inhomogeneous magnetic field can be used to concentrate the microcapsules in a particular location. At this moment, application of a stronger magnetic field will cause the ring of magnetic tops to open up to a chain [130] (see Fig. 3.5(b)). The imposed deformation of the cage will cause it to burst open, releasing the active substance at the desired location. Of course, the response of the system to the environmental variables such as tem- perature and magnetic field is critical for the successful application of this technique. There is significant freedom in selecting the system parameters, such as the diameter of the microcapsules, their surface tension, the diameter and the permanent magnetic- moment of the magnetic tops, the spatial variation and the strength of the externally 80 2R1 Figure 3.5: (a) In zero field, the equilibrium structure of the tops is a ring that fits into a spherical membrane of radius R0. (b) In nonzero field the ring opens up to a chain, thus deforming the membrane to an ellipsoid with long axis R1. The capsule will burst if R1 >> Ro.(c) Schematics of our model system. The outer shell of the microcapsule, consisting of mesh of 368 particles, contains 375 medication particles (grey) and six magnetic top of magnetite, which are represented by the large spheres with their magnetic dipole by the north (dark grey) and south (light grey) hemispheres. 81 applied magnetic field. In our model, the outer shell of the microcapsule, a cage, consisting of a mesh of 368 particles (white rods), contains 375 medication particles (grey), and six spherical magnetic tops of magnetite, which are represented by the large spheres with their magnetic dipole by the north (dark grey) and south (white grey) hemispheres shown in Fig. 35(0). The near spherical shape of the cage results from the internal pressure due to the repulsion between the enclosed molecules and the repulsion between these molecules and the cage. I have described the nonmagnetic interactions between molecules (microcapsule particles, medication particles, and magnetites) 1' and j by Lennard-Jones type po- tential 113?" = u“’”(r,-,-), an isotropic potential with a soft-core short-range repulsion and a weak, long-range attraction with the functional form: 1<><—><><>1 A crucial component of the microcapsule are six spherical magnetic tops of mag- netite, with a diameter a of 200 A and a large permanent magnetic moment [10 of 1.68x105 113. The potential energy U161 of magnetic particles with magnetic moment 11,- = 11011, [122] in an external magnetic field H is given by the interaction between each particle and the field, and pairwise interaction between the particles, as Utot— — —[.toZfli' H 'I' 201;? “i" un-m) . (3.21) j>i The dipole-dipole interaction uidf between two identical particles, separated by r,,-= r, — 1",, has the classical form [123] uaa = (Hg/7‘3) lfii 'fij — 3011 -f,-,-)([1,- ‘ 731)] - (3-22) 82 The nonmagnetic interaction between magnetites has been described by Eq. 3.20 using parameters of p1 = 5.0 A, p2 = 10.0 A, p1 = 4, p2 = 0, and e = 64 meV. In absence of the magnetic interaction, the particle aggregates minimize their surface tension by forming compact, spherical clusters with a near-constant equilib- rium inter-particle spacing Loza = 200.0 A. The dipole-dipole interaction, on the other hand, favors straight chains of aligned dipoles with the same separation Lo. Independent of the aggregate size, the equilibrium geometry should be a compact arrangement of deformed chains [121]. The optimum configuration for six magnetic particles has been studied using the conjugate gradient technique [139]. Instead of spanning whole configuration space, we reduce our configuration space to a volume of a hard sphere with radius m of 800.0 A, which physically represents a delivery cage in our system. This approach helps to efficiently search a relaxed geometry. A ring phase (as shown in uppermost figure in Fig. 3.6(b)) known as an optimum geometry for a small system [130] is obtained at zero magnetic field from an arbitrary geometry. On the other hand a chain phase is obtained under external magnetic field of 1500 Oe (as displayed in uppermost figure in Fig. 3.7(b)). To prevent the accidental delivery of medications under thermal or magnetic fluctuations, we need to understand the thermodynamic behavior of ferromagnetic particles. I study the thermodynamic behavior of six ferromagnetic particles confining in a hard sphere. I use microcanonical molecular dynamics (MD) simulations, where quaternion parameters [140] have been employed to properly describe the rotational motion of a magnetic top and to prevent the discontinuity and divergency as we solve equations of motion of the system. A fourth-order Runge—Kutta formalism is employed to integrate the Euler-Lagrange equations to obtain the trajectories of the particles [141] with a time step of 0.01 ns. I choose the optimum geometry at a given magnetic field (a ring for zero magnetic 83 (a) 400[m...m(b) ; . M . E. 01 f ~ LLI ‘ -400- . ‘1 1_—I ' g 1 E . 5; 05~ - 5° I . O: I I I: 800:- 3 g . I C - . Q: 400: +— 0;.n11._....... 0 50010001500 TlK] Figure 3.6: (a) Microcanonical molecular dynamics simulations have been performed to study the thermodynamic behavior of six ferromagnetic particles. Energy per par- ticle as a function of temperature is shown in the upper panel. The slop of energy changes continuously between 400 and 800 K, which may correspond to a critical points for an infinite system. At high temperature (T Z 800 K) the slop reaches at 3163. The total mean magnetic moment 11,0), an indicator to distinguish thermally equilibrium structures (rings and chains), is monitored with respect to maximum mag- netic moment of system (#101 = N110) as a function of temperature, which indicates the structural change from a ring (#101 / 11m 8 0) to chain segments (pm/11m >> 0) (middle panel). The average closest inter-particle distances has been also presented to monitor the structural changes (lower panel). (b) The snapshots of geometry at T x 300, 400, 500, 600, 700, and 1000 K. 84 A m V -1000 " -1400 ' E/N [meV] -1800 “lot/“max 0.8 800 ’ DnnIA] 400 I A111.LL.A 0 . . . . 1 O 500 1000 1500 TlKl Figure 3.7: (a) Microcanonical molecular dynamics simulations have been used to study the thermodynamic behavior of six ferromagnetic particles under external mag- netic field of 1500 Oe. The energy per particle as a function of temperature is displayed with error bars in energy and temperature (upper panel). The slope of energy contin- uously changes at the temperature range of 400 and 800 K, which may correspond to a critical points for an infinite system. The total magnetic moments pm is monitored with respect to maximum magnetic moments of system ,uma, (=N 110) as a function of temperature, which show fluctuations in the orientations of each magnetic tops under high temperature (middle panel). The closest inter—particle distances are displayed to monitor structural transitions as a function of temperature. The melting of a chain structure occurs around 800 K. (b) The snapshots of geometry at T x 300, 400, 500, 600, 700, and 1000 K from microcanonical molecular dynamics simulation discussed in (a). 85 field or a chain for high magnetic field) as a starting configuration of MD simulations. The described hard sphere (r0=800 A) has been used as a microcapsule containing six ferromagnetic particles. The system has been equilibrated at each given energy for 0.1 11 sec, corresponding to 10“ steps in simulations. The average energies per particles (solid circle) as a function of temperature are presented with error bars, standard deviations in temperature and energy (upper panel of Fig. 3.6(a) and Fig. 3.7(a)). Also, the total magnetic moments #101 of the system ratio to maximum magnetic moment 11m (=Npo) as a function of temperature are displayed in the middle panels of Fig. 3.6(a) and Fig. 3.7(a). The lower panels in Fig. 3.6(a) and Fig. 3.7 (a) show closest inter-particle distances Dun as a function of temperature. In case when no external magnetic field has been applied, the gradual change in energy slope (specific heat) occurs between 400 and 800 K (lower panel of Fig. 3.6(a)), which may correspond to a transition point for an infinite system. Based on the energy difference between a ring and a chain phase, 32.68 meV per particle, the approximate order of transition temperature of z 380 K can be estimated. To identify the structure at a given temperature I monitor the total magnetic moments and the closest inter- particle distances during MD simulations. Magnetic moment can be a good indicator to easily distinguish magnetic isomers since their equilibrium structure of small clusters (4 S N S 14) is either a ring or a chain. A ring phase can be identified by the magnetic moments ratio of 0 S mag/um << 1 and a broken ring structure (chain segments) can be also identified by 0 << 1.1M / pm S 1. The total magnetic moments ratio in the middle panel of Fig. 3.6(a) identify the expected transition from the energy curve: A ring phase to chain segments transitions occurs occurs between 400 and 800 K. Since no external magnetic field is applied, chain segments are not aligned in one direction, which results total magnetic moments of z 0.5, far below the value of aligned chains, 1.0. The closest inter-particle distance Dm, plot shown in the lower panel in Fig. 3.6(a) 86 confirms above observations. Dan for a perfect ring and a perfect chain is z 200.0 A. At high temperature (T m=T 2 1000 K) specific heat reaches to 3kg, which can be explained by equipartition theorem considering 6 degrees of freedom for each particle. This convergence in a specific heat indicates the structural melting (see downmost snapshot in Fig. 3.6(b)). Above Tm the average D,m reaches to 3 470.0 A due to spatial confinements and 11,0, / 11,,“ reaches to z 0.4. Snapshots in Fig. 3.6(b) show a structural evolution as temperature increase at 300, 400, 500, 600, 700, and 1000 K. The changes in energy slope also occurs at temperature range of 400 and 800 K as an external magnetic field of 1500 Oe has been applied (in the upper panel of Fig. 3.7(a)). Since high magnetic field has been applied, total magnetic moments can only indicate the rotational fluctuations in each top under thermal fluctuations. Considering the energy gain of each particle by interacting an external magnetic field, 1.46 eV, the rotational fluctuation of each magnetic top at low temperature can be estimated as 5.89 x 10’5 K‘1 in the middle panel of Fig. 3.7(a). To identify isomers at each temperature we also monitor the Dan. Dan shown in the lower panel of Fig. 3.7(a) shows same behavior as that of non magnetic field case, meaning a single chain to chain segments transitions occurs between 400 and 800 K. The specific heat reaches to 3kg above 100 K, indicating a melting of structure. Snapshots in Fig. 3.7(b) show a structural evolution as temperature increase at 300, 400, 500, 600, 700, and 1000 K. Unlike an infinite system, where a divergence in specific heat observed at a phase transition point, a transition point is expanded as a transition region in a finite system. Since the system properties (possible isomers) of a ferrofluid system strongly depend on its size, the method like finite size scaling cannot be applied to the system to study the nature of the transition. Instead, the nature of the transition between two ordered phases, rings and chains, in a finite system is explicitly analyzed and identified as a ”first-order” like transition using Metropolis Monte Carlo simulations [137]. 87 LY j ’ .;;.A~,, 11. "5.; ‘1',»‘a-w. ', ff)". €55.23: '3‘ .‘7 1‘73} 9.2737? . _;',.11\_« » {’5' "131‘ § 1‘ '3 _, .; 9 r4214“ 1r;— ,1 ..'T . I 52‘1“ '1. l = 0 t = 1008 1 = 30ns 111 111 $111 F'... .1". -. '3' ,"‘,‘. 2‘,“ " . F; 3 131‘ «.‘3 .-\ _~‘ :1 9‘ '1 _~~'-'. 9 1:: .. ', 9 f' ‘2' .13: - , 135-? '. ‘rc; ‘33,: , 9,53” l = 50ns l = 70ns t = 90115 Figure 3.8: Inter-atomic bonds in the membrane are shown by the white rods, and the fluid molecules by the small red spheres. The magnetite particles are represented by the large spheres and the orientation of their magnetic dipole by the north (blue) and south (yellow) hemispheres. The five snap shots of the geometry after switching on the magnetic field of 1500 Oe at t = 10, 30, 50, 70, and 90 ns. Finally, I investigate the targeted medication delivery using those magnetic par- ticles. There are obviously various ways to understand the bursting of the micro- capsule due to the structural transition in the magnetite aggregate in a quantitible fashion. My approach to this problem is to use MD simulations. In order to sim- ulate field-induced medication delivery in a human body, I apply a magnetic field of 1500 Oe [142] and equilibriated the system at 300 K starting from the optimum geometries at a given magnetic field, and follow the structural evolution for a total time of 100 ns with time step of 0.01 ns. Based on the thermodynamic properties of six ferromagnetic particles, we can make sure that the ring phase will be maintained inside a microcapsule up to 400 K, which is far higher than human blood temperature, 310 K. The results are shown in Fig. 3.8. The six consecutive snap shots illustrate the the time evolution of the bursting process. The magnetite preserve a ring configuration 88 until R520 ns. The whole system of magnetites is rotated along the direction of magnetic field with keeping its initial optimum structure from no magnetic field (see snap shots at 10 ns in Fig. 3.8). With the structural change of magnetite assembly to a chain, the first onset of the bursting can be observed after z30 ns, where several fluid particles escapes from the microcapsule through a small hole in the cage. Then I observe the hole size to increase very fast following the initial fracture. 90 us following the application of the magnetic field all fluid particles escape from the cage and the microcapsule collapses. In conclusion, we developed quaternion molecular dynamics to model ferromag- netic particles for the use of potential application of magnetic nanostructures for tar- geted medication delivery. First, I study the energetics and thermodynamic behavior of six ferromagnetic particles in a hard sphere, modelling a medication microcapsule. My study shows the stability of delivery process under thermal or magnetic fluctu- ations. The microcapsule, containing medication and a small number of magnetite, can be destroyed by the structural transition of magnetite inside from ring to chain under magnetic fields, thus releasing the medication. My molecular dynamics simula- tions shows the successful medication delivery within #100 us under magnetic fields of 1500 Oe. 89 3.4 Summary In this section, I discuss the ground state configuration for a ferromagnet string suspended in a viscous liquid. I suggested a continuum approach as an efficient tool to study structural transitions, which is in good agreement with numerical compu- tations. Based on the known configuration, chain and ring, we have investigated possible ground structures like chain and ring assembly, coil, multi-wall tube, and scroll structures. The structural phase transitions between the motifs were observed. To completely describe the system using our new approach, I consider the effect of fer- rofluid liquid and external magnetic field on structural phase transitions. The phase diagrams depending on those parameters were presented. I also use quaternion molecular dynamics to properly describe dynamic proper- ties of ferromagnetic particles in a viscous liquid. I suggest ferromagnetic particles for the use of potential application of magnetic nanostructures for targeted medication delivery. The microcapsule, containing medication and a small number of magnetite, can be destroyed by the structural transition of magnetite inside from ring to chain under magnetic fields, thus releasing the medication. Our molecular dynamics simu- lations shows the successful medication delivery within $21100 ns under magnetic fields of 1500 Oe. 90 Chapter 4 Quantum Transport through Molecules: Effect of Structural Changes 4.1 Theoretical Techniques In this section, I will review the formulas and techniques applied to study the transport properties of mesoscopic systems. In a mesoscopic system, where a system dimension is smaller than the mean-free path and the phase coherence length, carriers can travel an active region without scattering and moving elastically (ballistic transport) except for a possible reflection from a barrier. In this ballistic regime transport is determined in terms of reflection and transmission of carriers produced by elastic scattering from inhomogeneity or boundaries (barriers). Our conductance calculations are based on the Landauer-Biittiker formula [143], which relates the conductance of a system to the scattering problem in mesoscopic systems. To evaluate the transport coefficient and hence to calculate the conductance, 91 Figure 4.1: Illustration of a system for conductance calculation. Scattering region is sandwiched between left L and right electrode R which have chemical potentials of 111, and 113, respectively. An incoming electron from left-electrode has a probability of T to be transmitted and R to be reflected from a scattering region, where T + R=1. the general Green’s function formalism, developed by Sanvito et al. [145], has been applied. In this approach the electronic structure of a system is described by a tight- binding model. Also, the advance technique describing a non—equilibrium electronic situation based on Green’s function technique and ab initio density functional theory, as im- plemented in the TRANSIESTA code [148], is applied to calculate current-voltage characteristics. 4.1.1 Conductance Calculations: Green’s function technique Our interest is to calculate a conductance of a mesoscopic system comprising of a scattering region contacted to semi-infinite electrodes in both sides (see Fig. 4.1). Electrodes are assumed to be ballistic conductors and act as a thermal bath where there are no phase correlations between incoming and outgoing electrons. If we assume chemical potentials are 11;, for left-electrode L and 113 for right-electrode, where m, > HR, current flows take place entirely in the energy range between 111, and 113 at low temperature and low bias limit. Then, the current emitted from left—electrodes becomes [146] 611 1 = eve—EWL - #12). (4-1) where 611/613 is the electron density of states and v is a group velocity. 92 Since the scattering regions is sandwiched between two semi-infinite electrodes, each carrier propagating from left-electrode to the scattering region has a finite prob- ability T to be transmitted and R to be reflected, where T + R=1. Generally, when there exist NC scattering channels, carriers propagating from the n-th channel in the left-electrode has a probability Tmn(= [tmn[2) to be transmitted to the m-th channel in the right electrode. Therefore the total net current I from the left to the right becomes an N: 1 = eve—Em), — 11R);Tm,. (4.2) For one-dimensional case, we can decide the current and conductance using the relationship eAV = A11 and the density of state of 6n/8E=1/21rvh, 8 NC I (m. - #11): Tm... 3" I 82 N‘ G: EV = E27171", here the spin degeneracy is not considered and 262/}1 is called as a quantum conduc- tance Go (051:129 k9). The above equation which relates the conductance with transmission coeflicients is the Landauer-Biittiker formula [143]. The total trans- mission probability, ZZ; T mn=Tr{ttl}, can be calculated using scattering matrix S defined as: s = , (4.3) where r and t are reflection and transmission matrix of electrons injecting from left side and r’ and t’ are those from right side of a scattering region. To use the Landauer-Biittiker formula for conductance calculation first we need to calculate scattering matrix and hence transmission coefficient. In the following I will review the technique developed by Sanvito et al. [145] to calculate the scattering 93 H1 H1 H1 H1 --- H0 H0 H0 "- —9 2 Figure 4.2: Representation of a semi-infinite electrode by a periodic array of conduct- ing units called slices [145]. Ho represents the intra-slice interaction and H1 represents the coupling between adjacent slices. 2 indicates the transport direction. matrix of the system consisting of an arbitrary scattering region sandwiched between two semi-infinite crystalline electrodes. Let us consider an infinite system consisting of periodic slices and each unit is described by a intra-slice matrix Ho and a hopping matrix between a nearest neigh- boring inter-slice H1 (see Fig. 4.2), where H0 is Hermitian and H_1=H[. The total Hamiltonian of the system can be described by a tridigonalized matrix with diagonal components of Ho and off-diagonal components of H_1 and H1. The column vector corresponding to a slice at position 2 (transport direction) is HUI/(z + H—lzpz—l + lez-l-l = sz1 (4'4) where z is an integer in the units of inter-slice distance. If we allow N quantum numbers corresponding to the degree of freedom within a slice, the column vector can be identified as 111501 = 1, - - -,N). To solve the Schriidinger equation (Eq. 4.4) let us introduce the Bloch state, 1&2 = Aeik’zd’k (4-5) 3’ which makes it convenient to describe a periodic system. Here, k2 is the 1: component in transport direction, (0),, is a normalized column vector, and A is an arbitrary 94 constant. Then Eq. 4.4 can be rewritten as (H0 + H-113 + H1$_1)¢kz = E45, (4.6) where :r=e“k‘. We can determine all possible values of k, at a given energy E by solving det (H0 + H151: + H.1z‘l -— E) = 0. There are two sets of roots in Eq. 4.6: (1) N wave vectors of k: (11 = 1, . - - , N) of positive real value (right—moving solution) and of positive imaginary value (right- decaying solution) (2) E: (11 = 1,---,N) of negative real value (left-moving solution) and of positive imaginary value (left- decaying solution). From now on I will use a notation k“ and E” to represent k: and ES, respectively. For numerical purpose, it is more convenient to map Eq. 4.6 onto matrix H where I is a N «dimensional identity matrix and H1 is not singular. In the scattering problem it is useful to consider the retarded Green functions instead of wave functions [147]. Our goal is to describe the whole system, consisting of two semi-infinite electrodes and a scattering region. The semi-infinite electrodes can be described by an infinite electrode with imposing proper boundary conditions in Green’s function. Therefore, let us consider the case of infinite electrodes and derive the case of semi-infinite electrodes with proper boundary condition, and finally we will describe the whole system with the combination of scattering region, which will be described by an effective Hamiltonian. Finally, the total Green’s function can be obtained by solving a Dyson Equation. 95 Retarded Green’s function gal of infinite system satisfies [(E — H.) glzz’ : 622" (4.7) Therefore, the Green’s function is simply a wave function except at 223’: 217:1 ékyeik‘(z—ZI)XIW Z 2 Z, 9,, = N r 1 '1 _ ¢_ 8% p z—z _' Z < Z! 211—1 k” Xku — where the continuous condition holds at 2:2’ resulting, N N )2 ml. = Zaxl. (48) u a At z=z’, it satisfies N _. 211.. [¢k,,e"‘k“xk,.l - er,e""“‘xI-, ] =1 (4.9) u=1 using Eqs. 4.6 and 4.8. Combining continuity conditions and Eq. 4.9 we can finally get the solutions for My? and Xfpl’ 10.." = 01,5571,“ = 65%;". (4.10) where N - ~ .— ~ 6 = z H_1 [¢kpe—1kp¢£‘ _ (pipe‘lkp¢‘:cl] (4.11) 11:1 and 43km. = d3}, (10., = 5m Now the surface Green’s function for a semi-infinite electrode can be obtained by applying proper boundary conditions at the end of the electrodes. The left-electrode extended to zz—oo and terminated at z=z0 - 1 and the right-electrode extended to 96 z=+oo starting from 2:20 + 1, which requires the condition that Green’s function must vanish at z=zo. The surface Green’s functions of left (L) and right (R) electrodes which satisfy the boundary condition are 9L = 9(20—1)(zo—1)(Z) = [“2 3.1.11] 51 11.1 9R = 9(zo+l)(z0+l)(z) = l’ ‘ Z «111—<11] 5‘1 11,1 Our next step is to consider the coupling between surface Green’s function of the electrodes with a scatterer and hence to obtain the total Green’s function of the scatterer containing electrodes via Dyson’s equation. I will review the technique [145], recursive Green’s function technique, which reduce the internal degree of freedom of a scatterer so that describe the scatterer by an effective coupling matrix between the two surfaces. Let us consider the total Hamiltonian describing a system, H=HL+HL3+H3+HRS+HR, (4.12) In this Hamiltonian H L and H R are the Hamiltonians of isolated semi-finite left (L)- and right (R)-electrodes for each, HLS (H R3) is the Hamiltonian describing the coupling between left-electrode (right-electrode) and the scattering region, H s is for the scattering region. The decimation technique, recursive Green’s function technique, can be applied to reduce the internal degree of freedom of the scatterer so that the H LS + H3 + H RS can be described by an effective Hamiltonian He”. Suppose H LS + H5 + H as has N degrees of freedom and electrode surfaces have 97 M degrees of freedom. Using decimation method the internal degree of freedom of N x N can be reduced to M x M matrix by repeating such a step, (k-l) (k-l) H11: ij (k-l) ’ E" kk (k) _ (k-l) for k=M — N times. The resultant effective M x M Hamiltonian has the following form, HemE): ”3(5) H2313) , Hiu.(E) H;,(E) where HZ(E) and H M E) describe the intra-surface couplings in left and right surfaces and HZR(E) and HfiL(E) are for the effective coupling between these surfaces. Finally, we can determine the surface Green function of the whole system con- taining the scattering attached to semi-infinite electrodes by solving Dyson equation, 01E) = [91151-1 — Hefr(E)l". where 9L(E) 0 0 9R(E) 905‘) = When we consider the scatterer with length L, where perfect electrodes surfaces are located at 2:0 and z=L, one has the wave function of electrons which has the following form: 6 ._ ,- ezkp2fi¢ku + El ezkzzfifil Z S 0 ¢z = ‘ t Z! ezkizifi¢kl Z Z L By introducing the projector operator P,,(z’) = Eqbkfle’kflz'fi/vu, we obtain the com- 98 ponents of S matrix, 1" t’ S = , t 7" where _ 1'er ”1 ~‘l tip -— 8 U—¢k,GL0€¢k,,. p t, _ iEIL vi ~I G (u — e v—¢El 0L€¢Ep1 p ’01 ~ 7‘1). = v—¢II(GOO€—I)¢k" p ’1); ~ r1. = v—¢I,(GLL€‘I)¢E,,- 11 4.1.2 Non-equilibrium Green’s function technique In the previous section, I have discussed the equilibrium Green’s function tech- nique for calculating the scattering matrix and hence the conductance of a system based on the Landauer-Biittiker formula. In that calculation, we assume that our bias voltage is low enough to treat electrons in equilibrium. In fact, when finite bias voltage is applied to electrodes, the net current flows through the contact and the system is not in thermal equilibrium. Therefore, an advanced method is required to describe a non-equilibrium electronic structure of a nanostructure coupling to external electrodes with different chemical potentials. Also, the dissipation interaction such as electron-electron and electron-phonon interaction need to be described. The non-equilibrium Green’s function technique [146] combines quantum transport with a statistical description of such interaction, which has an analogy to Boltzmann equation in classical dynamics. I used commercially available program package, called as TRAN SIESTA [148], to calculate current-voltage characteristics of organic molecules sandwiched between semi—infinite electrodes. This program package makes the use of non-equilibrium 99 Figure 4.3: Illustration of a reduced system for conductance calculation. The left(L)- and right(R)—electrode Hamiltonians converge to the bulk electrode values. Scattering region is sandwiched between left(L)- and right(R)—electrode which have chemical potentials of 11;, and 113, respectively. The Hamiltonians of a scattering region and coupling between a scattering region and left- and right—electrode depend on the non- equilibrium electron density. Green’s function technique to calculate density matrix and conductance properties of a system under a finite bias voltage, which is implemented on SIESTA electronic structure approach [149]. Therefore, the system is described based on the density functional theory: exchange-correlation (XC) functional has been substituted to the locaLdensity approximation (LDA) and the effect of the core electrons is described by soft nonlocal norm-conserving pseudopotentials [150], whereas the linear combination of finite-range numerical atomic orbitals [151] are used to describe the valence states. The assumption in this method is that the commonly used XC functionals are able to describe the electrons in nonequilibrium situation where a current flow is present. This mean field like approximation is not able to describe inelasting scattering process during quantum transport. Figure 4.1 shows the systems of our interest. The physical system under our consideration is same as the previous one for equilibrium Green’s function technique. Our system consists of a conductor connected with two semi-infinite electrodes where a finite bias voltage AV is applied. So, the states starting in the left electrodes are filled up to the electrochemical potential of the left electrode 11;, and those in the right electrodes are filled up to 113. Here, eAV = 111, — 113 when 111, > 113. This is the situation which produces the ”non-equilibrium” condition in an electronic subsystem. The Hamiltonian of left (L)- and right (R)-electrode is assumed to be converged 100 to bulk B values and the coupling between the left and the right electrode takes place only through the scattering region. In order to study the transport properties, our interests are in the three regions, where the Hamiltonian is different from that of bulk values: coupling between left—electrode and a scattering region (L—S), right-electrode and a scattering region (R-S), and scattering region (S). We can reduce our infinite system to a finite L-S-R part (see Fig. 4.3). The L-S-R part of the Green’s function G can be obtained by inverting the finite matrix H, ELI-EL V1, 0 H = V; HS VR 1 0 V); HR+ZR G = [EI — H]_1, where I is the identity matrix. In H matrix, HL, HR, and H3 are the Hamiltonian matrices of left— and right-electrode and scattering region. The interaction between the scattering region and the left-electrode is described by VL and that of the right-electrode is V3. The coupling of L and R region with the rest of semi-infinite electrodes is fully described by the self-energies, EL and 23, respectively. H L + EL or H R + 2;; can be evaluated from the calculations for bulk of left and right electrodes. Since it is repeating in z direction, we can use Bloch theorem to evaluate this electrode components. The remaining part of Hamiltonian, VL, V3, and H5 depend on the non-equilibrium electron density and are determined by the self-consistent scheme. In the self-consistent density functional scheme, an effective device potential has following terms: Hartree, exchage-correlation, norm-conserving psedupotentials to describe an atomic core, and other external potentials which can be evaluated using electron density n(r). The electron density can be determined by the density matrix D. When we determine the density matrix of a system, we can solve the Schrédinger equation in conventional DFT scheme. Also, the current can be calculated using 101 Landauer-Biittiker formula for non-equilibrium condition. Let us discuss how to build the non-equilibrium density matrix D. The total density matrix D has two parts, the contribution from the left L and the one from the right R [148], 0,“, = [00 de [pfiu(e)nr(e - #L) + pfu(e)np(e - #3)] , —CXD where the spectral density matrix pL (pR) can be calculated from retarded self-energy 2L(e) and the retarded Green’s function: (0(e)rLG* 2.0 V. I find that the electronegative N02 radical induces a dipole moment in the central phenyl ring, normal to the long molecular axis. Nearly independent of the net molec- ular charge, the two oxygen atoms of this radical carry a net charge Qoz — 1.1 6 each. Consequently, we should consider the energetics within the entire SAM, con— sisting of an ordered layer of the oligo(phenylene ethynylene) molecules bridging the spacing between two aligned Au(001) surfaces [153, 154, 155]. The molecule ends are terminally attached to Au surface atoms. The close-packed ordered 2x2 overlayer, shown schematically in Fig. 4.5(a), has a nearest neighbor spacing a = 8.16 A, twice the lattice constant of Au. With the terminal phenyl rings anchored in the gold surface, an additional torque due to the surrounding charges, acting on the central phenyl ring carrying a large dipole, may change its orientation and thus modify the dihedral angle (,0, as shown in Fig. 4.5(a). For a given total chame QM of each molecule, which can be linked to the applied bias voltage, I consider a net charge Q0 = —1.1 e to reside on each of the oxygen atoms which are separated by z3.3 A from the molecular axis. The remaining charge Q, = QM — 2Q0 is distributed along the rest of the molecule. The energetics of this system of charges as a function of the dipole orientation (1) is depicted in Fig. 4.5(b) for different values of the total charge QM. Even though the energy of aligned point dipoles on a square lattice is independent of the dipole orientation 4), the assumption of point dipoles becomes invalid once the size of the physical dipole becomes comparable to the intermolecular spacing. In this case, I find that molecules carrying Qtot = 0 net charge tend to align along 43 = 0° or d) = 90° due to the dominating Coulomb attraction between the oxygens and 108 O 45 90 «1; (degrees) Figure 4.5: (a) Schematic top view of the SAM. The molecules, forming a square lattice on the electrodes, carry a net charge QM. Their orientation 43 relative to the lattice is strongly affected by the interaction of the dipoles consisting of the charge pair +2Qo, -2Qo, separated by the distance d, with the surrounding charges. (b) Potential energy of this dipole layer as a function of the dipole orientation ¢, presented for different values of the total charge QM. 109 AE (eV) o 45 90 p <9 (degrees) T Figure 4.6: Total energy of an oligo(phenylene ethynylene) molecule as a function of the torsional dihedral angle cp, containing both the intramolecular strain and dipole- dipole interactions within the SAM. the positively charged skeletons of the neighboring molecules. Combined with the fact that the P isomer with zero dihedral angle is most stable, I may assume that the neutral molecules become anchored in the gold electrodes with also the terminal phenyl rings aligned along along d) = 0° or (13 = 90°. I will also assume that once the terminal phenyl rings are attached in this way, they can no longer change their orientation. In this case, only the central phenyl rings may eventually rotate within the SAM. Only as the net charge QM exceeds two extra electrons, the interaction between the oxygens and the skeletons of the neighboring molecules becomes sufficiently re- pulsive to energetically favor the 43 = (p = 45° orientation of the dipoles. As the Coulomb energy gain associated with this rotation exceeds the energy cost of the intramolecular twist, the molecules within the SAM will twist to a nonzero dihedral angle cpz45°, thus disrupting the 7r electron system conjugation. The energetics of this transition is shown in Fig. 4.6. Should the extra charge be removed abruptly, the 110 twisted state becomes unstable, as indicated by the solid arrow in Fig. 4.6. In this case, the intramolecular Coulomb repulsion can take over and complete the internal twist, thus relaxing the system to the metastable T state. Transition from the T state back to the equilibrium P state is a thermally activated process that involves cross- ing the activation barrier depicted in Fig. 4.4(b). More important, this relaxation is sterically hindered by the presence of surrounding molecules within the SAM, thus causing a time delay which lies at the origin of the memory effect. My understanding of the bias-driven isomer transitions within the SAM derives from the interrelationship between the net charge QM on the molecules and the ap- plied bias voltage Vin-as. I find that when packed in a SAM, oligo(phenylene ethyny- lene) molecules carrying less than one extra electron remain in (and eventually return to) the equilibrium P state with an intact 7r electron system conjugation for cp = 0°. This situation is associated with bias voltages Vb,“ < 2 V. As the molecule acquires more than two extra electrons, corresponding to Vb,“ > 2 V, the intra- and inter- molecular Coulomb repulsion becomes dominant, thus inducing a transition to the metastable T state at «p = 90°. The deexcitation to the equilibrium ‘p = 0° state is a thermally activated process that is slowed down by the steric hindrance within the SAM. 4.2.3 Quantum Transport through Molecules After characterizing possible structural changes that may be induced in this two-level system by applying a bias voltage, I discuss in the following the effect of molecular structure changes on the conductance. We use the Landauer-Biittiker formalism [159, 143] to evaluate the electron transport through an oligo(phenylene ethylene) molecule in its two isomer states in the ballistic regime. In our calculation, we treat the molecule as a scattering region sandwiched between two semi-infinite gold leads. We use a recursive Greens function formalism to evaluate the transmission 111 ...... V_{bias)(V) Figure 4.7: Conductance G of the molecule in units of the conductance quantum G0 = 2e2/hz(12.9 k9)“1 as a function of bias voltage. The effect of Au leads is modeled by either a periodic chain of P isomers (denoted P(I)) or by model conductors which minimize the contact resistance (denoted P(II)). The P isomer conducts only for mes > 1.5 V, whereas the T isomer is always insulating. matrix t, describing the scattering of electrons of energy E = E1: + eVMa, from one semi-infinite lead to the other. The differential electrical conductance at the bias voltage Vb,“ is then related to the scattering properties by the Landauer—Biittiker formula[159, 143] G = GOTT {tit}. where G0 = 262/ h is the conductance quantum. In view of the fact that the conductance behavior of the system depends signif- icantly also on the (currently unknown) nature of the contacts, we use a simplified Hamiltonian to describe the electronic structure of the molecule. Our DF T calcula- tions show that the electronic states near the Fermi level, corresponding to the frontier orbitals, are dominated by the pp7r hybrids of the carbon skeleton. The Hiickel model we use to describe transport through these states, characterized by E1, = 0 eV and Vm = —2.67 eV, not only reproduces the electronic structure near E p, but also makes the conductance calculation numerically tractable. Moreover, the transparency of the model allows us to discuss the general behavior of related systems. 112 The results of our conductance calculation are presented in Fig. 4.7 as a function of the bias voltage Vb,“ for the P and T isomers. We have compared these results with those of a four-state Linear Combination of Atomic Orbitals Hamiltonian, with parameters based on ab initio calculations [162], and found very little difference, jus- tifying our approach. As mentioned above, the exact nature of the contacts between the molecule and the electrodes is unknown. In our calculation, which is based on two very different points of view, we model the effect of the Au leads by either a periodic chain of P isomers (denoted as P(I)) or by model conductors which minimize the contact resistance (denoted as P(II)). We notice that independent of the model used for the leads, the T isomer is insulating independent of the applied bias voltage. The same insulating behavior occurs also for the P isomer for bias voltages Vb,“ < 1.5 V, in agreement with results of Ref. [154]. Depending on the leads connected to the molecule, we also find a drastic conductance increase for bias voltages > 1.5 V, in good agreement with the experimental observation. At bias voltages exceeding 2 V, we explain the observed sudden increase in the impedance by the P—>T isomer transition. The persistence of the insulating state for many minutes is explained by the slow speed of the T—->P deexcitation. My description of the microscopic processes that are likely to occur in an oligo(phenylene ethylene) molecule sandwiched between gold electrodes has interesting consequences on its conductance behavior upon modifying the system. Adding neutral, nonpolar “spacer” molecules in the SAM should reduce the torque on the central phenyl ring that is induced by the surrounding molecules, thus requiring a higher bias voltage to initiate the intramolecular P—>T twist causing persistent conductance loss. Since the inter-molecular Coulomb interaction should also depend on the molecular arrange- ment within the SAM, a change from a square to a triangular packing, induced by using a Au(lll) surface instead of Au(001), should also modify the I-V characteristics 113 of the device. Adding an extra N 02 group opposite to the first N 02 group attached to the central phenyl ring should strongly reduce the dipole moment of this molecule and thus inhibit or delay the P—+T transition. The critical bias voltage required to induce an isomer transition should also be reduced when substituting N 02 by a less electronegative radical. Radicals creating a larger dipole moment normal to the main molecular axis, on the other hand, should reduce the critical bias voltage needed to switch the conductance state. Substituting hydrogen by neutral radicals at the phenyl rings should not modify the I-V characteristics significantly as long as these radicals do not perturb the 7r-electron system. Finally, similar switching and memory behavior is expected when using other TICT molecules to bridge the gap between two metal electrodes. 4.2.4 Current-Voltage (IV) Characteristics As I described above, the equilibrium Green’s function technique combined with tight-binding Hamiltonian has been used to calculate the conductance of oligo mole cules as a function of bias voltage. The geometry used for conductance calculations consisted of carbon atoms only, connected to Au electrodes. Since 1r electrons in the system plays an important role in the conductance properties, our results give insight into the observed switching behavior. An advanced technique is required in the following sense. First, to study the current-voltage characteristics of systems we should be able to describe the non- equilibrium situation where a finite bias voltage is applied. Also, a better description of the many-body system is required, especially to study the effects of side groups in our interest. Under this requirement I used the non-equilibrium Green’s function technique combining with the DFT scheme, as implemented in the TRAN SIESTA code [148]. This technique has been applied to many physical systems [164], which successfully describe the transport properties of system comparing to experimental 114 Vbias (V) Vbias (V) Figure 4.8: Current-Voltage (IV) characteristics of three phenyl ring molecules of planar (P) (a) and 90 degree twisted (T)(b) structure of middle ring. The results show the linear increase of current as a function of bias voltage for P geometry and it shows low conductance state for T geometry. observations. I considered the IV characteristics of three phenyl rings, which are connected to infinite phenyl rings. So, my electrode is described as semi-infinite phenyl rings instead of Au electrode, which was used in experiments. Figure 4.8 shows the IV characteristics of three phenyl ring molecules with planar (a) or 90 degree twisted (b) geometry. As observed experimentally, my results show that the planar geometry is in high conducting state and the twisted geometry is in low conducting state. In contrast to the results using equilibrium Green’s function combined with tight-hiding Hamiltonian, which show complete block of conductance channel in twisted geometry (Fig. 4.7), current results show low-conducting behavior (Fig. 4.8 (b)). Since only 1r electron contribute to the conductance in previous calculation, the rotating of middle ring will completely block the conductance channel, which results insulating state. Not only the three phenyl ring molecules but also the molecules with different side groups in the middle ring have been considered. Depending on the side group, NDR or memory effects has been strongly affected. Experimentally, the devices which contain nitroamine in the center and only nitro group show N DR and switching behavior. On the other hand, the devices with no nitro group, either only amino group or only 115 phenyl rings have no N DR and switching behavior. So, experimental group concludes that the nitro group is responsible for NDR and switching behavior [163]. To study the effects of side groups I calculated the transport IV characteristics of our oligo molecules with different side groups. In my calculation, semi-infinite elec- trodes are described as a infinite phenyl ring instead of Au electrode. The results are shown in Fig. 4.9. The devices containing nitro group (Fig. 4.9 (b)) and nitroamine (Fig. 4.9 (c)) show NDR in IV characteristics. On the other hand the device with amino group shows monotonic increase of current as a function of bias voltage. Also, as I rotate the middle ring of the structure, it becomes to be in a low-conducting state as shown in Fig. 4.9 (d). My results are quite well consistent with experimen- tal observations. Those behavior can be explained by monitoring the transmission coefficients as a function of bias voltage. 4.2.5 Summary In summary, I studied the electronic and transport properties of oligo (phenylene ethylene) molecules within a self-assembled monolayer, which experimentally found to be an interesting system which shows memory and switching behavior in current- voltage characteristics. I used various techniques, density functional theory and equilibrium and nonequi- librium Green’s function technique to determine the energetics and transport prop- erties of neutral and charged oligo (phenylene ethylene) molecules within a self- assembled monolayer. I found that a net charge transfer to the molecule, induced by an applied bias voltage, may shift the balance between the strength of the 7r-system conjugation, favoring a planar geometry, and and intra- and intermolecular Coulomb repulsion favoring an intramolecular twist. As the twisted geometry with a disrupted and hence non-conducting 1r-electron system constitutes a metastable state of the molecule, 116 (d)50 r .1 40.. A30r « g . " 20’- 10- - 0 film. 0 1 2 3 4 5 6 VMM vbiasM Figure 4.9: IV characteristics of three phenyl ring molecules with amino-group in the middle ring (a), nitro-group in the middle ring (b), and nitro and amino-group in the middle ring (c)-(d). The molecules without any nitro-group show linear increase of current as a function of bias voltage, whereas negative differential resistance (N DR) behavior has been observed for the molecules with nitro-groups in the middle ring. 117 the transition to the planar geometry takes place through thermal activation and is delayed due to the steric hindrance caused by the surrounding molecules within the SAM. The IV characteristics calculated from nonequilibrium Green’s function technique show that nitro group is responsible for N DR behavior, which is consistent with experimental observation. The side-group dependence behavior can be explained by monitoring the transmission coefficient as a function of bias voltage. 118 BIBLIOGRAPHY 119 BIBLIOGRAPHY [1] S. Berber, Y.-K. Kwon, and D. Tomanek, Phys. Rev. Lett. 88, 185502 (2002). [2] Y.—K. Kwon et al., Phys. Rev. Lett. 79, 2065 (1997). [3] Y.-K. Kwon, D. Tomanek, and S. Iijima, Phys. Rev. Lett. 82, 1470 (1999). [4] S. G. Kim and D. Tomanek, Phys. Rev. Lett. 72, 2418 (1994). [5] N. Troullier and J. L. Martins, Phys. Rev. B. 43, 1993 (1991). [6] L. Kleinman, and D. M. Bylander, Phys. Rev. Lett. 48, 1425 (1982). [7] O. F. Sankey and D. J. Niklewski, Phys. Rev. B 40, 3979 (1989). [8] N. Park et al., Phys. Rev. B 65, 121405 (2002). [9] P. Ordejon, E. Artacho and J. M. Soler, Phys. Rev. B 53, R10441 (2000); D. Sénchez-Portal, P. Ordején, E. Artacho, and J. M. Soler, Int. J. Quantum Chem. 65, 453 (1997). [10] R. H. Boyd, S. N. Sanwal, S. Shary-Tehrany, D. McNally, J. Phys. Chem. 75, 1264 (1971). [11] J. E. Dahl, S. G. Liu and R. M. K. Carlson, Science 299, 96 (2003). [12] J.Y. Raty , G. Galli, C. Bostedt, T.W. van Buuren, and L.J. Terminello, Phys. Rev. Lett. 90, 037401 (2003). [13] J.Y. Raty and G. Galli, Nat. Mat. 2, 792 (2003). [14] D. M. Ceperley, and B. J. Alder, Phys. Rev. Lett. 45, 566 (1980). [15] J. P. Perdew, and A. Zunger, Phys. Rev. B. 23, 5048 (1981). [16] o. F. Sankey and D. J. Niklewski, Phys. Rev. B. 40,3979 (1989). I [17] N. Park, K. Lee, 5. Han, J. Yu, and J. Ihm, Phys. Rev. B 65, 121405(R) (2002). ' [18] P. Hohenberg and W. Kohn, Phys. Rev. 136, B864 (1964). [19] W. Kohn and L. J. Sham, Phys. Rev. 140, A1133 (1965). 120 [20] N. W. Aschroft and N. D. Mermin, Solid State Physics, 2nd edition (Harcourt, Inc., 1976). [21] E. Hiickel, Z. Physik 70, 204 (1931). [22] W. Wolfsberg and L. Helmholtz, J. Chem. Phys. 20, 837 (1952). [23] R. Hoffmann, J. Chem. Phys. 39, 1397 (1963). [24] W. Zhong, D. Tomanek, and G. F. Bertsch, Solid State Comm. 86, 607 (1993). [25] J. D. Joannopoulos and F. Yndurain, Phys. Rev. B 10, 5164 (1974). [26] J. C. Slater and G. F. Koster, Phys. Rev. 94, 1498 (1954). [27] Y. K. Kwon, ph.D thesis, Michigan State University. [28] D. Tomanek and S. G. Louie, Phys. Rev. B 37, 8327 (1988). [29] D. Tomanek, C. Sun, N. Sharma, and L. Wang, Phys. Rev. B 39, 5361 (1989). [30] D. Tomanek and M. A. Schluter, Phys. Rev. Lett. 67, 2331 (1991). [31] M. Yoon, S. Berber, D. David Tomanek (submitted for publication) [32] H. W. Kroto et al., Nature (London) 318, 162 (1985). [33] S. Iijima, Nature (London) 354, 56 (1991). [34] B. W. Smith, M. Monthioux, and D. E. Luzzi, Nature 396, 323 (1998). [35] K. Hirahara et al., Phys. Rev. Lett. 85, 5384 (2000). [36] D. L. Dorset and J. R. Hyer, J. Phys. Chem. B 101, 3968 (1997); H. Kawada et al., Phys. Rev. B 51, 8723 (1995); Y. Saito et al., Phys. Rev. B 48, 9182 (1993); R. Beyers et al., Nature 370, 196 (1994). [37] B. W. Smith et al., J. Appl. Phys. 91, 9333 (2002). [38] K. Hirahara et al., Phys. Rev. B 64, 115420 (2001). [39] The encapsulation energy AE is defined as the energy of a single fullerene en- capsulated in a particular nanotube, with respect to the non-interacting reference system. Consequently, AE does not depend on the precise geometry or termination of the open nanotube edge. [40] S. Okada, S. Saito, A. Oshiyarna, Phys. Rev. Lett. 86, 3835 (2001). [41] H. Ulbricht, G. Moos, and T. Hertel, Phys. Rev. Lett. 90, 095501 (2003). [42] Yang Wang, D. Tomanek, and G. F. Bertsch, Phys. Rev. BR 44, 6562 (1991). 121 [43] D.M. Eigler and E.K. Schweizer, Nature 344, 524 (1990); J.A. Stroscio and D.M. Eigler, Science 254, 1319 (1991). [44] Y. Nakayama, H. Nishijima, S. Akita, K. I. Hohmura, S. H. Yoshimura, and K. Takeyasu, J. Vac. Sci. Techn. B 18, 661 (2000). [45] M. Hodak and L. A. Girifalco, Phys. Rev. B 67 , 075419 (2003). [46] S. Okada, M. Otani, and A. Oshiyama, Phys. Rev. B 67 , 205411 (2003). [47] B. W. Smith and D. E. Luzzi, Chem. Phys. Lett. 321, 169 (2000). [48] S. Bandow et al., Chem. Phys. Lett. 337, 48 (2001). [49] K. P. Meletov et al., Chem. Phys. Lett. 341, 435 (2001). [50] V. D. Blank et al., Phys. Lett. A 205, 208 (1995). [51] G. C. McIntosh M. Yoon, S. Berber, and D. Tomanek, Phys. RevB (2004). [52] S. Han, M. Yoon, S. Berber, N. Park, E. Osawa, J. Ihm, D. David Tomének (submitted for publication) Chem. Phys. Lett. 337, 48 (2001). [53] C. Ronchi et al., Int. J. of Thermophysics 13, 107 (1992). [54] T.L. Makarova et al., Nature (London) 413, 716 (2001). [55] Y.-H. Kim et al., Phys. Rev. B 68, 125420 (2003). [56] E. Osawa et al., J. Chem. Soc. Perk. 'Ifans. 2, 943 (1998). [57] Y.-H. Kim et al., Phys. Rev. Lett. 90, 65501 (2003). [58] A. J. Stone and D. J. Wales, Chem. Phys. Lett. 128, 501 (1986). [59] Y. F. Zhao, B. I. Yakobson, and R. E. Smalley, Phys. Rev. Lett. 88, 185501 (2002); Y. F. Zhao, R. E. Smalley, and B. I. Yakobson, Phys. Rev. B 66, 195409 (2002). [60] A. M. Rao et al., Appl. Phys. A 64, 231 (1997). [61] K. Honda et al., Fullerene Science and Technology 4, 819 (1996); S. Osawa, M. Sakai, and E. Osawa, J. Phys. Chem. A 101, 1378 (1997). [62] H. Ueno et al., Fullerene Science and Technology 6, 319 ( 1998). [63] W. E, W. Ren, and E. Vanden-Eijnden, Phys. Rev. B 66, 052301 (2002). 122 [64] For the sake of comparison, we also calculated the activation barrier for a GSW transformation in a graphene sheet and obtained a much larger value of z9 eV, which compares well with the published value of z8 eV based on ab initio calcula- tions of E. Kaxiras and K. C. Pandey, Phys. Rev. Lett. 61, 2693 (1988). [65] B. R. Eggen et al., Science 272, 87 (1996). [66] Z. Slanina et al., J. Organomet. Chem. 599, 57 (2000). [67] F. Banhart, Rep. Prog. Phys. 62, 1181 (1999). [68] HF. Bettinger, B.I. Yakobson, and GE. Scuseria, J. Am. Chem. Soc. 125, 5572 (2003). [69] R. L. Murry et al., Nature 366, 665 (1993); G. E. Scuseria, Science 271, 942 ( 1996); E. Osawa et al., Fullerene Science and Technology 6, 259 (1998). [70] The Arrhenius formula determines the reaction rate as v exp (—AE/k3T), where u is the attempt frequency, AE the activation barrier, k3 the Boltzmann constant, and T the temperature. [71] RA. Heiney, J .E. Fischer, A.R. McGhie, W.J. Romanow, A.M. Denenstein, J .P. McCauley Jr., A.B. Smith III, and DE. Cox, Phys. Rev. Lett. 66, 2911 (1991). [72] T. Okazaki et al., J. Am. Chem. Soc. 123, 9673 (2001). [73] M. Yoon et al. Phys. Rev. Lett. 92, 075504 (2004). [74] H. W. Kroto et al., Nature (London) 318, 162 (1985). [75] S. Iijima, Nature (London) 354, 56 (1991). [76] M. Terrones et al., Science 288, 1226 (2000). [77] S. Bandow et al., Chem. Phys. Lett. 337, 48 (2001). [78] H. Ueno, S. Osawa, E. Osawa, and K. Takeuchi, Fullerene Science and Technology 6,319 (1998). [79] E. Osawa and K. Honda, Fullerene Science and Technology 4, 939 (1996); A. J. Stone and D. J. Wales, Chem. Phys. Lett. 128, 501 (1986). [80] P. W. Fowler and D. E. Manolopoulos, An Atlas of Fullerenes (Clarendon, Ox- ford, 1995); E. Osawa et al., J. Chem. Soc. Perkin 'IYans. 2, 943 (1998). 81 Y. F. Zhao, B. I. Yakobson, and R. E. Smalley, Phys. Rev. Lett. 88, 185501 [ (2002); Y. F. Zhao, R. E. Smalley, and B. I. Yakobson, Phys. Rev. B 66, 195409 (2002). 123 [82] D. Tomanek, W. Zhong, and E. Krastev, Phys. Rev. B 48, 15461 (1993); D. H. Robertson, D. W. Brenner, and J. W. Mintmire, Phys. Rev. B 45, 12592 (1992) [83] D. Tomanek and M. A. Schluter, Phys. Rev. Lett. 67, 2331 (1991). [84] W. Press, T. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes (Cambridge Univ. Press, 1986). [85] W. E, W. Ren, and E. Vanden-Eijnden, Phys. Rev. B 66, 052301 (2002). [86] F. Banhart, Rep. Prog. Phys. 62, 1181 (1999). [87] B. R. Eggen et al., Science 272, 87 (1996). [88] Z. Slanina et al., J. Organometal. Chem. 599, 57 (2000). [89] E. Kaxiras and K. C. Pandey, Phys. Rev. Lett. 61, 2693 (1988). [90] M. Buongiorno Nardelli, B. I. Yakobson, and J. Bernholc, Phys. Rev. Lett. 81, 4656 (1998) and Phys. Rev. B 57, R4277 (1998); P. Zhang, P. E. Lammert, and V.H. Crespi, Phys. Rev. Lett. 81, 5346 (1998). [91] C. Journet et al., Nature 388, 756 (1997). [92] P. Nikolaev et al., Chem. Phys. Lett. 313, 91 (1999). [93] Y.-K. Kwon and D. Tomanek, Phys. Rev. Lett. 84, 1483 (2000). [94] Presentation of Richard P. Feynman at the annual meeting of the American Physical Society at the California Institute of Technology on December 29, 1959. Reprinted in “The Pleasure of Finding Things Out”, edited by J. Robbins (Perseus Books, 1999). [95] K. E. Drexler, Scientific American 285, 74 (2001). [96] G. I. Leach, R. C. Merkle, Nanotechonology 5, 168 (1994). [97] G. Leach, Nanotechnology 7, 197 ( 1996). [98] P. R. von Schleyer, J. Am. Chem. Soc. 79, 3292 (1957). [99] W. H. Bragg, W. L. Bragg, Nature 91, 554 (1913). [100] C. Cupas, P. R. von Schleyer, D. J. Decker, J. Am. Chem. Soc. 87, 917 (1965). [101] V. Z. Williams Jr., P. R. von Schleyer, G. J. Gleicher, L. B. Rodewald, J. Am. Chem. Soc. 88, 3362 (1966). [102] D. Farcasiu, H. Bohm, P. R. von Schleyer, J. Organic Chem. 42, 96 (1977). [103] G. A. Mansoori, J. Petrol. Science & Eng. 17, 101 (1997). 124 [104] D. Vazquez Gurrola, J. Escobedo, G. A. Mansoori, “Characterization of Crude Oils from Southern Mexican Oil Fields”, in EXITEP 1998 Proceedings, Mexico City, Mexico. [105] J. Reiser, E. McGregor, J. Jones, R. Enick, G. Holder, Fluid Phase Equilibria 117, 160 (1996). [106] J. E. Dahl, J. M. Moldowan, K. E. Peters, G. E. Claypool, M. A. Rooney, G. E. Michael, M. R. Mello, M. L. Kohnen, Nature 399, 54 (1999). [107] G. C. McIntosh, D. Tomanek, and Y. W. Park, Phys. Rev. B 67 , 125419 (2003). [108] We found a significant energy difference between the presently used double-zeta and the minimum basis set. Based on calculations for the related ethylene molecule, using a triple-zeta basis would change the binding energy per atom by 53 meV. [109] C. Kittel, Introduction to Solid State Physics (Wiley, New York, 1996). [110] To limit the computational effort associated with a large number of degrees of freedom, we kept the atomic arrangement within interacting diamondoids frozen in the optimum geometry when determining the total energy of the interacting system. Due to the suppression of relaxations, our estimates provide only a lower limit on the binding energy. [111] C. E. Nebel, Semicond. Sci. Technol. 18, S1 (2003). [112] We use the values E¢O¢(C)= —154.35 eV for the total energy of diamond per atom, and E¢o¢(H2)—-= —30.54 eV for the total energy of a hydrogen molecule, based on the same computational approach and basis as applied to the diamondoids. [113] Johan F. Prins, Semicond. Sci. Technol. 18, 8125 (2003). [114] To limit the computational effort, we chose diamantane as a representative of elongated building blocks. [115] D. Golberg, Y. Bando, W. Han, K. Kurashima, and T. Sato, Chem. Phys. Lett. 308, 337 (1999). [116] H. Zhang and M. Widom, Phys. Rev. E 49, R3591 (1994); J. Mag. Mag. Mat. 122, 119 (1993). [117] K. Raj, B. Moshowitz, and R. Casciari, J. Magn. Mag. Mat. 149, 174 (1995). [118] Hao Wang et al., Phys. Rev. Lett. 72, 1929 (1994). [119] A. J. Dickstein et al., Science 261, 1012 (1993). [120] Chin-Yih Hong et al., J. Appl. Phys. 81, 4275 (1997). [121] M. Yoon and D. Tomanek (in preparation). 125 [122] We define i as the the directional unit vector. [123] JD. Jackson, Classical Electrodynamics, 2nd edition (John Wiley, New York, 1975). [124] Dongqing Wei and G. N. Patey, Phys. Rev. Lett. 68, 2043 (1992); R. Tao and J. M. Sun, Phys. Rev. Lett. 67, 398 (1991); D. J. Klingenberg, Frank van Swol, and C. F. Zukoski, J. Chem. Phys. 91,7888 (1989). [125] P. Jund, S. G. Kim, D. Tomanek, and J. Hetherington Phys. Rev. Lett. 74, 3049 (1995). [126] A. S. Clarke and G. N. Patey J. Chem. Phys. 100, 2213 (1994). [127] C. F. Tejero, A. Daanoun, H. N. W. Lekkerkerker, and M. Baus, Phys. Rev. Lett. 73, 752 (1994). [128] M. Yoon, P. Borrmann, S. G. Kim, P. Jund, and D. Tomanek (in preparation). [129] Hao Wang et al., Phys. Rev. Lett. 72, 1929 (1994). [130] P. Jund, S. G. Kim, D. Tomanek, and J. Hetherington, Phys. Rev. Lett. 74, 3049 (1995). [131] Akiva J. Dickstein et al., Science 261, 1012 (1993). [132] J. J. Weis and D. Levesque, Phys. Rev. E 48, 3728 (1993); D. Levesque and J. J. Weis, Phys. Rev. E 49, 5131 (1994). [133] A. S. Clarke and G. N. Patey, J. Chem. Phys. 100, 2213 (1994). [134] Holly B. Lavender, Karthik A. Iyer, and Sherwin J. Singer, J. Chem. Phys. 101, 7856 (1994). [135] Thomas C. Halsey and Will Toor, Phys. Rev. Lett. 65, 2820 (1990); Thomas C. Halsey, James E. Martin, and Douglas Adolf, Phys. Rev. Lett. 68, 1519 (1992); Thomas C. Halsey, Phys. Rev. E 48, R673 (1993). [136] R. Tao and J. M. Sun, Phys. Rev. Lett. 67, 398 (1991). [137] Peter Borrmann et al., J. Chem. Phys. 111, 10689 (1999). [138] Patent application pending, Deutsches Patentamt, Munich (1996). [139] W. Press, T. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes (Cambridge Univ. Press, 1986). [140] H. Goldstein, Classical Mechanics, 2nd edition (Addison Wesley, 1980). [141] D. Tomanek et al., Z. Phys. D 40, 539 (1997). 126 [142] In clinical studies, magnetic fields commonly used in Magnetic Resonance Imag- ing (MRI), up to 2 x 104 Oe have not been found to produce adverse effects directly in humans. Instead, indirect hazards associated with B fields in MRI are due to the forces exhibited on ferromagnetic materials and on moving electrical charges. Hardy K, Meltz ML, and Glickman R, eds., Non-Ionizing Radiation: An Overview of the Physics and Biology (Medical Physics Publishing, Madison, 1997), p268. [143] M. Biittiker, Y. Imry, R. Landauer, and S. Pinhas, Phys. Rev. B 31, 6207 (1985) [144] C. H. Xu, C. Z. Wang, C. T. Chan, and K. M. Ho J. Phys. Condens. Matter. 4, 6047 (1992) [145] S. Sanvito, C. J. Lambert, J. H. Jefferson, and A. M. Bratkovsky, Phys. Rev. B 59, 11936 (1999). [146] S. Datta, Electronic Transport in Mesoscopic Systems, edited by H. Ahmed, M. Pepper, and A. Broers (Cambridge University Press, Cambridge, 1995). [147] Green’s function in Quantum Physics (Springer-Verlag, New York, 1983), 2nd ed. [148] M. Brandbyge, J. -L. Mozos, P. Ordejo’n, J. Taylor, and K. Stokbro, Phys. Rev. B 65, 165401 (2002). [149] D. Sanchez-Portal, P. Ordejon, E. Artacho, and J. M. Soler, Int. J. Quantum Chem. 65, 453 (1999). [150] N. 'Ii‘oullier and J. L. Martin, Phys. Rev. B 43, 1993 (1991). [151] O. F. Sankey and D. J. Niklewski, Phys. Rev. B 40, 3979 (1989); E. Artacho et al., Phys. Status Solidi B 215, 809 (1999). [152] D. Tomanek, M. Yoon, J. M. Pacheco, G. K. Gueorguiev, S. W. D. Bailey, and C. J. Lambert (in preparation). [153] Mark A. Reed and James M. Tour, Scientific American 282, Number 6, p. 86 (2000). [154] J. Chen, M.A. Reed, A.M. Rawlett, and J.M. Tour, Science 286, 1550-1552 (1999). [155] Jorge Seminario, Angelica G. Zacharias, and James M. Tour, J. Am. Chem. Soc. 122, 3015 (2000). [156] M. Di Ventra, S.T. Pantelides, and N .D. Lang, Phys. Rev. Lett. 84, 979 (2000). [157] M. Di Ventra, S.T. Pantelides, and ND. Lang, Appl. Phys. Lett. 76, 3448 (2000). 127 [158] We use the Amsterdam Density Functional (ADF) code, as documented in G. te Velde and E. J. Baerends, J. Comp. Phys. 99, 84 (1992). [159] R. Landauer, Phil. Mag. 21, 863 (1970). [160] J. Chen, W. Wang, M. A. Reed, A. M. Rawlett, D. W. Price, and J. M. Tour, Appl. Phys. Lett. 77, 1224 (2000). [161] Katsuhiko Okuyama, Yasushi N umata, Shino Odawara, and Isamu Suzaka, J. Chem. Phys. 109, 7185 (1998). [162] D. Tomanek and M. Schluter, Phys. Rev. Lett. 67, 2331 (1991). [163] Chen J, Su J, Wang W, and M. A. Reed, Physica E 16 17 (2003). [164] J. L. Mozos, P. Ordejon, M. Brandbyge, J. Taylor, and K. Stokbro, Nanotech- nology 13, 346 (2002); J. Taylor, M. Brandbyge, and K. Stokbro, Phys. Rev. Lett. 89, 138301 (2002); S. K. Nielsen, M. Brandbyge, K. Hansen, K. Stokbro, J. M. van Ruitenbeek, and F. Besenbacher, Phys. Rev. Lett. 89, 66804 (2002); M. Brandbyge, K. Stokbro, J. Taylor, J. -L. Mozos, and P. Ordejon, Phys. Rev. B 67, 193104 (2003). 128 IIIIIIIIIIIIIIIIIIIIIIIIIIIIII [Mill]lllllllllflllllll][I][lllllllllllllllll